text: 2 a classical laser ﬁeld with a frequency ω and coupling where L strength Ω. The classical laser ﬁeld and the quantized cavity mode are equally detuned from the corresponding − − − | | | | | | transition frequencies by ∆ = (E0 E2)/¯h ωL. The a(t) = 1 g 2/( Ω 2 + g 2) f(t) , V-type level conﬁguration cannot be considered as an b(t) = −Ω g∗/(|Ω|2 + |g|2) f(t) , ideal memory because, contrary to the Λ-type system, a − ∗ c(t) = ig /ν ei∆/2t sin ν t , (3) qubit is here encoded in a superposition of two excited (cid:112) states, and thus, the time of storage of quantum infor- mation is limited by the spontan|e(cid:105)ous em| i(cid:105)ssion. Sponta- | | | | neous emission rates from levels 0 and 1 we denote by with ν = (∆/2)2 + Ω 2 + g 2 and γ and γ , respectively. However, new technology shows 0 1 that it is possible to build devices where the coherent − coupling strength is much greater than the spontaneous f(t) = 1 + ei∆/2t(i∆/(2ν) sin ν t cos ν t) . (4) transition rate [64–66]. In [64] the coupling strength g be- tween quantum dot and the microcavity mode is 80 times greater than spontaneous emission rate γ for the same From (3) one can see that the population of the state | (cid:105) | (cid:105) transition. Therefore, the V-type level structure can be 10 will be fully transferred to the state 01 if three | | | | considered as a short-term memory useful in quantum conditions will be satisﬁed, i.e., g = Ω , sin ν t = 0 and information processing. Such large values of g compared f(t) = 2. Taking into account two ﬁrst conditions we can to γ are possible for microcavities since g is inversely express the third one in the form proportional to the square of the cavity mode volume. Unfortunately, the cavity decay rate κ increases with de- − creasing the cavity length, and therefore, κ > g in [64]. cos (∆/2 t ) cos ν t = 1 . (5) π π Nevertheless, it is reasonable to assume that the ﬁnesse of microcavities will be improved in the future and the coherent coupling will dominate all dissipative rates. Equation (5) leads to a discrete set of detunings [40] (cid:32) (cid:33) The Hamiltonian that describes the interaction of the atom (or the quantum dot) with the cavity ﬁeld mode is given by 2 ∆ ζ2 | | = 2 , (6) 2 g 2ζ + 1 − H = ∆σ + (Ωσ + gaσ + h.c.) 22 02 12 − † − − iκa a iγ0σ00 iγ1σ11 , (1) where ζ = k/θ, θ = 1, 3, 5, . . . is a natural odd number and k = 0, 1, 2, 3, . . . is a non-negative integer. So, the | (cid:105) | (cid:105) | (cid:105)(cid:104) | population is fully transferred from the state 10 to 01 where σ = i j denote the atomic ﬂip operators and ij if and only if the value of detuning satisﬁes condition (6) a denotes the annihilation operator of the cavity mode. and the operation time t is given by π III. NONEXISTENCE OF PERFECT MAPPING | | t = 2kπ/ ∆ = (θ + k)π/ν . (7) PULSE FOR V SYSTEMS π Let us consider the case of the unitary evolution of the − If we assume that Ω = g exp(iΦ) then it is seen system, i.e., we assume that κ, γ and γ are equal to 0 1 from (2) and (3) that such a perfect π pulse is given zero. The quantum information encoded in the atomic | (cid:105) | (cid:105) by state α 00 + β 10 has to be mapped perfectly and | (cid:105) | (cid:105) quickly onto the cavity ﬁeld state: α 00 + β 01 . Here, we have denoted a state of the system consisting of the | (cid:105) ≡ − | (cid:105) | (cid:105) | (cid:105) U 10 e iHtπ 10 = eiΦ 01 . (8) atomic state j and the cavity ﬁeld with n photons by π | (cid:105) jn . So, we have to perform quantum operation deﬁned | (cid:105) → | (cid:105) | (cid:105) → | (cid:105) by 10 01 and 00 00 . Is it possible to achieve The state-mapping operation requires also that the this task using the evolution governed by the Hamilto- | (cid:105) population of the state 00 has to remain unchanged. nian (1)? Let us investigate this problem. The evolution | (cid:105) | (cid:105) In the case of three-level Λ systems the state 00 experi- of the state 10 is given by ences no dynamics. Therefore, in such systems the per- | (cid:105) → | (cid:105) fect state-mapping operation (also deﬁned by 10 01 − | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) → | (cid:105) e iHt 10 = a(t) 10 + b(t) 01 + c(t) 21 , (2) and 00 00 ) can be easily achieved. The situation, however, is considerably more complicated for three-level V systems. In V systems the time evolution is given by ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: a

text: classical

text: laser

text: ﬁeld

text: with

text: a

text: frequency

text: ω

text: and

text: coupling

text: where

text: L

text: strength

text: Ω.

text: The

text: classical

text: laser

text: ﬁeld

text: and

text: the

text: quantized

text: cavity

text: mode

text: are

text: equally

text: detuned

text: from

text: the

text: corresponding

label: equation

text: transition

text: frequencies

text: by

text: ωL.

text: The

text: a(t)

text: g

text: Ω

text: g

text: f(t)

text: V-type

text: level

text: conﬁguration

text: cannot

text: be

text: considered

text: as

text: an

text: b(t)

text: −Ω

text: f(t)

text: ideal

text: memory

text: because,

text: contrary

text: to

text: the

text: Λ-type

text: system,

text: a

text: c(t)

text: ig

text: /ν

text: sin

text: ν

text: t

text: qubit

text: is

text: here

text: encoded

text: in

text: a

text: superposition

text: of

text: two

text: excited

text: states,

text: and

text: thus,

text: the

text: time

text: of

text: storage

text: of

text: quantum

text: infor-

text: mation

text: is

text: limited

text: by

text: the

text: em|

text: Sponta-

text: neous

text: emission

text: rates

text: from

text: levels

text: and

text: we

text: denote

text: by

text: with

text: ν

text: Ω

text: g

text: and

text: γ

text: and

text: γ

text: respectively.

text: However,

text: new

text: technology

text: shows

text: that

text: it

text: is

text: possible

text: to

text: build

text: devices

text: where

text: the

text: coherent

text: coupling

text: strength

text: is

text: much

text: greater

text: than

text: the

text: spontaneous

text: f(t)

text: sin

text: ν

text: t

text: cos

text: ν

text: t)

text: transition

text: rate

text: In

text: the

text: coupling

text: strength

text: g

text: be-

text: tween

text: quantum

text: dot

text: and

text: the

text: microcavity

text: mode

text: is

text: times

text: greater

text: than

text: spontaneous

text: emission

text: rate

text: γ

text: for

text: the

text: same

text: From

text: one

text: can

text: see

text: that

text: the

text: population

text: of

text: the

text: state

text: transition.

text: Therefore,

text: the

text: V-type

text: level

text: structure

text: can

text: be

text: will

text: be

text: fully

text: transferred

text: to

text: the

text: state

text: if

text: three

text: considered

text: as

text: a

text: short-term

text: memory

text: useful

text: in

text: quantum

text: conditions

text: will

text: be

text: satisﬁed,

text: i.e.,

text: g

text: Ω

text: sin

text: ν

text: t

text: and

text: information

text: processing.

text: Such

text: large

text: values

text: of

text: g

text: compared

text: f(t)

text: Taking

text: into

text: account

text: two

text: ﬁrst

text: conditions

text: we

text: can

text: to

text: γ

text: are

text: possible

text: for

text: microcavities

text: since

text: g

text: is

text: inversely

text: express

text: the

text: third

text: one

text: in

text: the

text: form

text: proportional

text: to

text: the

text: square

text: of

text: the

text: cavity

text: mode

text: volume.

text: Unfortunately,

text: the

text: cavity

text: decay

text: rate

text: κ

text: increases

text: with

text: de-

text: creasing

text: the

text: cavity

text: length,

text: and

text: therefore,

text: κ

text: g

text: in

text: cos

text: t

text: cos

text: ν

text: t

text: π

text: π

text: Nevertheless,

text: it

text: is

text: reasonable

text: to

text: assume

text: that

text: the

text: ﬁnesse

text: of

text: microcavities

text: will

text: be

text: improved

text: in

text: the

text: future

text: and

text: the

text: coherent

text: coupling

text: will

text: dominate

text: all

text: dissipative

text: rates.

text: Equation

text: leads

text: to

text: a

text: discrete

text: set

text: of

text: detunings

text: The

text: Hamiltonian

text: that

text: describes

text: the

text: interaction

text: of

text: the

text: atom

text: (or

text: the

text: quantum

text: dot)

text: with

text: the

text: cavity

text: ﬁeld

text: mode

text: is

text: given

text: by

text: ζ2

text: g

text: 2ζ

text: H

text: ∆σ

text: (Ωσ

text: gaσ

text: h.c.)

