text: 7 (cid:34) (cid:35) (cid:90) (cid:88) (31) is the desired dual representation of Eq. (30). We ∞ shall use both in the following. + β −  C. The DC Josephson current Q[ρ, q] = exp i dτJ(∂ q ρ) (cid:88) J(τ)=−∞ (cid:88)0 τ  Now we are able to compute the DC Josephson current. ∞ − T + N 1 − − When λ˜ = 0 or = 1, the action I˜ is nothing but l/r = −∞ex(cid:89)p i N(cid:88)→lim+∞ Jj(qj+1 bj+1 qj + bj) the one for a particle, of mass L/(πvF ) and unit charge, Jj= j=0 moving on a unit circle thread by a ﬂux φ, provided that − ∞ N 1 + we treat q as the azimuthal angle. In this sense, the − − = →lim∞ eiJj(qj+1 bj+1 qj+bj) , integer m isTthe winding number. Hence the partition N + −∞ function at = 1 can be written as j=0 Jj= l/r (cid:88) with A = A(τ ), ∆τ = β/N, and τ = j∆τ. More- ∞ j j j + − over, we have written ρ = ∂τb. By using the Poisson Z0(φ) = e β(cid:15)J , (33) summation formula −∞ (cid:88) (cid:88) J= ∞ ∞ − − 1 lead−ing to the free energy F0 = T ln Z0, where (cid:15)J = δ(x 2πn) = eimx . πvF [J φ/(2π)]2 is the energy of the zero mode with the −∞ 2π −∞ 2L n= m= quantum number J. The resulting DC Josephson current is of the form (cid:18) (cid:19) we get (cid:88) (cid:18) (cid:19) (cid:89) (cid:88) ∞ Q[ρ, q]= lim N−1 +∞δ qj+1 − bj+1 − qj + bj − l . I(0)(φ) = evLF +−∞ J − 2φπ PJ , (34) N→+∞ −∞ 2π J= j=0 l= − We see that Q[ρ, q] provides each time slice a δ-function where PJ = Z10 e β(cid:15)J| is(cid:105) the probability of the zero − − − mode at the eigenstate J . We notice that the whole which im(cid:80)poses the constraints bj+1 bj = qj+1 qj spectrum is 2π-periodic in φ, and thu−s I(0) −is a 2π- 2πlj. The integration over b in the pr−esence of these local periodic function of φ. Moreover, I(0)( φ) = I(0)(φ), constr−aints −leads to a global one b(β) b(0) = 2mπ where which implies that I(0)(0) = 0. The eigenvalue of Jˆ is m = N 1 l is an integer. Since ρ(β) = ρ(0), this is j=1 j the diﬀerence between the numbers of the right- and the possible only when b(τ) = q(τ) + 2mπτ/β. Because b left-movers. Hence, even and odd values of J correspond appears in the action only in the guise of ∂τb, we may to diﬀerent fermion parities. By taking into account the rewrite Q[ρ, q] as (cid:20) (cid:21) conservation of fermion parity, the allowed values of J (cid:88) become either odd or even integers. Consequently, I(0) ∞ turns into a 4π-periodic function of φ, a characteristic of + − − 2mπτ topological Josephson junctions. Q[ρ, q] = δ b(τ) q(τ) . −∞ β To proceed, it is more convenient to use the dual m= representation of Z [Eq. (31)]. By writing q(τ) = 0 q¯(τ) + 2mπτ/β with q¯(β) = q¯(0) and then performing Now we are in a position to integrate out b, and the the integration(cid:115)over q¯, we obtain Z partition function Z becomes 0 (cid:88) (cid:90) (cid:88) Z =m+=∞−∞ q(β)−q(0)=2mπD[q]e−I˜ , (31) Z0(φ) = v2FLβ m+=∞−∞ eimφ−2πvLFβ(2mπ)2 . (35) (cid:29) where(cid:90) (cid:20) (cid:21) Since T vF /L, the dominant con±tributions arise (cid:88) from the terms with m = 0 and m = 1. Hence, we ∞ may approximate Z(cid:115)0 as β L − φ + I (cid:16) (cid:17) I˜ = dτ (∂ q)2 i ∂ q + δ . (32) 2πv τ 2π τ n 0 F n=1 ≈ 2L − 2πL Z0(φ) v β 1 + 2e vFβ cos φ . (36) The eﬀects of the term 2mπτ/β have been taken into F account by changing the boundary condition for q(τ) − from the periodic one to q(β) q(0) = 2mπ. Equation Consequently, the DC Josephson current is given by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: the

text: desired

text: dual

text: representation

text: of

text: Eq.

text: We

text: shall

text: use

text: both

text: in

text: the

text: following.

text: β

text: C.

text: The

text: DC

text: Josephson

text: current

text: Q[ρ,

text: q]

text: exp

text: i

text: dτJ(∂

text: q

text: ρ)

text: J(τ)=−∞

text: τ

text: Now

text: we

text: are

text: able

text: to

text: compute

text: the

text: DC

text: Josephson

text: current.

text: T

text: N

text: When

text: λ˜

text: or

text: the

text: action

text: I˜

text: is

text: nothing

text: but

text: l/r

text: i

text: qj

text: bj)

text: the

text: one

text: for

text: a

text: particle,

text: of

text: mass

text: L/(πvF

text: and

text: unit

text: charge,

text: Jj=

text: moving

text: on

text: a

text: unit

text: circle

text: thread

text: by

text: a

text: ﬂux

text: φ,

text: provided

text: that

text: N

text: we

text: treat

text: q

text: as

text: the

text: azimuthal

text: angle.

text: In

text: this

text: sense,

text: the

text: →lim∞

text: qj+bj)

text: integer

text: m

text: isTthe

text: winding

text: number.

text: Hence

text: the

text: partition

text: N

text: function

text: at

text: can

text: be

text: written

text: as

text: Jj=

text: l/r

text: with

text: A

text: A(τ

text: ∆τ

text: β/N,

text: and

text: τ

text: j∆τ.

text: More-

text: j

text: j

text: j

text: over,

text: we

text: have

text: written

text: ρ

text: ∂τb.

text: By

text: using

text: the

text: Poisson

text: e

text: summation

text: formula

text: J=

text: lead−ing

text: to

text: the

text: free

text: energy

text: T

text: ln

text: where

text: δ(x

text: eimx

text: πvF

text: [J

text: is

text: the

text: energy

text: of

text: the

text: zero

text: mode

text: with

text: the

text: 2π

text: n=

text: m=

text: quantum

text: number

text: J.

text: The

text: resulting

text: DC

text: Josephson

text: current

text: is

text: of

text: the

text: form

text: we

text: get

text: Q[ρ,

text: q]=

text: lim

text: +∞δ

text: qj

text: bj

text: l

text: evLF

text: J

text: 2φπ

text: PJ

text: N→+∞

text: 2π

text: J=

text: l=

text: We

text: see

text: that

text: Q[ρ,

text: q]

text: provides

text: each

text: time

text: slice

text: a

text: δ-function

text: where

text: PJ

text: e

text: the

text: probability

text: of

text: the

text: zero

text: mode

text: at

text: the

text: eigenstate

text: J

text: We

text: notice

text: that

text: the

text: whole

text: which

text: the

text: constraints

text: bj

text: qj

text: spectrum

text: is

text: in

text: φ,

text: and

text: thu−s

text: −is

text: a

text: The

text: integration

text: over

text: b

text: in

text: the

text: pr−esence

text: of

text: these

text: local

text: periodic

text: function

text: of

text: φ.

text: Moreover,

text: φ)

text: constr−aints

text: −leads

text: to

text: a

text: global

text: one

text: b(β)

text: where

text: which

text: implies

text: that

text: The

text: eigenvalue

text: of

text: Jˆ

text: is

text: m

text: N

text: l

text: is

text: an

text: integer.

text: Since

text: ρ(β)

text: this

text: is

text: j

text: the

text: diﬀerence

text: between

text: the

text: numbers

text: of

text: the

text: right-

text: and

text: the

text: possible

text: only

text: when

text: b(τ)

text: q(τ)

text: Because

text: b

text: left-movers.

text: Hence,

text: even

text: and

text: odd

text: values

text: of

text: J

text: correspond

text: appears

text: in

text: the

text: action

text: only

text: in

text: the

text: guise

text: of

text: ∂τb,

text: we

text: may

text: to

text: diﬀerent

text: fermion

text: parities.

text: By

text: taking

text: into

text: account

text: the

text: rewrite

text: Q[ρ,

text: q]

text: as

text: conservation

text: of

text: fermion

text: parity,

text: the

text: allowed

text: values

text: of

text: J

text: become

text: either

text: odd

text: or

text: even

text: integers.

text: Consequently,

text: turns

text: into

text: a

text: function

text: of

text: φ,

text: a

text: characteristic

text: of

text: 2mπτ

text: topological

text: Josephson

text: junctions.

text: Q[ρ,

text: q]

text: δ

text: b(τ)

text: q(τ)

text: β

text: To

text: proceed,

text: it

text: is

text: more

text: convenient

text: to

text: use

text: the

text: dual

text: m=

text: representation

text: of

text: Z

text: [Eq.

text: By

text: writing

text: q(τ)

text: q¯(τ)

text: 2mπτ/β

text: with

text: q¯(β)

text: and

text: then

text: performing

text: Now

text: we

text: are

text: in

text: a

text: position

text: to

text: integrate

text: out

text: b,

text: and

text: the

text: the

text: q¯,

text: we

text: obtain

text: Z

text: partition

text: function

text: Z

text: becomes

text: Z

text: =m+=∞−∞

text: m+=∞−∞

text: Since

text: T

text: vF

text: /L,

text: the

text: dominant

text: con±tributions

text: arise

text: from

text: the

text: terms

text: with

text: m

text: and

text: m

text: Hence,

text: we

text: may

text: approximate

text: as

text: β

text: L

text: φ

text: I

text: I˜

text: dτ

text: i

text: q

text: δ

text: τ

text: 2π

text: τ

text: n

text: F

text: v

text: β

text: vFβ

text: cos

text: φ

text: The

text: eﬀects

text: of

text: the

text: term

text: 2mπτ/β

text: have

text: been

text: taken

text: into

text: F

text: account

text: by

text: changing

text: the

text: boundary

text: condition

text: for

text: q(τ)

text: from

text: the

text: periodic

text: one

text: to

text: q(β)

text: Equation

text: Consequently,

text: the

text: DC

text: Josephson

text: current

text: is

text: given

text: by

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: A USER GUIDE FOR SINGULARITY 13 Command/the default options Description PersistentDiagram(H, Vars) this function plots/animates bifurcation di- − agrams in x λ plane by passing through the parameter space. The default path is a circle around the singular point and it may include at most two parameters. · α ; for parameter space of dimension more than two, it chooses i three values for α , i.e., a negative, zero and a positive value for i − α . Then, it plots/animates bifurcation diagrams in x λ plane i by passing (circular path by default) through parameter space for ﬁxed parameter α . · i − f(ζ), g(ζ); this function animates bifurcation diagrams in x λ plane by passing through the given path (f(ζ), g(ζ)). · ShortList"; IntermediateList"; CompleteList"; either of these generates a list of points associated persistent bifurcation diagrams. · plot"; this option plots the persistent bifurcation diagrams associated with parameter points output of the previous option, i.e., ShortList"; IntermediateList"; CompleteList". · k; determines the truncation degree of the germ H. \end{itemize} Example 2.7.3. Now we present how the command PersistentDiagram works. − PersistentDiagram(x5 λ + α1x + α2x2 + α3x3, [x, λ] , plot, CompleteList) generates a list from which the list of inequivalent bifurcation diagrams are chosen in Figure 2. Extra sources of non-persistent is caused by singular boundary conditions of a parametric scalar map restricted to a bounded ⊂ R ⊂ R domain. Let W p be a closed disk and U, L be two closed intervals. Next, consider F(x, λ, α) = 0 ∈ ∈ × where α W and (x, λ) U L; see [7, Pages 154-158]. The new non-persistent sources are deﬁned by { ∈ | ∈ × } L := α W F(x, λ, α) = 0 for some (x, λ) ∂U ∂L , C { ∈ | ∈ × } L := α W F = F = 0 for some (x, λ) ∂U L , SH x { ∈ | ∈ × } L := α W F = F = 0 for some (x, λ) U ∂L , SV x { ∈ | ∈ × } L := α W F = F = 0 for some (x, λ) ∂U L , T λ { ∈ | ∈ × G := α W F = 0 at (x , λ, α) for some (x , λ) ∂U L, 1 0 0 (cid:54) ∈ × } x = x and F = F = 0 at (x, λ, α) for some (x, λ) U L , 0 x { ∈ | ∃ ∈ × (cid:54) G := α W (x , λ), (x , λ) ∂U L, x = x s.t F = 0 at 2 1 2 1 2 } (x , λ, α) for i = 1, 2 , i ##LTLine## ##LTLine## ##LTLine##

text: A

label: paragraph

text: USER

text: GUIDE

text: FOR

text: SINGULARITY

text: Command/the

label: list

text: default

text: options

text: Description

text: PersistentDiagram(H,

text: Vars)

text: this

text: function

text: plots/animates

text: bifurcation

text: di-

text: agrams

text: in

text: x

text: λ

text: plane

text: by

text: passing

text: through

text: the

text: parameter

text: space.

text: The

text: default

text: path

text: is

text: a

text: circle

text: around

text: the

text: singular

text: point

text: and

text: it

text: may

text: include

text: at

text: most

text: two

text: parameters.

text: α

text: for

text: parameter

text: space

text: of

text: dimension

text: more

text: than

text: two,

text: it

text: chooses

text: i

text: three

text: values

text: for

text: α

text: i.e.,

text: a

text: negative,

text: zero

text: and

text: a

text: positive

text: value

text: for

text: i

text: α

text: Then,

text: it

text: plots/animates

text: bifurcation

text: diagrams

text: in

text: x

text: λ

text: plane

text: i

text: by

text: passing

text: (circular

text: path

text: by

text: default)

text: through

text: parameter

text: space

text: for

text: ﬁxed

text: parameter

text: α

text: i

text: f(ζ),

text: g(ζ);

text: this

text: function

text: animates

text: bifurcation

text: diagrams

text: in

text: x

text: λ

text: plane

text: by

text: passing

text: through

text: the

text: given

text: path

text: (f(ζ),

text: g(ζ)).

text: ShortList";

text: IntermediateList";

text: CompleteList";

text: either

text: of

text: these

text: generates

text: a

text: list

text: of

text: points

text: associated

text: persistent

text: bifurcation

text: diagrams.

text: plot";

text: this

text: option

text: plots

text: the

text: persistent

text: bifurcation

text: diagrams

text: associated

text: with

text: parameter

text: points

text: output

text: of

text: the

text: previous

text: option,

text: i.e.,

text: ShortList";

text: IntermediateList";

text: CompleteList".

text: k;

text: determines

text: the

text: truncation

text: degree

text: of

text: the

text: germ

text: H.

text: \end{itemize}

text: Example

text: Now

text: we

text: present

text: how

text: the

text: command

text: PersistentDiagram

text: works.

text: λ

text: [x,

text: λ]

text: plot,

text: CompleteList)

text: generates

text: a

text: list

text: from

text: which

text: the

text: list

text: of

text: inequivalent

text: bifurcation

text: diagrams

text: are

text: chosen

text: in

text: Figure

text: Extra

text: sources

text: of

text: non-persistent

text: is

text: caused

text: by

text: singular

text: boundary

text: conditions

text: of

text: a

text: parametric

text: scalar

text: map

text: restricted

text: to

text: a

text: bounded

text: R

text: R

text: domain.

text: Let

text: W

text: p

text: be

text: a

text: closed

text: disk

text: and

text: U,

text: L

text: be

text: two

text: closed

text: intervals.

text: Next,

text: consider

text: F(x,

text: λ,

text: α)

text: where

text: α

text: W

text: and

text: (x,

text: λ)

text: U

text: L;

text: see

text: Pages

text: The

text: new

text: non-persistent

text: sources

text: are

text: deﬁned

text: by

text: L

text: α

text: W

text: F(x,

text: λ,

text: α)

text: for

text: some

text: (x,

text: λ)

text: ∂U

text: ∂L

text: C

text: L

text: α

text: W

text: F

text: F

text: for

text: some

text: (x,

text: λ)

text: ∂U

text: L

text: SH

text: x

text: L

text: α

text: W

text: F

text: F

text: for

text: some

text: (x,

text: λ)

text: U

text: ∂L

text: SV

text: x

text: L

text: α

text: W

text: F

text: F

text: for

text: some

text: (x,

text: λ)

text: ∂U

text: L

text: T

text: λ

text: G

text: α

text: W

text: F

text: at

text: (x

text: λ,

text: α)

text: for

text: some

text: (x

text: λ)

text: ∂U

text: L,

text: x

text: x

text: and

text: F

text: F

text: at

text: (x,

text: λ,

text: α)

text: for

text: some

text: (x,

text: λ)

text: U

text: L

text: x

text: G

text: α

text: W

text: (x

text: λ),

text: (x

text: λ)

text: ∂U

text: L,

text: x

text: x

text: s.t

text: F

text: at

text: (x

text: λ,

text: α)