text: iκa

text: a

text: where

text: ζ

text: k/θ,

text: θ

text: is

text: a

text: natural

text: odd

text: number

text: and

text: k

text: is

text: a

text: non-negative

text: integer.

text: So,

text: the

text: population

text: is

text: fully

text: transferred

text: from

text: the

text: state

text: to

text: where

text: σ

text: i

text: j

text: denote

text: the

text: atomic

text: ﬂip

text: operators

text: and

text: ij

text: if

text: and

text: only

text: if

text: the

text: value

text: of

text: detuning

text: satisﬁes

text: condition

text: a

text: denotes

text: the

text: annihilation

text: operator

text: of

text: the

text: cavity

text: mode.

text: and

text: the

text: operation

text: time

text: t

text: is

text: given

text: by

text: π

text: III.

text: NONEXISTENCE

text: OF

text: PERFECT

text: MAPPING

text: t

text: (θ

text: k)π/ν

text: PULSE

text: FOR

text: V

text: SYSTEMS

text: π

text: Let

text: us

text: consider

text: the

text: case

text: of

text: the

text: unitary

text: evolution

text: of

text: the

text: If

text: we

text: assume

text: that

text: Ω

text: g

text: exp(iΦ)

text: then

text: it

text: is

text: seen

text: system,

text: i.e.,

text: we

text: assume

text: that

text: κ,

text: γ

text: and

text: γ

text: are

text: equal

text: to

text: from

text: and

text: that

text: such

text: a

text: perfect

text: π

text: pulse

text: is

text: given

text: zero.

text: The

text: quantum

text: information

text: encoded

text: in

text: the

text: atomic

text: by

text: state

text: α

text: β

text: has

text: to

text: be

text: mapped

text: perfectly

text: and

text: quickly

text: onto

text: the

text: cavity

text: ﬁeld

text: state:

text: α

text: β

text: Here,

text: we

text: have

text: denoted

text: a

text: state

text: of

text: the

text: system

text: consisting

text: of

text: the

text: U

text: e

text: iHtπ

text: eiΦ

text: atomic

text: state

text: j

text: and

text: the

text: cavity

text: ﬁeld

text: with

text: n

text: photons

text: by

text: π

text: jn

text: So,

text: we

text: have

text: to

text: perform

text: quantum

text: operation

text: deﬁned

text: by

text: and

text: Is

text: it

text: possible

text: to

text: achieve

text: The

text: state-mapping

text: operation

text: requires

text: also

text: that

text: the

text: this

text: task

text: using

text: the

text: evolution

text: governed

text: by

text: the

text: Hamilto-

text: population

text: of

text: the

text: state

text: has

text: to

text: remain

text: unchanged.

text: nian

text: Let

text: us

text: investigate

text: this

text: problem.

text: The

text: evolution

text: In

text: the

text: case

text: of

text: three-level

text: Λ

text: systems

text: the

text: state

text: experi-

text: of

text: the

text: state

text: is

text: given

text: by

text: ences

text: no

text: dynamics.

text: Therefore,

text: in

text: such

text: systems

text: the

text: per-

text: fect

text: state-mapping

text: operation

text: (also

text: deﬁned

text: by

text: e

text: iHt

text: a(t)

text: b(t)

text: c(t)

text: and

text: can

text: be

text: easily

text: achieved.

text: The

text: situation,

text: however,

text: is

text: considerably

text: more

text: complicated

text: for

text: three-level

text: V

text: systems.

text: In

text: V

text: systems

text: the

text: time

text: evolution

text: is

text: given

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 (cid:112) (cid:112){| (cid:105) | (cid:105) | (cid:105)} {| (cid:105) | (cid:105)} ≤ spaces 10 , 01 , 21 and 00 , 20 ) evolve with where Re(f(t)) 2, and thus the probability that the | | | (cid:105) noncommensurate frequencies ν = (∆/2)2 + 2 g 2 and system will be found in other state than 10 is limited (cid:48) | | | | | | ν = (∆/2)2 + g 2 (for Ω = g ). The operations by (cid:16) (cid:17) | (cid:105) → | (cid:105) | (cid:105) → | (cid:105) 10 01 and 00 00 require the π pulse and the (cid:48) | | | | 2π pulse, respectively. Since ν and ν are noncommen- 4 g 2 2 g 2 surate, the duration times of these pulses are always dif- (cid:15) < | | | | < | | . (15) Ω 2 + g 2 Ω ferent, and therefore, there is no rectangular pulse which | (cid:105) → | (cid:105) | (cid:105) → | (cid:105) can perform operations 10 01 and 00 00 si- | | multaneously. Therefore, Ω/g > 633 i−s necessary to get the error We can om{it| th(cid:105)is| pr(cid:105)o}blem shifting the evolution in probability smaller than 10 5. | (cid:105) the subspace 0{0| , (cid:105)20| (cid:105)wi|th (cid:105)r}espect to the evolution | T(cid:105)he eﬀect of this operation is given by U1(t1) 10 = in the subspace 10 , 01 , 21 before performing the 10 and (cid:2) perfect π pulse opera|tio(cid:105)n. By ’shifting the evolution’ we (cid:3) mean that the state 10 experiences no dynamics while | (cid:105) (cid:48) (cid:48) − (cid:48) | (cid:105) | (cid:105) U (t ) 00 = ei∆/2t1 (2ν cos ν t i∆ sin ν t ) 00 the state 00 is transformed into such a special state 1 1 1 1 1 1 1 | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) − ∗ (cid:48) | (cid:105) (cid:48) Φ = η 00 + η 20 that U Φ = 00 . The main (cid:112) i2Ω sin ν t 20 /(2ν ) , (16) η 0 1 π η 1 1 1 (cid:48) | | where ν = (∆/2)2 + Ω 2. From (16) it is seen that 1 | (cid:105) we can set arbitrary populations of the states 00 and | (cid:105) | | (cid:29) | | 20 for Ω ∆ . It is also seen that we can easily give | (cid:105) an arbitrary phase shift to the state 20 with respect to | (cid:105) the state 00 just by setting proper argument of Ω = | | Ω exp(iφ ), where Ω = Ω . So we can always produce 1 |Ω (cid:105) 1 the state Φ for large enough Ω . η 1 We can calculate amplitudes η and η by premulti- | (cid:105) | (cid:105) − 0 1 plying Uπ Φη = 00 by Uπ 1 = eiHtπ. We also assume that in the second stage of the protocol, i.e., during the perfect π pulse we change the intensity and the phase of − the laser ﬁeld that Ω(cid:2)= g exp(iΦ). In this way we get (cid:3) | (cid:105) − (cid:48) (cid:48) (cid:48) | (cid:105) | (cid:105) | (cid:105) Φη = e i∆/2tπ (2νπ cos νπ tπ + i∆ sin νπ tπ) 00 FIG. 3: Populations of the states 20 (solid line), 21 (dot- − − ∗ (cid:48) | (cid:105) (cid:48) | (cid:105) ie (cid:112)iΦ2g sin ν t 20 /(2ν ) , (17) ted line) and 10 (dashed line) during the two-stage state- π π π mapping protocol. At the end these three populations are close to zero, and thus, the ﬁdelity can be close to one. Inset: (cid:48) | | where ν = (∆/2)2 + g 2. A comparison of the the pulses sequence. π moduli of the amp(cid:16)litudes (16) and (17) lea(cid:17)ds to idea of such a two-stage state-mapping protocol is illus- trated in ﬁgure 3. Thus, in the ﬁrst stage of this protocol − |g|2ν(cid:48)2 (cid:48) (cid:48) we need an operation, wh{i|ch(cid:105)ch|an(cid:105)g}es only the state that t1 = arccos 1 2 Ω2ν(cid:48)12 (sin νπ tπ)2 /(2ν1) . (18) belongs to the subspace 00 , 20 and leaves the state 1 π belonging to the other subspace unchanged. In order to ﬁnd the proper argument of Ω we write (16) This needed operation is just the intense laser pulse op- and (17) in terms of the moduli and the arguments of η eration. It is clearly seen from equations (2), (3) and (9) 0 | | (cid:29) | | that when the laser pulse is very intensive Ω g then and η1 (cid:0) (cid:1) ≈ | (cid:105) a(t) 1, and therefore, the state 10 experiences almost | (cid:105) (cid:0) (cid:1) no dynamics while the population of the state 00 oscil- | (cid:105) | | | (cid:105) | | | (cid:105) U 00 = η eiθ0 00 + η eiθ1 20 ei∆/2t1 , (19) lates with a very high frequency. In the following, we will 1 0 1 | | (cid:29) | | | (cid:105) | | | (cid:105) | | | (cid:105) − assume that a(t) = 1 for Ω g . Since we need the Φ = η eiφ0 00 + η eiφ1 20 e i∆/2tπ ,(20) − − η 0 1 ﬁdelity greater than 1 10 5, we have to estimate the − | | error (cid:15) = 1 a(t) 2 introduced by this approximation. From (3) we see that where | | | | | | − g 2 | | (cid:0)g | (cid:48) | (cid:1) a(t) 2 > 1 (|Ω|2 + |g|2) 2 Re(f(t1)) , (14) η1 = νπ(cid:48) sin νπ t2π , | | | | (cid:48) (cid:48) η = g 2 cos2 ν t + ∆2/4 1/2/ν . (21) 0 π π π ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: spaces

text: and

text: evolve

text: with

text: where

text: Re(f(t))

text: and

text: thus

text: the

text: probability

text: that

text: the

text: noncommensurate

text: frequencies

text: ν

text: g

text: and

text: system

text: will

text: be

text: found

text: in

text: other

text: state

text: than

text: is

text: limited

text: ν

text: g

text: (for

text: Ω

text: g

text: The

text: operations

text: by

text: and

text: require

text: the

text: π

text: pulse

text: and

text: the

text: 2π

text: pulse,

text: respectively.