text: for

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 MAJID GAZOR AND MAHSA KAZEMI 3.3. Standard basis Now we describe how to compute a standard basis for a set of germs in either of the following local rings: fractional germs R R [x, λ](cid:104) (cid:105), formal power series [[x, λ]], and ring of smooth x,λ germs E ; see [4]. The command StandardBasis follows the table and options described below. Note that Vars denotes an order list of variables and the germs in G are in terms of the variables in Vars. The anti-lexicographical ordering is here used while it is determined by the ordering of variables in Vars. Command Description StandardBasis(G, Vars) computes the standard basis of the polynomial germs in G in the fractional germ ring. Default computational ring fractional germ ring. Default input the default germs in G must be polynomial germs. For the cases of non-polynomials, it needs a trun- cation degree k. · k; this determines the truncation degree. · Consider the cases that the option k is used. A warning note is given when the truncation degree k is not suﬃciently high to guarantee that the output is correct. A warning note is given like “The truncation degree is not suﬃciently high and thus, the fol- lowing results might be wrong.” In this case, for an appropriate truncation degree l is given. · Formal"; SmoothGerms"; Fractional"; this determines the computational germ ring and computes the standard basis accordingly. Example 3.3.1. For an example, the command StandardBasis([x5+ − x2 sin(λ+x)+λ2, x3λ2+cos(λ)x, λ6+x4 λx], [x, λ], 7, SmoothGerms) computes the standard basis of the set of germs { − } x5 + x2 sin(λ + x) + λ2, x3λ2 + cos(λ)x, λ6 + x4 λx { } as x, λ2 . 3.4. Colon ideals (cid:104) (cid:105) The colon ideal I : g refers to the ideal deﬁned by (cid:104) (cid:105) { ∈ (cid:104) (cid:105) ⊆ } I : g E = f E : f g E I . (cid:104) (cid:105) (cid:104) | Using the arguments on [4, Page 22], we have I : g E = hi i = (cid:105) g 1,∩. .(cid:104). ,(cid:105)n E , where h1, h2, . . . , hn is a standard basis for the ideal I g R . The command ColonIdeal follows this. [x,λ] Command Description ColonIdeal(I, g) computes the colon ideal I : g for g in E . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MAJID

text: GAZOR

text: AND

text: MAHSA

text: KAZEMI

label: section

text: Standard

text: basis

text: Now

text: we

text: describe

text: how

text: to

text: compute

text: a

text: standard

text: basis

text: for

text: a

text: set

text: of

text: germs

text: in

text: either

text: of

text: the

text: following

text: local

text: rings:

text: fractional

text: germs

text: R

text: R

text: [x,

text: formal

text: power

text: series

text: [[x,

text: λ]],

text: and

text: ring

text: of

text: smooth

text: x,λ

text: germs

text: E

text: see

text: The

text: command

text: StandardBasis

text: follows

text: the

text: table

text: and

text: options

text: described

text: below.

text: Note

text: that

text: Vars

text: denotes

text: an

text: order

text: list

text: of

text: variables

text: and

text: the

text: germs

text: in

text: G

text: are

text: in

text: terms

text: of

text: the

text: variables

text: in

text: Vars.

text: The

text: anti-lexicographical

text: ordering

text: is

text: here

text: used

text: while

text: it

text: is

text: determined

text: by

text: the

text: ordering

text: of

text: variables

text: in

text: Vars.

text: Command

text: Description

text: StandardBasis(G,

text: Vars)

text: computes

text: the

text: standard

text: basis

text: of

text: the

text: polynomial

text: germs

text: in

text: G

text: in

text: the

text: fractional

text: germ

text: ring.

text: Default

text: computational

text: ring

text: fractional

text: germ

text: ring.

text: Default

text: input

text: the

text: default

text: germs

text: in

text: G

text: must

text: be

text: polynomial

text: germs.

text: For

text: the

text: cases

text: of

text: non-polynomials,

text: it

text: needs

text: a

text: trun-

text: cation

text: degree

text: k.

text: k;

text: this

text: determines

text: the

text: truncation

text: degree.

text: Consider

text: the

text: cases

text: that

text: the

text: option

text: k

text: is

text: used.

text: A

text: warning

text: note

text: is

text: given

text: when

text: the

text: truncation

text: degree

text: k

text: is

text: not

text: suﬃciently

text: high

text: to

text: guarantee

text: that

text: the

text: output

text: is

text: correct.

text: A

text: warning

text: note

text: is

text: given

text: like

text: “The

text: truncation

text: degree

text: is

text: not

text: suﬃciently

text: high

text: and

text: thus,

text: the

text: fol-

text: lowing

text: results

text: might

text: be

text: wrong.”

text: In

text: this

text: case,

text: for

text: an

text: appropriate

text: truncation

text: degree

text: l

text: is

text: given.

text: Formal";

text: SmoothGerms";

text: Fractional";

text: this

text: determines

text: the

text: computational

text: germ

text: ring

text: and

text: computes

text: the

text: standard

text: basis

text: accordingly.

text: Example

text: For

text: an

text: example,

text: the

text: command

text: λx],

text: [x,

text: λ],

text: SmoothGerms)

text: computes

text: the

text: standard

text: basis

text: of

text: the

text: set

text: of

text: germs

text: sin(λ

text: x)

text: cos(λ)x,

text: λ6

text: λx

text: as

text: x,

text: λ2

text: Colon

text: ideals

text: The

text: colon

text: ideal

text: I

text: g

text: refers

text: to

text: the

text: ideal

text: deﬁned

text: by

text: I

text: g

text: E

text: f

text: E

text: f

text: g

text: E

text: I

text: Using

text: the

text: arguments

text: on

text: Page

text: we

text: have

text: I

text: g

text: E

text: hi

text: i

text: g

text: E

text: where

text: hn

text: is

text: a

text: standard

text: basis

text: for

text: the

text: ideal

text: I

text: g

text: R

text: The

text: command

text: ColonIdeal

text: follows

text: this.

text: [x,λ]

text: Command

text: Description

text: ColonIdeal(I,

text: g)

text: computes

text: the

text: colon

text: ideal

text: I

text: g

text: for

text: g

text: in

text: E

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 MAJID GAZOR AND MAHSA KAZEMI Command/option Description Intrinsic(A, Vars) computes intrinsic part of the ideal generated by A. It remarks when the ideal is of inﬁnite codimension. Intrinsic(A, B, Vars) computes intrinsic part of a vector space given by (cid:104) (cid:105) (cid:104) (cid:105) A E + B R. It remarks when the vector space is of inﬁnite codimension. Computational ring The default computational ring is the ring of frac- tional germs. Verify/Warning/Suggestion By default, Intrinsic checks and veriﬁes whether the fractional germ ring is suﬃcient for computation of the intrinsic part of the ideal or vector space spanned by A (and B). If fractional germ ring is not suﬃcient, it gives a warning note of possible errors and a suggestion to circumvent the problem. Intrinsic remarks when the problem might be of inﬁnite codimension. Despite possible warning errors, Intrinsic computes the intrinsic part using the fractional germ ring. · k; this option enforces that the computations are performed mod- ulo degree k. · Formal"; SmoothGerms"; Polynomial"; Fractional"; this determines the computational germ ring. It checks and verifies the computations. It gives warning notes of possible errors and alternative suggestions when it finds them necessary. Example 4.1.1. For an example Intrinsic([x3λ+λ2, 3x3λ, 3x2λ2], [x, λ], InfCodim, Fractional) leads to (cid:104) (cid:105) (cid:104) (cid:105) M3 λ + λ2 . As for a second example Intrinsic([x5+λx3+λ2, 5x5+3x3λ, 5x4λ+ 3x2λ2], [λx3 + 2λ2, x3 + 2λ, x4 + 3λx2, λ2, x5], [x, λ]) results in 5 (cid:104) (cid:105) (cid:104) (cid:105) M5 + M3 λ + λ2 . 4.2. Algebraic objects Singularity theory deﬁnes and uses many algebraic objects in the bifurcation analysis of zeros of smooth germs. These include re- stricted tangent space RT, tangent space T, high order term ideal P, smallest intrinsic ideal associated with a singular germ S , a basis for complement of the tangent space E /T, and low order ⊥ terms S ; see [4, 7]. These can be computed in Singularity using the command AlgObjects as well as the individual com- mands RT, T, P, and TangentPerp. The individual commands RT, T, and P have the same default and non-default options as AlgObjects has as follows. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MAJID

text: GAZOR

text: AND

text: MAHSA

text: KAZEMI

text: Command/option

text: Description

text: Intrinsic(A,

text: Vars)

text: computes

text: intrinsic

text: part

text: of

text: the

text: ideal

text: generated

text: by

text: A.

text: It

text: remarks

text: when

text: the

text: ideal

text: is

text: of

text: inﬁnite

text: codimension.

text: Intrinsic(A,

text: B,

text: Vars)

text: computes

text: intrinsic

text: part

text: of

text: a

text: vector

text: space

text: given

text: by

text: A

text: E

text: B

text: R.

text: It

text: remarks

text: when

text: the

text: vector

text: space

text: is

text: of

text: inﬁnite

text: codimension.

text: Computational

text: ring

text: The

text: default

text: computational

text: ring

text: is

text: the

text: ring

text: of

text: frac-

text: tional

text: germs.

text: Verify/Warning/Suggestion

text: By

text: default,

text: Intrinsic

text: checks

text: and

text: veriﬁes

text: whether

text: the

text: fractional

text: germ

text: ring

text: is

text: suﬃcient

text: for

text: computation

text: of

text: the

text: intrinsic

text: part

text: of

text: the

text: ideal

text: or

text: vector

text: space

text: spanned

text: by

text: A

text: (and

text: B).

text: If

text: fractional

text: germ

text: ring

text: is

text: not

text: suﬃcient,

text: it

text: gives

text: a

text: warning

text: note

text: of

text: possible

text: errors

text: and

text: a

text: suggestion

text: to

text: circumvent

text: the

text: problem.

text: Intrinsic

text: remarks

text: when

text: the

text: problem

text: might

text: be

text: of

text: inﬁnite

text: codimension.