text: Since

text: ν

text: and

text: ν

text: are

text: noncommen-

text: g

text: g

text: surate,

text: the

text: duration

text: times

text: of

text: these

text: pulses

text: are

text: always

text: dif-

text: Ω

text: g

text: Ω

text: ferent,

text: and

text: therefore,

text: there

text: is

text: no

text: rectangular

text: pulse

text: which

text: can

text: perform

text: operations

text: and

text: si-

text: multaneously.

text: Therefore,

text: Ω/g

text: i−s

text: necessary

text: to

text: get

text: the

text: error

text: We

text: can

text: om{it|

text: shifting

text: the

text: evolution

text: in

text: probability

text: smaller

text: than

text: the

text: subspace

text: to

text: the

text: evolution

text: eﬀect

text: of

text: this

text: operation

text: is

text: given

text: by

text: in

text: the

text: subspace

text: before

text: performing

text: the

text: and

text: perfect

text: π

text: pulse

text: By

text: ’shifting

text: the

text: evolution’

text: we

text: mean

text: that

text: the

text: state

text: experiences

text: no

text: dynamics

text: while

text: U

text: (t

text: cos

text: ν

text: t

text: i∆

text: sin

text: ν

text: t

text: the

text: state

text: is

text: transformed

text: into

text: such

text: a

text: special

text: state

text: Φ

text: η

text: η

text: that

text: U

text: Φ

text: The

text: main

text: sin

text: ν

text: t

text: η

text: π

text: η

text: where

text: ν

text: Ω

text: From

text: it

text: is

text: seen

text: that

text: we

text: can

text: set

text: arbitrary

text: populations

text: of

text: the

text: states

text: and

text: for

text: Ω

text: It

text: is

text: also

text: seen

text: that

text: we

text: can

text: easily

text: give

text: an

text: arbitrary

text: phase

text: shift

text: to

text: the

text: state

text: with

text: respect

text: to

text: the

text: state

text: just

text: by

text: setting

text: proper

text: argument

text: of

text: Ω

text: Ω

text: exp(iφ

text: where

text: Ω

text: Ω

text: So

text: we

text: can

text: always

text: produce

text: |Ω

text: the

text: state

text: Φ

text: for

text: large

text: enough

text: Ω

text: η

text: We

text: can

text: calculate

text: amplitudes

text: η

text: and

text: η

text: by

text: premulti-

text: plying

text: Uπ

text: Φη

text: by

text: Uπ

text: eiHtπ.

text: We

text: also

text: assume

text: that

text: in

text: the

text: second

text: stage

text: of

text: the

text: protocol,

text: i.e.,

text: during

text: the

text: perfect

text: π

text: pulse

text: we

text: change

text: the

text: intensity

text: and

text: the

text: phase

text: of

text: the

text: laser

text: ﬁeld

text: that

text: g

text: exp(iΦ).

text: In

text: this

text: way

text: we

text: get

text: Φη

text: e

text: (2νπ

text: cos

text: νπ

text: tπ

text: i∆

text: sin

text: νπ

text: tπ)

text: FIG.

text: Populations

text: of

text: the

text: states

text: (solid

text: line),

text: (dot-

text: ie

text: sin

text: ν

text: t

text: ted

text: line)

text: and

text: (dashed

text: line)

text: during

text: the

text: two-stage

text: state-

text: π

text: π

text: π

text: mapping

text: protocol.

text: At

text: the

text: end

text: these

text: three

text: populations

text: are

text: close

text: to

text: zero,

text: and

text: thus,

text: the

text: ﬁdelity

text: can

text: be

text: close

text: to

text: one.

text: Inset:

text: where

text: ν

text: g

text: A

text: comparison

text: of

text: the

text: the

text: pulses

text: sequence.

text: π

text: moduli

text: of

text: the

text: and

text: to

text: idea

text: of

text: such

text: a

text: two-stage

text: state-mapping

text: protocol

text: is

text: illus-

text: trated

text: in

text: ﬁgure

text: Thus,

text: in

text: the

text: ﬁrst

text: stage

text: of

text: this

text: protocol

text: we

text: need

text: an

text: operation,

text: only

text: the

text: state

text: that

text: arccos

text: (sin

text: νπ

text: belongs

text: to

text: the

text: subspace

text: and

text: leaves

text: the

text: state

text: π

text: belonging

text: to

text: the

text: other

text: subspace

text: unchanged.

text: In

text: order

text: to

text: ﬁnd

text: the

text: proper

text: argument

text: of

text: Ω

text: we

text: write

text: This

text: needed

text: operation

text: is

text: just

text: the

text: intense

text: laser

text: pulse

text: op-

text: and

text: in

text: terms

text: of

text: the

text: moduli

text: and

text: the

text: arguments

text: of

text: η

text: eration.

text: It

text: is

text: clearly

text: seen

text: from

text: equations

text: and

text: that

text: when

text: the

text: laser

text: pulse

text: is

text: very

text: intensive

text: Ω

text: g

text: then

text: and

text: η1

text: a(t)

text: and

text: therefore,

text: the

text: state

text: experiences

text: almost

text: no

text: dynamics

text: while

text: the

text: population

text: of

text: the

text: state

text: oscil-

text: U

text: η

text: η

text: lates

text: with

text: a

text: very

text: high

text: frequency.

text: In

text: the

text: following,

text: we

text: will

text: assume

text: that

text: a(t)

text: for

text: Ω

text: g

text: Since

text: we

text: need

text: the

text: Φ

text: η

text: η

text: e

text: η

text: ﬁdelity

text: greater

text: than

text: we

text: have

text: to

text: estimate

text: the

text: error

text: a(t)

text: introduced

text: by

text: this

text: approximation.

text: From

text: we

text: see

text: that

text: where

text: g

text: a(t)

text: η1

text: sin

text: νπ

text: η

text: g

text: ν

text: t

text: π

text: π

text: π

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 YEVGENY LIOKUMOVICH On the other hand, consider the 3-legged starﬁsh example on Figure 1. As it has been observed in [11], for any r > 1, if the tentacles are suﬃciently 3 thin and long, the length of the shortest simple closed curve subdividing M into ≥ | | two regions of area r M can be arbitrarily large. A. Nabutovsky asked ([8]) the ∈ following question: what is the maximal value of r [1, 1] such that for some c(r) 4 3 ≥ each Riemannian 2-sphere of area 1 can be subdivided into two discs of area r by ≤ (cid:112) a simple closed curve of length c(r)? Our ﬁrst result provides an answer for this question. ≤ | | Theorem 1. There exists a simple closed curve γ of length 26 M subdividing ≥ | | M into two subdiscs of area 1 M . 3 (cid:112) To prove Theorem 1 we obtain the following result of independent interest. Theorem 2. There exists a map f from M into a trivalent tree T , such that ﬁbers ≤ | | of f have length 26 M and controlled topology: preimage of every interior point is a simple closed curve, preimage of every terminal vertex is a point and preimage of every vertex of degree 3 is homeomorphic to the greek letter θ. Theorem 2 follows from a more general Theorem 10 in Section 3 for spheres with ≥ k 0 holes. Using diﬀerent methods Guth [6] proved existence of a map from M (cid:112) into a trivalent tree with lengths of ﬁbers bounded in terms of hypersphericity of M. It follows from Theorem 0.3 in [6] that there exists such a map with ﬁbers of length ≤ | | 34 M . − The second main result of this paper is about slicing M by 1 cycles. If instead (cid:112) of simple closed curves we allow subdividison by 1-cycles, we show that M can be subdivided into two regions with arbitrary prescribed ratio of areas by a 1-cycle of ≤ | | (cid:112) length 52 M . In fact, we prove the following → R ≤ Theorem 3. There exists a Morse function f : M with ﬁbers of length | | 52 M . (cid:112) This improves the result of Balacheﬀ and Sabourau [3] that there exists a sweep-out − ≤ | | of M by 1 cycles of length 108 M . Theorem 3 follows from a more general result in Section 4, where we prove existence of a Morse function on a sphere with k holes, such that the function is constant on each connected component of the boundary and the length of ﬁbers are bounded in terms of area and boundary length. Alvarez Paiva, Balacheﬀ and Tzanev [1] show that existence of a Morse function on a Riemannian 2-sphere with bounded ﬁbers yields a length-area inequality for the shortest periodic geodesic on a Finsler 2-sphere (for both reversible and non- reversible metrics). Note that arguments used by Croke [4] to prove the length-area Date: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YEVGENY

text: LIOKUMOVICH

text: On

text: the

text: other

text: hand,

text: consider

text: the

text: starﬁsh

text: example

text: on

text: Figure

text: As

text: it

text: has

text: been

text: observed

text: in

text: for

text: any

text: r

text: if

text: the

text: tentacles

text: are

text: suﬃciently

text: thin

text: and

text: long,

text: the

text: length

text: of

text: the

text: shortest

text: simple

text: closed

text: curve

text: subdividing

text: M

text: into

text: two

text: regions

text: of

text: area

text: r

text: M

text: can

text: be

text: arbitrarily

text: large.

text: A.

text: Nabutovsky

text: asked

text: the

text: following

text: question:

text: what

text: is

text: the

text: maximal

text: value

text: of

text: r

text: such

text: that

text: for

text: some

text: c(r)

text: each

text: Riemannian

text: of

text: area

text: can

text: be

text: subdivided

text: into

text: two

text: discs

text: of

text: area

text: r

text: by

text: a

text: simple

text: closed

text: curve

text: of

text: length

text: c(r)?

text: Our

text: ﬁrst

text: result

text: provides

text: an

text: answer

text: for

text: this

text: question.

text: Theorem

text: There

text: exists

text: a

text: simple

text: closed

text: curve

text: γ

text: of

text: length

text: M

text: subdividing

text: M

text: into

text: two

text: subdiscs

text: of

text: area

text: M

text: To

text: prove

text: Theorem

text: we

text: obtain

text: the

text: following

text: result

text: of

text: independent

text: interest.

text: Theorem

text: There

text: exists

text: a

text: map

text: f

text: from

text: M

text: into

text: a

text: trivalent

text: tree

text: T

text: such

text: that

text: ﬁbers

text: of

text: f

text: have

text: length

text: M

text: and

text: controlled

text: topology:

text: preimage

text: of

text: every

text: interior

text: point

text: is

text: a

text: simple

text: closed

text: curve,

text: preimage

text: of

text: every

text: terminal

text: vertex

text: is

text: a

text: point

text: and

text: preimage

text: of

text: every

text: vertex

text: of

text: degree

text: is

text: homeomorphic

text: to

text: the

text: greek

text: letter

text: θ.

text: Theorem

text: follows

text: from

text: a

text: more

text: general

text: Theorem

text: in

text: Section

text: for

text: spheres

text: with

text: k

text: holes.

text: Using

text: diﬀerent

text: methods

text: Guth

text: proved

text: existence

text: of

text: a

text: map

text: from

text: M

text: into

text: a

text: trivalent

text: tree

text: with

text: lengths

text: of

text: ﬁbers

text: bounded

text: in

text: terms

text: of

text: hypersphericity

text: of

text: M.

text: It

text: follows

text: from

text: Theorem

text: in

text: that

text: there

text: exists

text: such

text: a

text: map

text: with

text: ﬁbers

text: of

text: length

text: M

text: The

text: second

text: main

text: result

text: of

text: this

text: paper

text: is

text: about

text: slicing

text: M

text: by

text: cycles.

text: If

text: instead

text: of

text: simple

text: closed

text: curves

text: we

text: allow

text: subdividison

text: by

text: we

text: show

text: that

text: M

text: can

text: be

text: subdivided

text: into

text: two

text: regions

text: with

text: arbitrary

text: prescribed

text: ratio

text: of

text: areas

text: by

text: a

text: of

text: length

text: M

text: In

text: fact,

text: we

text: prove

text: the

text: following

text: R

text: Theorem

text: There

text: exists

text: a

text: Morse

text: function

text: f

text: M

text: with

text: ﬁbers

text: of

text: length

text: M

text: This

text: improves

text: the

text: result

text: of

text: Balacheﬀ

text: and

text: Sabourau

text: that

text: there

text: exists

text: a

text: sweep-out

text: of

text: M

text: by

text: cycles

text: of

text: length

text: M

text: Theorem

text: follows

text: from

text: a

text: more

text: general

text: result

text: in

text: Section

text: where

text: we

text: prove

text: existence

text: of

text: a

text: Morse

text: function

text: on

text: a

text: sphere

text: with

text: k

text: holes,

text: such

text: that

text: the

text: function

text: is

text: constant

text: on

text: each

text: connected

text: component

text: of

text: the

text: boundary

text: and

text: the

text: length

text: of

text: ﬁbers

text: are

text: bounded

text: in

text: terms

text: of

text: area

text: and

text: boundary

text: length.

text: Alvarez

text: Paiva,

text: Balacheﬀ

text: and

text: Tzanev

text: show

text: that

text: existence

text: of

text: a

text: Morse

text: function

text: on

text: a

text: Riemannian

text: with

text: bounded

text: ﬁbers

text: yields

text: a

text: length-area

text: inequality

text: for

text: the

text: shortest

text: periodic

text: geodesic

text: on

text: a

text: Finsler

text: (for

text: both

text: reversible

text: and

text: non-

text: reversible

text: metrics).

text: Note

text: that

text: arguments

text: used

text: by

text: Croke

text: to

text: prove

text: the

text: length-area

text: Date:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 YEVGENY LIOKUMOVICH Figure 6. Inductive step of the proof of Lemma 14 − T maps on these smaller regions to obtain one map from M with the desired bound on lengths of ﬁbers. (cid:100) | | (cid:101) Fix (cid:15) > 0 and let A be as in Lemma 14. Let N = log ( M ) . We claim that ⊂ | | | | ≤ | | 4/3 A − for every M M with (3)n+1 M < M (3)n M there exists a T map with k p 4 p k 4 p ﬁbers of length (cid:88) (cid:112) √ − ≤ | | − N 1 3 | | (1) ∂M + 4(N n + k) A + c(r)( )i/2 M + (cid:15) k k 4 i=n When n = N the inequality (1) is true by Lemma 14. We assume it to be true for ≤ n + 1 N and prove it for n. ∈ ≤ (cid:48) By Proposition 7 there exists γ S(M , r) of length c(r) + (cid:15) . k We have two possibilities. Case 1. γ is a simple closed curve. | | ≤ | | In this case γ separates M into two regions N and N of area N (3)n+1 M . k 1 2 i 4 The number of connected components of ∂N is at most k + 1. i − By inductive assumption N and N admit T maps into trees T and T respec- 1 2 1 2 tively. We construct T by identifying the terminal vertex f (γ) of T with the terminal − 1 1 vertex f (γ) of T . T map f is deﬁned by setting it equal to f when restricted to 2 2 i N . A simple calculation shows that lengths of ﬁbers of f satisfy the desired bound. i Case 2. γ is a collection of arcs with endpoints on ∂M . { } k Let S , ..., S be connected components of ∂M that intersect arcs of γ. For a 1 j k suﬃciently small δ the normal neighbourhood N (S ) is foliated by closed curves of | | δ i length very close to S , which are transverse to the arcs of γ. In particular, each i ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YEVGENY

text: LIOKUMOVICH

text: Figure

text: Inductive

label: caption

text: step

text: of

text: the

text: proof

text: of

text: Lemma

text: T

text: maps

text: on

text: these

text: smaller

text: regions

text: to

text: obtain

text: one

text: map

text: from

text: M

text: with

text: the

text: desired

text: bound

text: on

text: lengths

text: of

text: ﬁbers.

text: Fix

text: and

text: let

text: A

text: be

text: as

text: in

text: Lemma

text: Let

text: N

text: log

text: M

text: We

text: claim

text: that

text: A

text: for

text: every

text: M

text: M

text: with

text: M

text: M

text: M

text: there

text: exists

text: a

text: T

text: map

text: with

text: k

text: p

text: p

text: k

text: p

text: ﬁbers

text: of

text: length

text: N

text: ∂M

text: n

text: k)

text: A

text: c(r)(

text: M

text: k

text: k

text: i=n

text: When

text: n

text: N

text: the

text: inequality

text: is

text: true

text: by

text: Lemma

text: We

text: assume

text: it

text: to

text: be

text: true

text: for

text: n

text: N

text: and

text: prove

text: it

text: for

text: n.

text: By

text: Proposition

text: there

text: exists

text: γ

text: S(M

text: r)

text: of

text: length

text: c(r)

text: k

text: We

text: have

text: two

text: possibilities.

text: Case

text: γ

text: is

text: a

text: simple

text: closed

text: curve.

text: In

text: this

text: case

text: γ

text: separates

text: M

text: into

text: two

text: regions

text: N

text: and

text: N

text: of

text: area

text: N

text: M

text: k

text: i

text: The

text: number

text: of

text: connected

text: components

text: of

text: ∂N

text: is

text: at

text: most

text: k

text: i

text: By

text: inductive

text: assumption

text: N

text: and

text: N

text: admit

text: T

text: maps

text: into

text: trees

text: T

text: and

text: T

text: respec-

text: tively.