text: Despite

text: possible

text: warning

text: errors,

text: Intrinsic

text: computes

text: the

text: intrinsic

text: part

text: using

text: the

text: fractional

text: germ

text: ring.

text: k;

text: this

text: option

text: enforces

text: that

text: the

text: computations

text: are

text: performed

text: mod-

text: ulo

text: degree

text: k.

text: Formal";

text: SmoothGerms";

text: Polynomial";

text: Fractional";

text: this

text: determines

text: the

text: computational

text: germ

text: ring.

text: It

text: checks

text: and

text: verifies

text: the

text: computations.

text: It

text: gives

text: warning

text: notes

text: of

text: possible

text: errors

text: and

text: alternative

text: suggestions

text: when

text: it

text: finds

text: them

text: necessary.

text: Example

text: For

text: an

text: example

text: [x,

text: λ],

text: InfCodim,

text: Fractional)

text: leads

text: to

text: λ

text: λ2

text: As

text: for

text: a

text: second

text: example

text: [x,

text: λ])

text: results

text: in

text: λ

text: λ2

text: Algebraic

text: objects

text: Singularity

text: theory

text: deﬁnes

text: and

text: uses

text: many

text: algebraic

text: objects

text: in

text: the

text: bifurcation

text: analysis

text: of

text: zeros

text: of

text: smooth

text: germs.

text: These

text: include

text: re-

text: stricted

text: tangent

text: space

text: RT,

text: tangent

text: space

text: T,

text: high

text: order

text: term

text: ideal

text: P,

text: smallest

text: intrinsic

text: ideal

text: associated

text: with

text: a

text: singular

text: germ

text: S

text: a

text: basis

text: for

text: complement

text: of

text: the

text: tangent

text: space

text: E

text: /T,

text: and

text: low

text: order

text: terms

text: S

text: see

text: These

text: can

text: be

text: computed

text: in

text: Singularity

text: using

text: the

text: command

text: AlgObjects

text: as

text: well

text: as

text: the

text: individual

text: com-

text: mands

text: RT,

text: T,

text: P,

text: and

text: TangentPerp.

text: The

text: individual

text: commands

text: RT,

text: T,

text: and

text: P

text: have

text: the

text: same

text: default

text: and

text: non-default

text: options

text: as

text: AlgObjects

text: has

text: as

text: follows.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A USER GUIDE FOR SINGULARITY 3 computer, the above list is accessible by using the command with(Singularity). In fact, Maple enlists all the commands in the above two tables as its output. Part 2. Singularity theory The terminology “singularity theory” has been used to deal with many diﬀerent problems in diﬀerent mathematical disciplines. Singularity theory here refers the methodologies in dealing with the qualitative behavior of local zeros (2.0.1) g(x, λ) = 0, ∈ R where x is a state variable and λ is a distinguished parameter. The cases of multi-dimensional parameters are dealt with through the notions of unfolding. Singularity will be soon enhanced to deal with the cases of multi-dimensional state variables. In many real life problems at certain solutions, g(x, λ) is singular, i.e., g(x, λ) = g (x, λ) = 0. x A singular germ g(x, λ) subjected to smooth changes demonstrates surprising changes in the qualitative properties of the solutions, e.g., changes in the number of solutions. This phenomenon is called a bifurcation. 2.1. Qualitative properties We deﬁne the qualitative properties as the invariance of an equivalence relation. The equivalence relation used in Singularity is contact equivalence and is deﬁned by ∼ ⇔ (2.1.1) f g f(x, λ) = S(x, λ)g(X(x, λ), Λ(λ)) where S(x, λ) > 0 while (X, Λ) is locally a diﬀeomorphism such that X (x, λ) (cid:48) x and Λ (λ) > 0. 2.2. Ring and truncation degree In this section we describe how to determine the permissible computational ring and the truncation degree. In fact, each smooth germ with a nonzero-inﬁnite Tay- lor series expansion must be truncated at certain degree. Further, there are four diﬀerent options in Singularity for computational rings, i.e., polynomial, frac- tional, formal power series and smooth germ rings, that they each can be used by commands in Singularity for bifurcation analysis of each singular germ. The command Verify(g) derives the following information about the singular germ g for correct and eﬃcient computations: (1) the permissible computational rings for the germ g. (2) the least permissible truncation degree k for computations involving the germ g. In other words, the computations modulo degrees of higher than (but not equal to) k does not lead to error. (3) our recommended computational ring.

text: A

text: USER

text: GUIDE

text: FOR

text: SINGULARITY

text: computer,

text: the

text: above

text: list

text: is

text: accessible

text: by

text: using

text: the

text: command

text: with(Singularity).

text: In

text: fact,

text: Maple

text: enlists

text: all

text: the

text: commands

text: in

text: the

text: above

text: two

text: tables

text: as

text: its

text: output.

text: Part

text: Singularity

text: theory

text: The

text: terminology

text: “singularity

text: theory”

text: has

text: been

text: used

text: to

text: deal

text: with

text: many

text: diﬀerent

text: problems

text: in

text: diﬀerent

text: mathematical

text: disciplines.

text: Singularity

text: theory

text: here

text: refers

text: the

text: methodologies

text: in

text: dealing

text: with

text: the

text: qualitative

text: behavior

text: of

text: local

text: zeros

text: g(x,

label: equation

text: λ)

text: R

text: where

text: x

text: is

text: a

text: state

text: variable

text: and

text: λ

text: is

text: a

text: distinguished

text: parameter.

text: The

text: cases

text: of

text: multi-dimensional

text: parameters

text: are

text: dealt

text: with

text: through

text: the

text: notions

text: of

text: unfolding.

text: Singularity

text: will

text: be

text: soon

text: enhanced

text: to

text: deal

text: with

text: the

text: cases

text: of

text: multi-dimensional

text: state

text: variables.

text: In

text: many

text: real

text: life

text: problems

text: at

text: certain

text: solutions,

text: g(x,

text: λ)

text: is

text: singular,

text: i.e.,

text: g(x,

text: λ)

text: g

text: (x,

text: λ)

text: x

text: A

text: singular

text: germ

text: g(x,

text: λ)

text: subjected

text: to

text: smooth

text: changes

text: demonstrates

text: surprising

text: changes

text: in

text: the

text: qualitative

text: properties

text: of

text: the

text: solutions,

text: e.g.,

text: changes

text: in

text: the

text: number

text: of

text: solutions.

text: This

text: phenomenon

text: is

text: called

text: a

text: bifurcation.

text: Qualitative

text: properties

text: We

text: deﬁne

text: the

text: qualitative

text: properties

text: as

text: the

text: invariance

text: of

text: an

text: equivalence

text: relation.

text: The

text: equivalence

text: relation

text: used

text: in

text: Singularity

text: is

text: contact

text: equivalence

text: and

text: is

text: deﬁned

text: by

text: f

text: g

text: f(x,

text: λ)

text: S(x,

text: λ)g(X(x,

text: λ),

text: Λ(λ))

text: where

text: S(x,

text: λ)

text: while

text: (X,

text: Λ)

text: is

text: locally

text: a

text: diﬀeomorphism

text: such

text: that

text: X

text: (x,

text: λ)

text: x

text: and

text: Λ

text: (λ)

text: Ring

text: and

text: truncation

text: degree

text: In

text: this

text: section

text: we

text: describe

text: how

text: to

text: determine

text: the

text: permissible

text: computational

text: ring

text: and

text: the

text: truncation

text: degree.

text: In

text: fact,

text: each

text: smooth

text: germ

text: with

text: a

text: nonzero-inﬁnite

text: Tay-

text: lor

text: series

text: expansion

text: must

text: be

text: truncated

text: at

text: certain

text: degree.

text: Further,

text: there

text: are

text: four

text: diﬀerent

text: options

text: in

text: Singularity

text: for

text: computational

text: rings,

text: i.e.,

text: polynomial,

text: frac-

text: tional,

text: formal

text: power

text: series

text: and

text: smooth

text: germ

text: rings,

text: that

text: they

text: each

text: can

text: be

text: used

text: by

text: commands

text: in

text: Singularity

text: for

text: bifurcation

text: analysis

text: of

text: each

text: singular

text: germ.

text: The

text: command

text: Verify(g)

text: derives

text: the

text: following

text: information

text: about

text: the

text: singular

text: germ

text: g

text: for

text: correct

text: and

text: eﬃcient

text: computations:

text: the

text: permissible

text: computational

text: rings

text: for

text: the

text: germ

text: g.

text: the

text: least

text: permissible

text: truncation

text: degree

text: k

text: for

text: computations

text: involving

text: the

text: germ

text: g.

text: In

text: other

text: words,

text: the

text: computations

text: modulo

text: degrees

text: of

text: higher

text: than

text: (but

text: not

text: equal

text: to)

text: k

text: does

text: not

text: lead

text: to

text: error.

text: our

text: recommended

text: computational

text: ring.


text: 6 MAJID GAZOR AND MAHSA KAZEMI Warning: The polynomial germ ring is not suitable for normal form computa- tions. Suggestion: Use the command Verify to ﬁnd the appropriate computational ring. The following output might be wrong. The germ is an inﬁnite codimensional germ. In fact the above statement is wrong since the high order term ideal contains M M (cid:104) (cid:105) M(cid:104) (cid:105) 6 + 4 λ + λ2 . Now the command Verify(x5 + x3λ + sin(λ2), [x, λ]) gives rise to Fractional germ ring; Formal power series ring; Smooth germ ring. Thus, we use Normalform(x5 +x3λ+sin(λ2), [x, λ], 10, Fractional) to obtain x5 + x3λ + λ2. 2.4. Universal unfolding Generally in dealing with singular problems, extra complications are experi- enced in the laboratory data than what are predicted by the modeling theo- retical analysis. The problem here is due to modeling imperfections; natural phenomena can not be perfectly modeled by a mathematical model. In fact one usually neglects the impact of many factors like friction, pressure, and/or temperature, etc., to get a manageable mathematical model. Otherwise one will end up with a mathematical modeling problem with too many or inﬁ- nite number of parameters. The imperfections around singular points may cause dramatic qualitative changes in the solution set of the model. Universal unfolding gives us a natural way to circumvent the problem of imperfections. Deﬁnition 2.4.1. A parametric germ G(x, λ, α) is called an unfolding for g(x, λ) when G(x, λ, 0) = g(x, λ). An unfolding G(x, λ, α) for g is called a versal unfolding when for each un- folding H(x, λ, β) of g(x, λ) there is a smooth germ α(β) so that H is contact- equivalent to G(x, λ, α(β)). Roughly speaking, a versal unfolding is a paramet- ric germ that contains a contact-equivalent copy of all small perturbations of g(x, λ). A versal unfolding with insigniﬁcant parameters is not suitable for the bifurcation analysis. So, we are interested in a versal unfolding that has a minimum possible number of parameters, that is called universal unfolding. In other words, universal unfolding has the minimum possible number of pa- rameters so that they accommodate all possible qualitative types that small perturbations of g(x, λ) may experience.

text: MAJID

text: GAZOR

text: AND

text: MAHSA

text: KAZEMI

text: Warning:

text: The

text: polynomial

text: germ

text: ring

text: is

text: not

text: suitable

text: for

text: normal

text: form

text: computa-

text: tions.

text: Suggestion:

text: Use

text: the

text: command

text: Verify

text: to

text: ﬁnd

text: the

text: appropriate

text: computational

text: ring.

text: The

text: following

text: output

text: might

text: be

text: wrong.

text: The

text: germ

text: is

text: an

text: inﬁnite

text: codimensional

text: germ.

text: In

text: fact

text: the

text: above

text: statement

text: is

text: wrong

text: since

text: the

text: high

text: order

text: term

text: ideal

text: contains

text: M

text: M

text: λ

text: λ2

text: Now

text: the

text: command

text: [x,

text: λ])

text: gives

text: rise

text: to

text: Fractional

text: germ

text: ring;

text: Formal

text: power

text: series

text: ring;

text: Smooth

text: germ

text: ring.

text: Thus,

text: we

text: use

text: [x,

text: λ],

text: Fractional)

text: to

text: obtain

text: Universal

text: unfolding

text: Generally

text: in

text: dealing

text: with

text: singular

text: problems,

text: extra

text: complications

text: are

text: experi-

text: enced

text: in

text: the

text: laboratory

text: data

text: than

text: what

text: are

text: predicted

text: by

text: the

text: modeling

text: theo-

text: retical

text: analysis.

text: The

text: problem

text: here

text: is

text: due

text: to

text: modeling

text: imperfections;

text: natural

text: phenomena

text: can

text: not

text: be

text: perfectly

text: modeled

text: by

text: a

text: mathematical

text: model.

text: In

text: fact

text: one

text: usually

text: neglects

text: the

text: impact

text: of

text: many

text: factors

text: like

text: friction,

text: pressure,

text: and/or

text: temperature,

text: etc.,

text: to

text: get

text: a

text: manageable

text: mathematical

text: model.

text: Otherwise

text: one

text: will

text: end

text: up

text: with

text: a

text: mathematical

text: modeling

text: problem

text: with

text: too

text: many

text: or

text: inﬁ-

text: nite

text: number

text: of

text: parameters.

text: The

text: imperfections

text: around

text: singular

text: points

text: may

text: cause

text: dramatic

text: qualitative

text: changes

text: in

text: the

text: solution

text: set

text: of

text: the

text: model.

text: Universal

text: unfolding

text: gives

text: us

text: a

text: natural

text: way

text: to

text: circumvent

text: the

text: problem

text: of

text: imperfections.

text: Deﬁnition

text: A

text: parametric

text: germ

text: G(x,

text: λ,

text: α)

text: is

text: called

text: an

text: unfolding

text: for

text: g(x,

text: λ)

text: when

text: G(x,

text: λ,

text: g(x,

text: λ).

text: An

text: unfolding

text: G(x,

text: λ,

text: α)

text: for

text: g

text: is

text: called

text: a

text: versal

text: unfolding

text: when

text: for

text: each

text: un-

text: folding

text: H(x,

text: λ,

text: β)

text: of

text: g(x,

text: λ)

text: there

text: is

text: a

text: smooth

text: germ

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text: JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 4 While in Fig. 2(b) and Fig. 2(c), the correct frequencies are 150 found. The results accord with our previous analysis. 100 150 50 100 0 0 10 20 30 40 50 60 50 f (a) 0 0 10 20 30 40 50 60 150 f (a) 100 150 50 100 0 0 10 20 30 40 50 60 50 f (b) 0 Fig. 3. The pseudo-spectrum in two cases. (a) S1,S2; (b) S1,S3. 0 10 20 30 40 50 60 f (b) 1 150 0.9 0.8 100 0.7 50 0.6 0 0.5 0 10 20 30 40 50 60 0.4 f (c) 0.3 Proposed method Conventional ESPRIT 0.2 Fig. 2. The pseudo-spectrum of the eligible frequencies. (a) S1,S2; (b) S1,S3; 1 2 3 4 5 6 7 8 (c) S2,S3. K Fig. 4. The probabilities of success for different numbers of frequency In the second simulation, we set K = 3 and f1 = components. 25Hz, f = 33Hz, f = 50Hz. The results obtained from 2 3 S1,S2 and S1,S3 are shown in Fig. 3(a) and Fig. 3(b) respec- tively. Although neither of the results is right, their intersection Finally, we compare the errors of the frequencies with indicates the correct frequencies. This experiment validates our conventional ESPRIT at different SNR values. K = 3 is method in the previous section. ﬁxed and SNR varies from 10dB to 30dB. Average values are Then we test the capacity of the algorithms when the obtained from 500 trials for each SNR. As shown in Fig. 5, frequencies distribute uniformly in (0, 100Hz). The number the proposed method has the same level of accuracy with of the frequencies K varies from 1 to 8. The minimum conventional ESPRIT. This experiment demonstrates that the interval between the frequencies is larger than 0.1Hz and the proposed method can achieve enough accuracy. corresponding amplitudes are in [0.1, 1]. a = 7, b = 8, c = 9 are ﬁxed and 100 snapshots are used for each sub-Nyquist 0.7 sequence. We do 500 trials for each K in the noisy environ- Proposed method (cid:115) (cid:44) ment with SNR=20dB. The mean square errors of estimated 0.6 Conventional ESPRIT (cid:80) frequencies f different from true frequencies are computed 0.5 es K − by MSE = (f f )2 K. In this simulation, if the 0.4 es k k=1 0.3 MSE of one trial is smaller than 0.05, we say that this is a successful trial. For the sake of comparison, we also sample 0.2 the signals at the rate fS = fH and use conventional ESPRIT 0.1 to estimate the frequencies. The success probabilities of the 0 proposed method and conventional ESPRIT for different K 10 15 20 25 30 SNR are shown in Fig. 4. The probabilities of success are higher ≤ than 90% when K 5 and roughly equivalent to conventional Fig. 5. The average MSE for different SNR values. ESPRIT. P P P UM UM UM sseccus fo ytilibaborP ESM P P UM UM ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: JOURNAL

text: OF

text: LATEX

text: CLASS

text: FILES,

text: VOL.

text: NO.

text: DECEMBER

text: While

text: in

text: Fig.

text: and

text: Fig.

text: the

text: correct

text: frequencies

text: are

font: default

text: found.

text: The

text: results

text: accord

text: with

text: our

text: previous

text: analysis.

text: f

text: (a)

text: f

text: (a)

text: f

text: (b)

text: Fig.

text: The

text: pseudo-spectrum

text: in

text: two

text: cases.

text: (a)

text: (b)

text: f

text: (b)

text: f

text: (c)

text: Proposed

text: method

text: Conventional

text: ESPRIT

text: Fig.

text: The

text: pseudo-spectrum

text: of

text: the

text: eligible

text: frequencies.

text: (a)

text: (b)

text: (c)

text: K

text: Fig.

text: The

text: probabilities

text: of

text: success

text: for

text: different

text: numbers

text: of

text: frequency

text: In

text: the

text: second

text: simulation,

text: we

text: set

text: K

text: and

text: components.

text: f

text: f

text: The

text: results

text: obtained

text: from

text: and

text: are

text: shown

text: in

text: Fig.