text: We

text: construct

text: T

text: by

text: identifying

text: the

text: terminal

text: vertex

text: f

text: (γ)

text: of

text: T

text: with

text: the

text: terminal

text: vertex

text: f

text: (γ)

text: of

text: T

text: T

text: map

text: f

text: is

text: deﬁned

text: by

text: setting

text: it

text: equal

text: to

text: f

text: when

text: restricted

text: to

text: i

text: N

text: A

text: simple

text: calculation

text: shows

text: that

text: lengths

text: of

text: ﬁbers

text: of

text: f

text: satisfy

text: the

text: desired

text: bound.

text: i

text: Case

text: γ

text: is

text: a

text: collection

text: of

text: arcs

text: with

text: endpoints

text: on

text: ∂M

text: k

text: Let

text: S

text: S

text: be

text: connected

text: components

text: of

text: ∂M

text: that

text: intersect

text: arcs

text: of

text: γ.

text: For

text: a

text: j

text: k

text: suﬃciently

text: small

text: δ

text: the

text: normal

text: neighbourhood

text: N

text: (S

text: is

text: foliated

text: by

text: closed

text: curves

text: of

text: δ

text: i

text: length

text: very

text: close

text: to

text: S

text: which

text: are

text: transverse

text: to

text: the

text: arcs

text: of

text: γ.

text: In

text: particular,

text: each

text: i

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 (cid:90) (cid:90) (cid:88) Introducing x = v2 (k2 + k2), we can rewrite this as F x y ∞ (cid:113) − e2q π/d Πinter(q, iΩ) = dk dx Ωt (x, k ) 8π2 − z z z t π/d 0 × − − [1 n ( x + m2(k ) (cid:15) )], (15) F t z F where − m (k ) Ωt (x, k ) = t z , (16) z z 2[x + m2(k )]3/2 t z − is the z-component of the Berry curvature of the s = , t bands. The ﬁrst term in Eq. (15) is the contribution to − Πinter(q, iΩ) of the two s = bands, which are completely ﬁ(cid:90)lled when (cid:15) > 0. The second term is the contribution (cid:88) F of the incompletely ﬁlled s = + bands. Integrating over x, we obtain Πinte(cid:88)r(q(cid:90), iΩ) = e2q π/d d(cid:20)k sign[m (k(cid:21))] 8π2 − z t z t π/d | | − e2q π/d − m (k ) dk sign[m (k )] 1 t z 8π2 − z t z (cid:15) × −t | π/d | F Θ((cid:15) m (k ) ), (17) F t z − where Θ(x) is the Heaviside step function. The ﬁrst term in Eq. (17) is the contribution of the completely ﬁlled s = bands. The second term is the contribution of incompletely ﬁlled s = + bands and, as ﬁrst pointed out by Haldane,22 can be expressed in terms of an integral of the Berry connection ﬁeld over the Fermi surface. Indeed, components of the Berry connection for the s = + bands are given by − (cid:104) | | (cid:105) − v2 k At (k) = i v+t ∂ v+t = F y , x k kx k 2(cid:15) (k)[(cid:15) (k) + m (k )] t t t z − (cid:104) | | (cid:105) v2 k At (k) = i v+t ∂ v+t = F x . (18) y k ky k 2(cid:15) (k)[(cid:15) (k) + m (k )] t t t z Integrating the Berry connection over a 2D s(cid:73)ection of the Fermi surface, corresponding to a ﬁxed k , i.e. v2 (k2 +k2) = − z F x y (cid:15)2 m2(k ), we obtain F t z · − φ (k ) = dk A+t(k) = π [1 m (k )/(cid:15) ] t z t z F − − = π [sign[m (k )] m (k )/(cid:15) ] + π[1 sign[m (k )]], t z t z F t z (19) where φ (k ) is the total Berry phase, accumulated along(cid:90)the corresponding section of the Fermi surface. Using this, t z (cid:88) we can rewrite Eq. (17) in the following way Πinte(cid:88)r(q(cid:90), iΩ) = e2q π/d dk sign[m (k )] 8π2 − z t z t π/d − e2q (cid:88) (cid:90) π/d dk φt(kz)Θ((cid:15) − |m (k )|) 8π2 − z π F t z t π/d e2q π/d − − | | + dk [1 sign[m (k )]] Θ((cid:15) m (k ) ). 8π2 − z t z F t z t π/d (20) The interpretation of Eq. (20) is, unfortunately, not unambiguous due to the fact that the Berry phase φ (k ) is t z only deﬁned modulo 2π. Due to this ambiguity, one has some freedom in how to group various terms in Eq. (20) and assign meaning to them. In particular, note that both(cid:90)the ﬁrst and the last term in Eq. (20) contributes either 0 or (cid:88) 2π to φ . Thus we could absorb both terms into φ and obtain t t − e2q π/d φ (k ) − | | Πinter(q, iΩ) = dk t z Θ((cid:15) m (k ) ), (21) 8π2 − z π F t z t π/d ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Introducing

text: x

text: we

text: can

text: rewrite

text: this

text: as

text: F

text: x

text: y

text: π/d

text: Πinter(q,

text: iΩ)

text: dk

text: dx

text: Ωt

text: (x,

text: k

text: z

text: z

text: z

text: t

text: π/d

text: n

text: x

text: F

text: t

text: z

text: F

text: where

text: m

text: (k

text: Ωt

text: (x,

text: k

text: t

text: z

text: z

text: z

text: t

text: z

text: is

text: the

text: z-component

text: of

text: the

text: Berry

text: curvature

text: of

text: the

text: s

text: t

text: bands.

text: The

text: ﬁrst

text: term

text: in

text: Eq.

text: is

text: the

text: contribution

text: to

text: Πinter(q,

text: iΩ)

text: of

text: the

text: two

text: s

text: bands,

text: which

text: are

text: completely

text: when

text: The

text: second

text: term

text: is

text: the

text: contribution

text: F

text: of

text: the

text: incompletely

text: ﬁlled

text: s

text: bands.

text: Integrating

text: over

text: x,

text: we

text: obtain

text: iΩ)

text: π/d

text: sign[m

text: z

text: t

text: z

text: t

text: π/d

text: π/d

text: m

text: (k

text: dk

text: sign[m

text: (k

text: t

text: z

text: z

text: t

text: z

text: −t

text: π/d

text: F

text: m

text: (k

text: F

text: t

text: z

text: where

text: Θ(x)

text: is

text: the

text: Heaviside

text: step

text: function.

text: The

text: ﬁrst

text: term

text: in

text: Eq.

text: is

text: the

text: contribution

text: of

text: the

text: completely

text: ﬁlled

text: s

text: bands.

text: The

text: second

text: term

text: is

text: the

text: contribution

text: of

text: incompletely

text: ﬁlled

text: s

text: bands

text: and,

text: as

text: ﬁrst

text: pointed

text: out

text: by

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: an

text: integral

text: of

text: the

text: Berry

text: connection

text: ﬁeld

text: over

text: the

text: Fermi

text: surface.

text: Indeed,

text: components

text: of

text: the

text: Berry

text: connection

text: for

text: the

text: s

text: bands

text: are

text: given

text: by

text: k

text: At

text: (k)

text: i

text: v+t

text: v+t

text: F

text: y

text: x

text: k

text: kx

text: k

text: (k)

text: m

text: (k

text: t

text: t

text: t

text: z

text: k

text: At

text: (k)

text: i

text: v+t

text: v+t

text: F

text: x

text: y

text: k

text: ky

text: k

text: (k)

text: m

text: (k

text: t

text: t

text: t

text: z

text: Integrating

text: the

text: Berry

text: connection

text: over

text: a

text: of

text: the

text: Fermi

text: surface,

text: corresponding

text: to

text: a

text: ﬁxed

text: k

text: i.e.

text: z

text: F

text: x

text: y

text: we

text: obtain

text: F

text: t

text: z

text: φ

text: (k

text: dk

text: A+t(k)

text: π

text: m

text: (k

text: t

text: z

text: t

text: z

text: F

text: π

text: [sign[m

text: (k

text: m

text: (k

text: sign[m

text: (k

text: t

text: z

text: t

text: z

text: F

text: t

text: z

text: where

text: φ

text: (k

text: is

text: the

text: total

text: Berry

text: phase,

text: accumulated

text: corresponding

text: section

text: of

text: the

text: Fermi

text: surface.

text: Using

text: this,

text: t

text: z

text: we

text: can

text: rewrite

text: Eq.

text: in

text: the

text: following

text: way

text: iΩ)

text: π/d

text: dk

text: sign[m

text: (k

text: z

text: t

text: z

text: t

text: π/d

text: π/d

text: dk

text: |m

text: (k

text: z

text: π

text: F

text: t

text: z

text: t

text: π/d

text: π/d

text: dk

text: sign[m

text: (k

text: m

text: (k

text: z

text: t

text: z

text: F

text: t

text: z

text: t

text: π/d

text: The

text: interpretation

text: of

text: Eq.