text: and

text: Fig.

text: respec-

text: tively.

text: Although

text: neither

text: of

text: the

text: results

text: is

text: right,

text: their

text: intersection

text: Finally,

text: we

text: compare

text: the

text: errors

text: of

text: the

text: frequencies

text: with

text: indicates

text: the

text: correct

text: frequencies.

text: This

text: experiment

text: validates

text: our

text: conventional

text: ESPRIT

text: at

text: different

text: SNR

text: values.

text: K

text: is

text: method

text: in

text: the

text: previous

text: section.

text: ﬁxed

text: and

text: SNR

text: varies

text: from

text: to

text: Average

text: values

text: are

text: Then

text: we

text: test

text: the

text: capacity

text: of

text: the

text: algorithms

text: when

text: the

text: obtained

text: from

text: trials

text: for

text: each

text: SNR.

text: As

text: shown

text: in

text: Fig.

text: frequencies

text: distribute

text: uniformly

text: in

text: The

text: number

text: the

text: proposed

text: method

text: has

text: the

text: same

text: level

text: of

text: accuracy

text: with

text: of

text: the

text: frequencies

text: K

text: varies

text: from

text: to

text: The

text: minimum

text: conventional

text: ESPRIT.

text: This

text: experiment

text: demonstrates

text: that

text: the

text: interval

text: between

text: the

text: frequencies

text: is

text: larger

text: than

text: and

text: the

text: proposed

text: method

text: can

text: achieve

text: enough

text: accuracy.

text: corresponding

text: amplitudes

text: are

text: in

text: a

text: b

text: c

text: are

text: ﬁxed

text: and

text: snapshots

text: are

text: used

text: for

text: each

text: sub-Nyquist

text: sequence.

text: We

text: do

text: trials

text: for

text: each

text: K

text: in

text: the

text: noisy

text: environ-

text: Proposed

text: method

text: ment

text: with

text: The

text: mean

text: square

text: errors

text: of

text: estimated

text: Conventional

text: ESPRIT

text: frequencies

text: f

text: different

text: from

text: true

text: frequencies

text: are

text: computed

text: es

text: K

text: by

text: MSE

text: (f

text: f

text: K.

text: In

text: this

text: simulation,

text: if

text: the

text: es

text: k

text: MSE

text: of

text: one

text: trial

text: is

text: smaller

text: than

text: we

text: say

text: that

text: this

text: is

text: a

text: successful

text: trial.

text: For

text: the

text: sake

text: of

text: comparison,

text: we

text: also

text: sample

text: the

text: signals

text: at

text: the

text: rate

text: fS

text: fH

text: and

text: use

text: conventional

text: ESPRIT

text: to

text: estimate

text: the

text: frequencies.

text: The

text: success

text: probabilities

text: of

text: the

text: proposed

text: method

text: and

text: conventional

text: ESPRIT

text: for

text: different

text: K

text: SNR

text: are

text: shown

text: in

text: Fig.

text: The

text: probabilities

text: of

text: success

text: are

text: higher

text: than

text: when

text: K

text: and

text: roughly

text: equivalent

text: to

text: conventional

text: Fig.

text: The

text: average

text: MSE

text: for

text: different

text: SNR

text: values.

text: ESPRIT.

text: P

text: P

text: P

text: UM

text: UM

text: UM

text: sseccus

text: fo

text: ytilibaborP

text: ESM

text: P

text: P

text: UM

text: UM

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text: | (cid:105) | (cid:105) | (cid:105) | (cid:105) ⊗ ⊗ | (cid:105) ⊗ ⊗ where ψ = a 1 + beiδ 6 with a2 + b2 = 1. Since ψ = 11 11 σ ψ , (11 11 II I x II ⊗ ⊗ † σ )ρˆT (0)(11 11 σ ) has a GHZ-symmetry provided that a2 = b2 = 1/2 and δ = 0. x II x Using Eqs. (2.12) and (2.13) one can show that the spectral decomposition of ρˆT (t) II becomes (cid:2) (cid:3) | (cid:105)(cid:104) | − − | (cid:105)(cid:104) | − | (cid:105)(cid:104) | | (cid:105)(cid:104) | ρˆT (t) = λ φ φ +(1 P 2) a2 + b2(1 P 2) 0 0 +b2P 2(1 P 2) ( 2 2 + 4 4 ) (3.12) II 2 II II t t t t where (cid:0) (cid:1) λ = P 2(cid:112)(a2 + b2P 2) (3.13) 2 t t | (cid:105) 1 | (cid:105) | (cid:105) φ = a 1 + bP eiδ 6 . II t a2 + b2P 2 t Unlike the case of type I ρˆT (t) is rank four tensor. From Eq. (3.12) one can derive the II upper bound of τ for ρˆT (t), which is ABC II ≤ 4a2b2P 4 τ t . (3.14) ABC a2 + b2P 2 t N N N The negativities , , and of ρˆT (t) can be computed by making use A(BC) B(AC) (AB)C II of Eq. (1.6). The ﬁnal expressions are (cid:113) (cid:113) N N − − − (cid:2) (cid:3) = = b4P 4(1 P 2)2 + 4a2b2P 6 b2P 2(1 P 2) (3.15) A(BC) B(AC) t t t t t N − − − − − = (1 P 2)2 [a2 + b2(1 P 2)]2 + 4a2b2P 6 (1 P 2) a2 + b2(1 P 2) . (AB)C t t t t t N N N It is also easy to show = = = 0. Thus the π-tangle of ρˆT (t) is AB AC (cid:2) BC (cid:3) II 1 N N πII (t) = 2 2 + 2 . (3.16) GHZ 3 A(BC) (AB)C 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

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text: a

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text: has

text: a

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text: provided

text: that

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text: δ

text: x

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text: Using

text: Eqs.

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text: can

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text: becomes

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text: Eq.

text: one

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text: the

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text: ρˆT

text: (t),

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text: negativities

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text: of

text: ρˆT

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text: be

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text: making

text: use

text: A(BC)

text: B(AC)

text: (AB)C

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text: of

text: Eq.

text: The

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text: of

text: ρˆT

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text: is

text: AB

text: AC

text: BC

text: II

text: N

text: N

text: πII

text: (t)

text: GHZ

text: A(BC)

text: (AB)C

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text: → ∞ When t = 0, πII (0) becomes 4a2b2 and it reduces to zero as t . Of course, GHZ · · · the entanglement of ρˆT (t) is completely disentangled at t = t (n = 1, 2, ) in the non- II n Markovian regime. C. Type III Let us choose the initial state in a form | (cid:105)(cid:104) | ρˆT (0) = ψ ψ (3.17) III III III | (cid:105) | (cid:105) | (cid:105) | (cid:105) ⊗ ⊗ | (cid:105) ⊗ ⊗ where ψ = a 3 + beiδ 4 with a2 + b2 = 1. Since ψ = 11 σ σ ψ , (11 σ III I x x III x ⊗ ⊗ † σ )ρˆT (0)(11 σ σ ) has a GHZ-symmetry provided that a2 = b2 = 1/2 and δ = 0. x III x x Using Eqs. (2.12) and (2.13) one can show that the spectral decomposition of ρˆT (t) III becomes (cid:2) (cid:3) | (cid:105)(cid:104) | − − | (cid:105)(cid:104) | − | (cid:105)(cid:104) | | (cid:105)(cid:104) | ρˆT (t) = λ φ φ +(1 P 2) a2(1 P 2) + b2 0 0 +a2P 2(1 P 2) ( 1 1 + 2 2 ) III 3 III III t t t t (3.18) where (cid:0) (cid:1) λ = P 2((cid:112)a2P 2 + b2) (3.19) 3 t t | (cid:105) 1 | (cid:105) | (cid:105) φ = aP 3 + beiδ 4 . III t a2P 2 + b2 t Unlike the case of type I ρˆT (t) is rank four tensor. From Eq. (3.18) one can derive the III upper bound of τ for ρˆT (t), which is ABC III ≤ 4a2b2P 4 τ t . (3.20) ABC a2P 2 + b2 t N N N The negativities , , and of ρˆT (t) can be computed by making use A(BC) B(AC) (AB)C III of Eq. (1.6), whose explicit expressions are 14 ##LTLine## ##LTLine## ##LTLine##

text: When

text: t

text: πII

text: becomes

text: and

text: it

text: reduces

text: to

text: zero

text: as

text: t

text: Of

text: course,

text: GHZ

text: the

text: entanglement

text: of

text: ρˆT

text: (t)

text: is

text: completely

text: disentangled

text: at

text: t

text: t

text: (n

text: in

text: the

text: non-

text: II

text: n

text: Markovian

text: regime.

text: C.

text: Type

text: III

text: Let

text: us

text: choose

text: the

text: initial

text: state

text: in

text: a

text: form

text: ρˆT

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text: ψ

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text: a

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text: provided

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text: and

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text: x

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text: x

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text: Using

text: Eqs.

text: and

text: one

text: can

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text: the

text: spectral

text: decomposition

text: of

text: ρˆT

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text: III

text: III

text: t

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text: the

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text: (t)

text: is

text: rank

text: four

text: tensor.

text: From

text: Eq.