text: is,

text: unfortunately,

text: not

text: unambiguous

text: due

text: to

text: the

text: fact

text: that

text: the

text: Berry

text: phase

text: φ

text: (k

text: is

text: t

text: z

text: only

text: deﬁned

text: modulo

text: Due

text: to

text: this

text: ambiguity,

text: one

text: has

text: some

text: freedom

text: in

text: how

text: to

text: group

text: various

text: terms

text: in

text: Eq.

text: and

text: assign

text: meaning

text: to

text: them.

text: In

text: particular,

text: note

text: that

text: ﬁrst

text: and

text: the

text: last

text: term

text: in

text: Eq.

text: contributes

text: either

text: or

text: 2π

text: to

text: φ

text: Thus

text: we

text: could

text: absorb

text: both

text: terms

text: into

text: φ

text: and

text: obtain

text: t

text: t

text: π/d

text: φ

text: (k

text: Πinter(q,

text: iΩ)

text: dk

text: t

text: z

text: m

text: (k

text: z

text: π

text: F

text: t

text: z

text: t

text: π/d

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: 13 [1] H. Andersson and T. Britton, Stochastic epidemic models and[9th]eAir. sMta.tiZsutibcaklova,nTalhyesiosr,yVoofl.P4ro(bSapbriinligtyer&NIetws AYporpkli,c2a0ti0o0n)s. 17, 174 (1972). [2] E. Ben-Naim and P. Krapivsky, The European Physical Jour[n1a0l] BN.8B5,ai1le(y2,0T1h2e).elements of stochastic processes with applications to the natural sciences (Wiley-Interscience, 1990). [3] J. P. Sethna, Statistical mechanics: entropy, order parameter[s1,1a]nSd. cOonmgp,leMxiettyrik(Oa x4f3o,rd22U1n(i1v9e9rs6i)t.y Press Oxford, 2006). [4] F. Brauer, P. Van den Driessche, J. Wu, and L. Allen, Math[e1m2]atDic.aGl e.pKideenmdaiolll,ogTyh(eSAprninnaglesr,o2f 0M08a)t.hematical Statistics , 338 (1953). [5] T. Antal and P. Krapivsky, Journal of Statistical Mechanics:[1T3h]eDor.ySahnadnbEhxapg,erJiomuernntal2o0f1A2,pPpl0i7ed01P8r(o2b0a1b2i)li.ty 3, 274 (1966). [6] J. Lloyd-Smith, D. George, K. Pepin, V. Pitzer, J. Pulliam,[1A4.]DJo. bVsiornta, mP.o,HLuedcstounr,e NanotdesB,.HGelrseinnfkeillU, nSicvieenrsciety3o2f6T, e1c3h6n2ology (2005). (2009). [15] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN 77: Volume 1, [7] P. Rohani, R. Breban, D. E. Stallknecht, and J. M. Drake, ProceVeodliunmgseo1f othf eFoNratrtaionnaNluAmceardiceamlyRoecfiSpecise:nTcehse1A0r6t,o1f0S36ci5entiﬁc Computing, Vol. 1 (Cambridge university press, 1992). (2009). [16] P. Flajolet and R. Sedgewick, Analytic combinatorics (cambridge University press, 2009). [8] S. Singh, D. J. Schneider, and C. R. Myers, arXiv preprint a[1rX7]ivK:1.3C07h.r4is6t2e8nse(2n0a1n3d). N. R. Moloney, Complexity and criticality, Vol. 1 (Imperial College Press, 2005). ##LTLine## ##LTLine## ##LTLine##

label: reference

text: H.

text: Andersson

text: and

text: T.

text: Britton,

text: Stochastic

text: epidemic

text: models

text: E.

text: Ben-Naim

text: and

text: P.

text: Krapivsky,

text: The

text: European

text: Physical

text: of

text: stochastic

text: processes

text: with

text: applications

text: to

text: the

text: natural

text: sciences

text: (Wiley-Interscience,

text: J.

text: P.

text: Sethna,

text: Statistical

text: mechanics:

text: entropy,

text: order

text: cOonmgp,leMxiettyrik(Oa

text: Press

text: Oxford,

text: F.

text: Brauer,

text: P.

text: Van

text: den

text: Driessche,

text: J.

text: Wu,

text: and

text: L.

text: Allen,

text: Statistics

text: T.

text: Antal

text: and

text: P.

text: Krapivsky,

text: Journal

text: of

text: Statistical

text: J.

text: Lloyd-Smith,

text: D.

text: George,

text: K.

text: Pepin,

text: V.

text: Pitzer,

text: J.

text: bVsiornta,

text: mP.o,HLuedcstounr,e

text: NanotdesB,.HGelrseinnfkeillU,

text: W.

text: H.

text: Press,

text: B.

text: P.

text: Flannery,

text: S.

text: A.

text: Teukolsky,

text: and

text: W.

text: T.

text: Vetterling,

text: Numerical

text: Recipes

text: in

text: FORTRAN

text: Volume

text: P.

text: Rohani,

text: R.

text: Breban,

text: D.

text: E.

text: Stallknecht,

text: and

text: J.

text: M.

text: Drake,

text: othf

text: Computing,

text: Vol.

text: (Cambridge

text: university

text: press,

text: P.

text: Flajolet

text: and

text: R.

text: Sedgewick,

text: Analytic

text: combinatorics

text: (cambridge

text: University

text: press,

text: S.

text: Singh,

text: D.

text: J.

text: Schneider,

text: and

text: C.

text: R.

text: Myers,

text: arXiv

text: preprint

text: N.

text: R.

text: Moloney,

text: Complexity

text: and

text: criticality,

text: Vol.

text: (Imperial

text: College

text: Press,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Gaussian mixture model. distribution p(x ), which is represented by a GMM. We ge- n 2. SYSTEM MODEL neralise both uncertainty propagation and parameter estimati- on to the case where p(x − ) is given by a GMM. In Secti- n 1 We consider nonlinear dynamical systems on , we propose a method for pruning the number of mixture components in a GMM to avoid their exponential increase in number of components. In Section , we propose the resulting ∼ N x = f(x − ) + w , w (0, Q), (1) ﬁltering algorithm, which exploits the results from Sections –. n n 1 n n ∼ N 3.1. Estimation of the Gaussian Mixture Parameters y = h(x ) + v , v (0, R), (2) n n n n (cid:80) Let the mean and the variance of the state distribution p(x − ) be given by µ, Σ, respectively. Then, we can re- n 1 where f and h are the non-linear transition and measure- present p(x − ) by a GMM p(x − ) = 2D δ ϕ (x − ), ment function, respectively. The noise processes wn and vn such that thne m1ean and the variannce1of the api=pr0oxiimiatende1n- are i.i.d. zero mean Gaussian with covariances Q and R, re- sity p(x − ) equal the mean µ and variance Σ of p(x − ). spectively. We denote the D dimensional{state by xn}, and yn This repnres1entation is achieved by the closed-form relatinon1s is the E dimensional observation. Y = y , . . . , y repres- j 1 j ents all observations up to time step j. We deﬁne the state | estimation problem as determining the density p(xn Yj). Fil- δi = (cid:16)1/(2D + 1(cid:17)), tering and prediction are deﬁned for j = n and j < n, re- µ0 = µ, µj = µ + σj, µj+D = µ − σj, (5) spectively. We deﬁne a GMM representation of a state distri- − butions as Σi = 1 2α Σ, (cid:88) D+1 | M where i = 0, . . . , 2D and j = 1, . . . , D, where D is the pϕ(x(nx Y|j)) ::== N(xi=0| δ|µiϕii|(x, Σn|ij|) ,) , ((34)) dniomteesnDsiornoawlistyorocfotlhuemsntastferovmaritahbelemxantr−ix1.sqTuhaerev√arorioatb±le√σαdΣe-. i n j n j n j n(cid:80)j From (5), we can see that we need to calculate Σ only on- ce for all 2D + 1 Gaussians ϕ (x − ). To ensure that Σi is i n 1 with weights δ ∈ [0, 1] and requiring δ = 1, which positive semi-≤deﬁnite, the scaling factor α should be chosen ensures that p(x |Yi ) is a valid probability disitriibution. such that 2α (2D + 1), see (5). For 2α = 2D + 1 in (5), n j the equations above reduce to scaled sigma points. Hence, the 3. MULTI-MODAL FILTERING GMM representation in (5) can be considered a generalisati- on of the classical sigma point representation of densities em- In the following, we devise a closed-form ﬁltering algo- ployed by the Unscented Transform, where each sigma point rithm with multi-modal representations of the state distribu- becomes an improper probability distribution. tions. Our algorithm is inspired by the following observation 3.2. Propagation of Uncertainty made by Julier and Uhlmann [3]: “Given only the mean and the variance of the underlying distribution, and, in absence A key step in ﬁltering is the uncertainty propagation step, of any a priori information, any distribution (with the same i.e. estimating the probability distribution of random varia- (cid:82) mean and variance) used to calculate the transformed mean ble, which has been transformed by means of the transition and variance is trivially optimal.” This observation was the function f. Given p(x − ) and the system dynamics (1), we n 1 | basis to derive the Unscented transform and the UKF. Ho- determine p(x ) by evaluating p(x x − )p(x − )dx − . n n n 1 n 1 n 1 wever, predictions based on the Unscented Transform often For non-linear functions f, the integral above can only rare- under-estimate the true predictive uncertainty, which can re- ly be solved in closed form. Thus, approximate solutions are sult in incoherent state estimation and divergent tracking per- required. formance. Uncertainty propagation in non-linear systems can be To address this issue, we use a different (optimal) repre- achieved by approximate methods, employing linearisation sentation of the underlying distribution, which still matches or deterministic sampling as in the EKF and UKF. In such ap- the mean and variance: We propose to represent each sigma proaches, the state distribution p(x − ) and the approximate n 1 point in the Unscented transform by a Gaussian centred at this predictive density p(x ) are well represented by Gaussians. If n sigma point. This approximation of the original distribution is the state distribution p(x − ) is a Gaussian mixture as in (5), n 1 effectively a GMM with 2D + 1 components. we can estimate the predictive distribution p(x ) similarly, n In Section , we derive an optimal GMM representation e.g. by applying such an approximate update to each mixture of a state distribution p(x − ) of which only the mean and component in the GMM, see (4). In this paper, we propagate n 1 variance are known. In Section , we detail how to map this each mixture component ϕ (x − ) of the GMM through f i n 1 GMM through a non-linear function to obtain a predictive and approximate p(x ) by n ##LTLine## ##LTLine## ##LTLine##

text: Gaussian

text: mixture

text: model.

text: distribution

text: p(x

text: which

text: is

text: represented

text: by

text: a

text: GMM.

text: We

text: ge-

text: n

label: section

text: SYSTEM

text: MODEL

text: neralise

text: both

text: uncertainty

text: propagation

text: and

text: parameter

text: estimati-

text: on

text: to

text: the

text: case

text: where

text: p(x

text: is

text: given

text: by

text: a

text: GMM.

text: In

text: Secti-

text: n

text: We

text: consider

text: nonlinear

text: dynamical

text: systems

text: on

text: we

text: propose

text: a

text: method

text: for

text: pruning

text: the

text: number

text: of

text: mixture

text: components

text: in

text: a

text: GMM

text: to

text: avoid

text: their

text: exponential

text: increase

text: in

text: number

text: of

text: components.

text: In

text: Section

text: we

text: propose

text: the

text: resulting

text: N

text: x

text: f(x

text: w

text: w

text: Q),

text: ﬁltering

text: algorithm,

text: which

text: exploits

text: the

text: results

text: from

text: Sections

text: n

text: n

text: n

text: n

text: N

text: Estimation

text: of

text: the

text: Gaussian

text: Mixture

text: Parameters

text: y

text: h(x

text: v

text: v

text: R),

text: n

text: n

text: n

text: n

text: Let

text: the

text: mean

text: and

text: the

text: variance

text: of

text: the

text: state

text: distribution

text: p(x

text: be

text: given

text: by

text: µ,

text: Σ,

text: respectively.

text: Then,

text: we

text: can

text: re-

text: n

text: where

text: f

text: and

text: h

text: are

text: the

text: non-linear

text: transition

text: and

text: measure-

text: present

text: p(x

text: by

text: a

text: GMM

text: p(x

text: δ

text: ϕ

text: (x

text: ment

text: function,

text: respectively.

text: The

text: noise

text: processes

text: wn

text: and

text: vn

text: such

text: that

text: thne

text: and

text: the

text: the

text: are

text: i.i.d.

text: zero

text: mean

text: Gaussian

text: with

text: covariances

text: Q

text: and

text: R,

text: re-

text: sity

text: p(x

text: equal

text: the

text: mean

text: µ

text: and

text: variance

text: Σ

text: of

text: p(x

text: spectively.

text: We

text: denote

text: the

text: D

text: dimensional{state

text: by

text: xn},

text: and

text: yn

text: This

text: is

text: achieved

text: by

text: the

text: closed-form

text: is

text: the

text: E

text: dimensional

text: observation.

text: Y

text: y

text: y

text: repres-

text: j

text: j

text: ents

text: all

text: observations

text: up

text: to

text: time

text: step

text: j.

text: We

text: deﬁne

text: the

text: state

text: estimation

text: problem

text: as

text: determining

text: the

text: density

text: p(xn

text: Yj).

text: Fil-

text: δi

text: tering

text: and

text: prediction

text: are

text: deﬁned

text: for

text: j

text: n

text: and

text: j

text: n,

text: re-

text: µ0

text: µ,

text: µj

text: µ

text: σj,

text: µj+D

text: µ

text: σj,

text: spectively.

text: We

text: deﬁne

text: a

text: GMM

text: representation

text: of

text: a

text: state

text: distri-

text: butions

text: as

text: Σi

text: 2α

text: Σ,

text: M

text: where

text: i

text: and

text: j

text: D,

text: where

text: D

text: is

text: the

text: pϕ(x(nx

text: Y|j))

text: δ|µiϕii|(x,

text: Σn|ij|)

text: i

text: n

text: j

text: n

text: j

text: n

text: j

text: From

text: we

text: can

text: see

text: that

text: we

text: need

text: to

text: calculate

text: Σ

text: only

text: on-

text: ce

text: for

text: all

text: Gaussians

text: ϕ

text: (x

text: To

text: ensure

text: that

text: Σi

text: is

text: i

text: n

text: with

text: weights

text: δ

text: and

text: requiring

text: δ

text: which

text: positive

text: semi-≤deﬁnite,

text: the

text: scaling

text: factor

text: α

text: should

text: be

text: chosen

text: ensures

text: that

text: p(x

text: |Yi

text: is

text: a

text: valid

text: probability

text: disitriibution.

text: such

text: that

text: 2α

text: see

text: For

text: 2α

text: in

text: n

text: j

text: the

text: equations

text: above

text: reduce

text: to

text: scaled

text: sigma

text: points.

text: Hence,

text: the

text: MULTI-MODAL

text: FILTERING

text: GMM

text: representation

text: in

text: can

text: be

text: considered

text: a

text: generalisati-

text: on

text: of

text: the

text: classical

text: sigma

text: point

text: representation

text: of

text: densities

text: em-

text: In

text: the

text: following,

text: we

text: devise

text: a

text: closed-form

text: ﬁltering

text: algo-

text: ployed

text: by

text: the

text: Unscented

text: Transform,

text: where

text: each

text: sigma

text: point

text: rithm

text: with

text: multi-modal

text: representations

text: of

text: the

text: state

text: distribu-

text: becomes

text: an

text: improper

text: probability

text: distribution.

text: tions.

text: Our

text: algorithm

text: is

text: inspired

text: by

text: the

text: following

text: observation

text: Propagation

text: of

text: Uncertainty

text: made

text: by

text: Julier

text: and

text: Uhlmann

text: “Given

text: only

text: the

text: mean

text: and

text: the

text: variance

text: of

text: the

text: underlying

text: distribution,

text: and,

text: in

text: absence

text: A

text: key

text: step

text: in

text: ﬁltering

text: is

text: the

text: uncertainty

text: propagation

text: step,

text: of

text: any

text: a

text: priori

text: information,

text: any

text: distribution

text: (with

text: the

text: same

text: i.e.

text: estimating

text: the

text: probability

text: distribution

text: of

text: random

text: varia-

text: mean

text: and

text: variance)

text: used

text: to

text: calculate

text: the

text: transformed

text: mean

text: ble,

text: which

text: has

text: been

text: transformed

text: by

text: means

text: of

text: the

text: transition

text: and

text: variance

text: is

text: trivially

text: optimal.”

text: This

text: observation

text: was

text: the

text: function

text: f.

text: Given

text: p(x

text: and

text: the

text: system

text: dynamics

text: we

text: n

text: basis

text: to

text: derive

text: the

text: Unscented

text: transform

text: and

text: the

text: UKF.

text: Ho-

text: determine

text: p(x

text: by

text: evaluating

text: p(x

text: x

text: )p(x

text: )dx

text: n

text: n

text: n

text: n

text: n

text: wever,

text: predictions

text: based

text: on

text: the

text: Unscented

text: Transform

text: often

text: For

text: non-linear

text: functions

text: f,

text: the

text: integral

text: above

text: can

text: only

text: rare-

text: under-estimate

text: the

text: true

text: predictive

text: uncertainty,

text: which

text: can

text: re-

text: ly

text: be

text: solved

text: in

text: closed

text: form.

text: Thus,

text: approximate

text: solutions

text: are

text: sult

text: in

text: incoherent

text: state

text: estimation

text: and

text: divergent

text: tracking

text: per-

text: required.

text: formance.