text: one

text: can

text: derive

text: the

text: III

text: upper

text: bound

text: of

text: τ

text: for

text: ρˆT

text: (t),

text: which

text: is

text: ABC

text: III

text: τ

text: t

text: ABC

text: t

text: N

text: N

text: N

text: The

text: negativities

text: and

text: of

text: ρˆT

text: (t)

text: can

text: be

text: computed

text: by

text: making

text: use

text: A(BC)

text: B(AC)

text: (AB)C

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text: Eq.

text: whose

text: explicit

text: expressions

text: are

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text: and πIII (t) should exhibit an oscillatory behavior with rapid decrease of amplitude due to GHZ P in Eq. (2.11). This can be seen in Fig. 2, where πII (t) and πIII (t) are plotted as a t GHZ GHZ function of dimensionless parameter γ t and a2. We choose λ as a λ = 0.01γ . As expected 0 0 the tripartite entanglement reduces to zero with increasing time with oscillatory behavior. FIG. 3: (Color online) The γ t dependence of πI (t) (red solid), πII (t) (black dashed), and 0 GHZ GHZ πIII (t) (blue dotted) when (a) a2 = 0.1 and (b) a2 = 0.9. We choose λ as a λ = 0.001γ . GHZ 0 The π-tangles πI (t) , πII (t) , and πIII (t) are compared in Fig. 3 when λ/γ = 0.001. GHZ GHZ GHZ 0 They are represented by red solid, black dashed, and blue dotted lines respectively. Fig. 3(a) and Fig. 3(b) correspond to a2 = 0.1 and a2 = 0.9. Both ﬁgures clearly show the revival of the tripartite entanglement, after a ﬁnite period of time of complete disappearance. The revival phenomenon seems to be mainly due to the memory eﬀect of the non-Markovian ≥ ≥ environment. It is of interest to note that while πIII (t) πII (t) πI (t) when GHZ GHZ GHZ ≥ ≥ a2 = 0.1, the order is changed as πI (t) πII (t) πIII (t) when a2 = 0.9. GHZ GHZ GHZ IV. ENTANGLEMENT DYNAMICS OF W-TYPE INITIAL STATES In this section we examine the tripartite entanglement dynamics when the initial states are two W-type states. Both initial states are LU to each other. However, their entanglement dynamics are diﬀerent due to Eqs. (2.12) and (2.13). A. Type I In this subsection we choose the initial state as 16 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: πIII

text: (t)

text: should

text: exhibit

text: an

text: oscillatory

text: behavior

text: with

text: rapid

text: decrease

text: of

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text: can

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text: Fig.

text: where

text: πII

text: (t)

text: and

text: πIII

text: (t)

text: are

text: plotted

text: as

text: a

text: t

text: GHZ

text: GHZ

text: function

text: of

text: dimensionless

text: parameter

text: γ

text: t

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text: λ

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text: expected

text: the

text: tripartite

text: entanglement

text: reduces

text: to

text: zero

text: with

text: increasing

text: time

text: with

text: oscillatory

text: behavior.

text: FIG.

text: (Color

label: caption

text: online)

text: The

text: γ

text: t

text: dependence

text: of

text: πI

text: (t)

text: (red

text: solid),

text: πII

text: (t)

text: (black

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text: and

text: GHZ

text: GHZ

text: πIII

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text: (blue

text: dotted)

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text: λ

text: as

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text: π-tangles

text: πI

text: (t)

text: πII

text: (t)

text: and

text: πIII

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text: Fig.

text: when

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text: GHZ

text: GHZ

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text: after

text: a

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text: πIII

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text: GHZ

text: GHZ

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text: IV.

text: ENTANGLEMENT

text: DYNAMICS

text: OF

text: W-TYPE

text: INITIAL

text: STATES

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text: the

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text: where (cid:34) 1 (cid:0) | (cid:105)(cid:104) | | (cid:1)(cid:105)(cid:104) | (cid:0) | (cid:105)(cid:104) | (cid:1) σ (t) = (b2 + c2) 1 1 +(a2 + c2) 2 2 +(a2 + b2) 4 4 II 2 (cid:35) | (cid:105)(cid:104) | (cid:0) − | (cid:105)(cid:104) | | (cid:105)(cid:104) | −(cid:1) | (cid:105)(cid:104) | +ab eiδ1 1 2 +e iδ1 2 1 + ac eiδ2 1 4 +e iδ2 4 1 (4.9) − − | (cid:105)(cid:104) | − | (cid:105)(cid:104) | +bc e i(δ1 δ2) 2 4 +ei(δ1 δ2) 4 2 . The spectral decomposition of σ (t) cannot be derived analytically. Also, analytic com- II putation of π-tangle for ρˆW (t) is impossible. Thus, we have to reply on the numerical II approach for computation of π-tangle. 0.5 0.4 0.3 0.2 0.1 γ 0.0 0.0 0.5 1.0 1.5 2.0 t 0 | (cid:105) FIG. 4: (Color online) The γ t dependence of πI (red line) and πII (blue line) when W and 0 W W 1 | (cid:105) W interact with the Markovian environment. We choose λ = 3γ and a2 = b2 = c2 = 1/3. 2 0 However, some special cases allow the analytic computation. In this paper we consider a special case a2 = b2 = c2 = 1/3. In this case the spectral decomposition of σ (t) can be II derived as 2| (cid:105)(cid:104) | 1| (cid:105)(cid:104) | 1| (cid:105)(cid:104) | σ (t) = α α + α α + α α (4.10) II 1 1 2 2 3 3 3 6 6 where 19 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: σ

text: (t)

text: II

text: +ab

text: +e

text: ac

text: +e

text: +bc

text: e

text: The

text: spectral

text: decomposition

text: of

text: σ

text: (t)

text: cannot

text: be

text: derived

text: analytically.

text: Also,

text: analytic

text: com-

text: II

text: putation

text: of

text: π-tangle

text: for

text: ρˆW

text: (t)

text: is

text: impossible.

text: Thus,

text: we

text: have

text: to

text: reply

text: on

text: the

text: numerical

text: II

text: approach

text: for

text: computation

text: of

text: π-tangle.

text: γ

text: t

text: FIG.

text: (Color

text: online)

text: The

text: γ

text: t

text: dependence

text: of

text: πI

text: (red

text: line)

text: and

text: πII

text: (blue

text: line)

text: when

text: W

text: and

text: W

text: W

text: W

text: interact

text: with

text: the

text: Markovian

text: environment.

text: We

text: choose

text: λ

text: 3γ

text: and

text: However,

text: some

text: special

text: cases

text: allow

text: the

text: analytic

text: computation.

text: In

text: this

text: paper

text: we

text: consider

text: a

text: special

text: case

text: In

text: this

text: case

text: the

text: spectral

text: decomposition

text: of

text: σ

text: (t)

text: can

text: be

text: II

text: derived

text: as

text: σ

text: (t)

text: α

text: α

text: α

text: α

text: α

text: α

text: II

text: where

text: ##LTFigure##

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text: alone need not necessarily ensure its clousure to be topologically equivalent to the closed ball, the fact that a domain D of convergence of some power series, say with D bounded for sim- BN plicity, is complete Reinhardt ensures that its closure D is homeomorphic to the closed ball . However, this does not mean that domains of convergence of power series are holomorphically equivalent to the ball and thereby to each other. Indeed, two of the simplest logarithmically convex complete Reinhardt domains (which is to say, domains of convergence of power series) U namely, the polydisc N whose logarithmic image has its support function ﬁnite-valued on all { ∈ C | | } of P S and the unbounded domain H = z N : z . . . z < 1 obtained as the inverse N 1 N image of a half-space under the logarithmic map λ with the support function of H ﬁnite only at a single point of P S , are biholomorphically inequivalent. One way to see this non-equivalence N B U of H with N or N is via a theorem due to H. Cartan about biholomorphic mappings be- B U tween circular domains, in conjunction with the fact that the automorphism group of N or N act transitively on their respective domains. This failure of the Riemann mapping theorem of complex analysis in dimension one, persists even if we restrict ourselves to bounded domains of convergence of power series in any higher dimension. Indeed, two of the simplest topologically B U trivial bounded domains N and N are not biholomorphically equivalent. Cartan’s method- ology towards establishing these inequivalences, is an excellent example of how the local power series representation of holomorphic functions suﬃces to provide an elementary and (yet!) neat proof of, the just alluded to failure, of the Riemann mapping theorem in every dimension N > 1. We mention in passing, as a matter of (a non-trivial!) fact that any pair of such domains (of convergence of some power series) will generically fail to be biholomorphically equivalent. Let us get back from the detour about gaining an understanding of the topology of domains of convergence, to our question: what are the characterizing features of the logarithmic images of domains of convergence of power series? A little contemplation will help convince that what is ⊂ R required here is, to determine a condition to be imposed on the given convex domain G N − which ensures the completeness of λ 1(G), without involving its pull back into the absolute space. An answer to this requirement here would be that the characteristic/recession cone of G contain −R the standard backward cone ( )N. This means that the domain G is unbounded in N-many + −R independent directions; to be precise, in every direction of ( )N. While what constitutes + a satisfactory answer to our question at hand is subjective, saying that the characteristic cone −R contain ( )N cannot be considered satisfactory; for, one is often ‘given’ a domain by its + deﬁning function and it remains therefore to ﬁgure out ways to ﬁnd out the characteristic cone from the deﬁning function. Instead of taking up this task in all its generality, we shall directly ⊂ R lay down the condition in our setting. Namely, given a convex domain G N with deﬁning { } function ψ, write down a condition to decide if the characteristic cone of G = s : ψ(s) < 0 −R contains ( )N. As this is easily seen to be equivalent to requiring that the gradient vector + of ψ at all boundary points of G i.e., the outer normal vector ﬁeld along ∂G points into the R standard cone N, the sought-after condition on ψ is that it satisfy + ∇ (cid:104) ψ √1 (cid:105) √1 | | , (1, . . . , 1) < ψ N 2 which can be rew(cid:0)ritten as: (cid:1) (cid:0) (cid:1) | | | | (2.24) 2 ∂ψ/∂z + . . . + ∂ψ/∂z 2 < N ∂ψ/∂z 2 + . . . + ∂ψ/∂z 2 , 1 N 1 N R where all derivatives are to be evaluated at points s in N where ψ(s) = 0. This is the ana- { } lytic condition for a convex function ψ(s) to satisfy, for the convex domain G = s : ψ(s) < 0 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: alone

text: need

text: not

text: necessarily

text: ensure

text: its

text: clousure

text: to

text: be

text: topologically

text: equivalent

text: to

text: the

text: closed

text: ball,

text: the

text: fact

text: that

text: a

text: domain

text: D

text: of

text: convergence

text: of

text: some

text: power

text: series,

text: say

text: with

text: D

text: bounded

text: for

text: sim-

text: BN

text: plicity,

text: is

text: complete

text: Reinhardt

text: ensures

text: that

text: its

text: closure

text: D

text: is

text: homeomorphic

text: to

text: the

text: closed

text: ball

text: However,

text: this

text: does

text: not

text: mean

text: that

text: domains

text: of

text: convergence

text: of

text: power

text: series

text: are

text: holomorphically

text: equivalent

text: to

text: the

text: ball

text: and

text: thereby

text: to

text: each

text: other.

text: Indeed,

text: two

text: of

text: the

text: simplest

text: logarithmically

text: convex

text: complete

text: Reinhardt

text: domains

text: (which

text: is

text: to

text: say,

text: domains

text: of

text: convergence

text: of

text: power

text: series)

text: U

text: namely,

text: the

text: polydisc

text: N

text: whose

text: logarithmic

text: image

text: has

text: its

text: support

text: function

text: ﬁnite-valued

text: on

text: all

text: C

text: of

text: P

text: S

text: and

text: the

text: unbounded

text: domain

text: H

text: z

text: N

text: z

text: z

text: obtained

text: as

text: the

text: inverse

text: N

text: N

text: image

text: of

text: a

text: half-space

text: under

text: the

text: logarithmic

text: map

text: λ

text: with

text: the

text: support

text: function

text: of

text: H

text: ﬁnite

text: only

text: at

text: a

text: single

text: point

text: of

text: P

text: S

text: are

text: biholomorphically

text: inequivalent.

text: One

text: way

text: to

text: see

text: this

text: non-equivalence

text: N

text: B

text: U

text: of

text: H

text: with

text: N

text: or

text: N

text: is

text: via

text: a

text: theorem

text: due

text: to

text: H.

text: Cartan

text: about

text: biholomorphic

text: mappings

text: be-

text: B

text: U

text: tween

text: circular

text: domains,

text: in

text: conjunction

text: with

text: the

text: fact

text: that

text: the

text: automorphism

text: group

text: of

text: N

text: or

text: N

text: act

text: transitively

text: on

text: their

text: respective

text: domains.

text: This

text: failure

text: of

text: the

text: Riemann

text: mapping

text: theorem

text: of

text: complex

text: analysis

text: in

text: dimension

text: one,

text: persists

text: even

text: if

text: we

text: restrict

text: ourselves

text: to

text: bounded

text: domains

text: of

text: convergence

text: of

text: power

text: series

text: in

text: any

text: higher

text: dimension.

text: Indeed,

text: two

text: of

text: the

text: simplest

text: topologically

text: B

text: U

text: trivial

text: bounded

text: domains

text: N

text: and

text: N

text: are

text: not

text: biholomorphically

text: equivalent.

text: Cartan’s

text: method-

text: ology

text: towards

text: establishing

text: these

text: inequivalences,

text: is

text: an

text: excellent

text: example

text: of

text: how

text: the

text: local

text: power

text: series

text: representation

text: of

text: holomorphic

text: functions

text: suﬃces

text: to

text: provide

text: an

text: elementary

text: and

text: (yet!)

text: neat

text: proof

text: of,

text: the

text: just

text: alluded

text: to

text: failure,

text: of

text: the

text: Riemann

text: mapping

text: theorem

text: in

text: every

text: dimension

text: N

text: We

text: mention

text: in

text: passing,

text: as

text: a

text: matter

text: of

text: (a

text: non-trivial!)

text: fact

text: that

text: any

text: pair

text: of

text: such

text: domains

text: (of

text: convergence

text: of

text: some

text: power

text: series)

text: will

text: generically

text: fail

text: to

text: be

text: biholomorphically

text: equivalent.

text: Let

text: us

text: get

text: back

text: from

text: the

text: detour

text: about

text: gaining

text: an

text: understanding

text: of

text: the

text: topology

text: of

text: domains

text: of

text: convergence,

text: to

text: our

text: question:

text: what

text: are

text: the

text: characterizing

text: features

text: of

text: the

text: logarithmic

text: images

text: of

text: domains

text: of

text: convergence

text: of

text: power

text: series?

text: A

text: little

text: contemplation

text: will

text: help

text: convince

text: that

text: what

text: is

text: R

text: required

text: here

text: is,

text: to

text: determine

text: a

text: condition

text: to

text: be

text: imposed

text: on

text: the

text: given

text: convex

text: domain

text: G

text: N

text: which

text: ensures

text: the

text: completeness

text: of

text: λ

text: without

text: involving

text: its

text: pull

text: back

text: into

text: the

text: absolute

text: space.

text: An

text: answer

text: to

text: this

text: requirement

text: here

text: would

text: be

text: that

text: the

text: characteristic/recession

text: cone

text: of

text: G

text: contain

text: −R

text: the

text: standard

text: backward

text: cone

text: )N.

text: This

text: means

text: that

text: the

text: domain

text: G

text: is

text: unbounded

text: in

text: N-many

text: −R

text: independent

text: directions;

text: to

text: be

text: precise,

text: in

text: every

text: direction

text: of

text: )N.

text: While

text: what

text: constitutes

text: a

text: satisfactory

text: answer

text: to

text: our

text: question

text: at

text: hand

text: is

text: subjective,

text: saying

text: that

text: the

text: characteristic

text: cone

text: −R

text: contain

text: )N

text: cannot

text: be

text: considered

text: satisfactory;

text: for,

text: one

text: is

text: often

text: ‘given’

text: a

text: domain

text: by

text: its

text: deﬁning

text: function

text: and

text: it

text: remains

text: therefore

text: to

text: ﬁgure

text: out

text: ways

text: to

text: ﬁnd

text: out

text: the

text: characteristic

text: cone

text: from

text: the

text: deﬁning

text: function.

text: Instead

text: of

text: taking

text: up

text: this

text: task

text: in

text: all

text: its

text: generality,

text: we

text: shall

text: directly

text: R

text: lay

text: down

text: the

text: condition

text: in

text: our

text: setting.

text: Namely,

text: given

text: a

text: convex

text: domain

text: G

text: N

text: with

text: deﬁning

text: function

text: ψ,

text: write

text: down

text: a

text: condition

text: to

text: decide

text: if

text: the

text: characteristic

text: cone

text: of

text: G

text: s

text: ψ(s)

text: −R

text: contains

text: )N.

text: As

text: this

text: is

text: easily

text: seen

text: to

text: be

text: equivalent

text: to

text: requiring

text: that

text: the

text: gradient

text: vector

text: of

text: ψ

text: at

text: all

text: boundary

text: points

text: of

text: G

text: i.e.,

text: the

text: outer

text: normal

text: vector

text: ﬁeld

text: along

text: ∂G

text: points

text: into

text: the

text: R

text: standard

text: cone

text: N,

text: the

text: sought-after

text: condition

text: on

text: ψ

text: is

text: that

text: it

text: satisfy

text: ψ

text: √1

text: √1

text: ψ

text: N

text: which

text: can

text: be

text: as:

text: ∂ψ/∂z

text: ∂ψ/∂z

text: N

text: ∂ψ/∂z

text: ∂ψ/∂z

text: N

text: N

text: R

text: where

text: all

text: derivatives

text: are

text: to

text: be

text: evaluated

text: at

text: points

text: s

text: in

text: N

text: where

text: ψ(s)

text: This

text: is

text: the

text: ana-

text: lytic

text: condition

text: for

text: a

text: convex

text: function

text: ψ(s)

text: to

text: satisfy,

text: for

text: the

text: convex

text: domain

text: G

text: s

text: ψ(s)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##