text: Uncertainty

text: propagation

text: in

text: non-linear

text: systems

text: can

text: be

text: To

text: address

text: this

text: issue,

text: we

text: use

text: a

text: different

text: (optimal)

text: repre-

text: achieved

text: by

text: approximate

text: methods,

text: employing

text: linearisation

text: sentation

text: of

text: the

text: underlying

text: distribution,

text: which

text: still

text: matches

text: or

text: deterministic

text: sampling

text: as

text: in

text: the

text: EKF

text: and

text: UKF.

text: In

text: such

text: ap-

text: the

text: mean

text: and

text: variance:

text: We

text: propose

text: to

text: represent

text: each

text: sigma

text: proaches,

text: the

text: state

text: distribution

text: p(x

text: and

text: the

text: approximate

text: n

text: point

text: in

text: the

text: Unscented

text: transform

text: by

text: a

text: Gaussian

text: centred

text: at

text: this

text: predictive

text: density

text: p(x

text: are

text: well

text: represented

text: by

text: Gaussians.

text: If

text: n

text: sigma

text: point.

text: This

text: approximation

text: of

text: the

text: original

text: distribution

text: is

text: the

text: state

text: distribution

text: p(x

text: is

text: a

text: Gaussian

text: mixture

text: as

text: in

text: n

text: effectively

text: a

text: GMM

text: with

text: components.

text: we

text: can

text: estimate

text: the

text: predictive

text: distribution

text: p(x

text: similarly,

text: n

text: In

text: Section

text: we

text: derive

text: an

text: optimal

text: GMM

text: representation

text: e.g.

text: by

text: applying

text: such

text: an

text: approximate

text: update

text: to

text: each

text: mixture

text: of

text: a

text: state

text: distribution

text: p(x

text: of

text: which

text: only

text: the

text: mean

text: and

text: component

text: in

text: the

text: GMM,

text: see

text: In

text: this

text: paper,

text: we

text: propagate

text: n

text: variance

text: are

text: known.

text: In

text: Section

text: we

text: detail

text: how

text: to

text: map

text: this

text: each

text: mixture

text: component

text: ϕ

text: (x

text: of

text: the

text: GMM

text: through

text: f

text: i

text: n

text: GMM

text: through

text: a

text: non-linear

text: function

text: to

text: obtain

text: a

text: predictive

text: and

text: approximate

text: p(x

text: by

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CONTRIBUTORS OF CARBON DIOXIDE IN THE ATMOSPHERE IN EUROPE: THE SURFACE RESPONSE ANALYSIS IULIANA TEODORESCU AND CHRIS TSOKOS Abstract. This paper is a continuation of the statistical modeling of the nonlinear relationship between atmospheric CO and attributable 2 variables that can account for emissions, based on data from EU coun- tries, in order to compare the relevant ﬁndings to those obtained in the case of US data, in [1, 2]. The current study was initiated in [3], lead- ing to the optimal second-order model, based on three linear terms and ﬁve second-order terms. We conclude this study in the present work, by ﬁnding the canonical decomposition of the nonlinear model, and by computing the speciﬁc two-dimensional conﬁdence regions that it leads to. We then use the model in order to quantify the net eﬀect of various risk factors, and compare to the results obtained in the US case. 1. Introduction This article contains the second part of the statistical modeling of the nonlinear relationship between atmospheric CO and various contributor 2 variables that can account for emissions, based on data from EU countries. The ﬁrst part [3] indicated the model-building procedure, including linear terms, quadratic terms, and mixed (interacting) terms, and produced rank- ings for the most signiﬁcant attributable variables (or their interactions). In the current paper, we start from the second-order model developed in [3] and perform its surface reponse analysis, leading to canonical two- dimensional conﬁdence regions, and to speciﬁc comparisons between canon- ical variables, much as it was done in [2], in the case of US data. As indicated in the previous studies, the response variable is the CO in 2 the atmosphere and is given in parts per million by volume (ppmv), obtained from yearly data collected from 1959 to 20081. The CO emission data for 2 the EU countries listed below was obtained from Carbon Dioxide Informa- tion Analysis Center (CDIAC) during the same period: Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Luxembourg, Malta, Netherlands, Poland, Portugal, Romania, Slovakia, Spain, Sweden, United Kingdom (as of June 2013, there are 27 member states of the EU. Slovenia and the Baltic states were excluded from the present study since there was no individual data available for the period during which they were part of former Yugoslavia, 1Year 1964 was ignored due to incomplete records. 1 ##LTLine##

text: CONTRIBUTORS

text: OF

text: CARBON

text: DIOXIDE

text: IN

text: THE

text: ATMOSPHERE

text: IN

text: EUROPE:

text: THE

text: SURFACE

text: RESPONSE

text: ANALYSIS

text: IULIANA

text: TEODORESCU

text: AND

text: CHRIS

text: TSOKOS

text: Abstract.

text: This

label: abstract

text: paper

text: is

text: a

text: continuation

text: of

text: the

text: statistical

text: modeling

text: of

text: the

text: nonlinear

text: relationship

text: between

text: atmospheric

text: CO

text: and

text: attributable

text: variables

text: that

text: can

text: account

text: for

text: emissions,

text: based

text: on

text: data

text: from

text: EU

text: coun-

text: tries,

text: in

text: order

text: to

text: compare

text: the

text: relevant

text: ﬁndings

text: to

text: those

text: obtained

text: in

text: the

text: case

text: of

text: US

text: data,

text: in

text: The

text: current

text: study

text: was

text: initiated

text: in

text: lead-

text: ing

text: to

text: the

text: optimal

text: second-order

text: model,

text: based

text: on

text: three

text: linear

text: terms

text: and

text: ﬁve

text: second-order

text: terms.

text: We

text: conclude

text: this

text: study

text: in

text: the

text: present

text: work,

text: by

text: ﬁnding

text: the

text: canonical

text: decomposition

text: of

text: the

text: nonlinear

text: model,

text: and

text: by

text: computing

text: the

text: speciﬁc

text: two-dimensional

text: conﬁdence

text: regions

text: that

text: it

text: leads

text: to.

text: We

text: then

text: use

text: the

text: model

text: in

text: order

text: to

text: quantify

text: the

text: net

text: eﬀect

text: of

text: various

text: risk

text: factors,

text: and

text: compare

text: to

text: the

text: results

text: obtained

text: in

text: the

text: US

text: case.

text: Introduction

text: This

text: article

text: contains

text: the

text: second

text: part

text: of

text: the

text: statistical

text: modeling

text: of

text: the

text: nonlinear

text: relationship

text: between

text: atmospheric

text: CO

text: and

text: various

text: contributor

text: variables

text: that

text: can

text: account

text: for

text: emissions,

text: based

text: on

text: data

text: from

text: EU

text: countries.

text: The

text: ﬁrst

text: part

text: indicated

text: the

text: model-building

text: procedure,

text: including

text: linear

text: terms,

text: quadratic

text: terms,

text: and

text: mixed

text: (interacting)

text: terms,

text: and

text: produced

text: rank-

text: ings

text: for

text: the

text: most

text: signiﬁcant

text: attributable

text: variables

text: (or

text: their

text: interactions).

text: In

text: the

text: current

text: paper,

text: we

text: start

text: from

text: the

text: second-order

text: model

text: developed

text: in

text: and

text: perform

text: its

text: surface

text: reponse

text: analysis,

text: leading

text: to

text: canonical

text: two-

text: dimensional

text: conﬁdence

text: regions,

text: and

text: to

text: speciﬁc

text: comparisons

text: between

text: canon-

text: ical

text: variables,

text: much

text: as

text: it

text: was

text: done

text: in

text: in

text: the

text: case

text: of

text: US

text: data.

text: As

text: indicated

text: in

text: the

text: previous

text: studies,

text: the

text: response

text: variable

text: is

text: the

text: CO

text: in

text: the

text: atmosphere

text: and

text: is

text: given

text: in

text: parts

text: per

text: million

text: by

text: volume

text: (ppmv),

text: obtained

text: from

text: yearly

text: data

text: collected

text: from

text: to

text: The

text: CO

text: emission

text: data

text: for

text: the

text: EU

text: countries

text: listed

text: below

text: was

text: obtained

text: from

text: Carbon

text: Dioxide

text: Informa-

text: tion

text: Analysis

text: Center

text: (CDIAC)

text: during

text: the

text: same

text: period:

text: Austria,

text: Belgium,

text: Bulgaria,

text: Cyprus,

text: Czech

text: Republic,

text: Denmark,

text: Finland,

text: France,

text: Germany,

text: Greece,

text: Hungary,

text: Ireland,

text: Italy,

text: Luxembourg,

text: Malta,

text: Netherlands,

text: Poland,

text: Portugal,

text: Romania,

text: Slovakia,

text: Spain,

text: Sweden,

text: United

text: Kingdom

text: (as

text: of

text: June

text: there

text: are

text: member

text: states

text: of

text: the

text: EU.

text: Slovenia

text: and

text: the

text: Baltic

text: states

text: were

text: excluded

text: from

text: the

text: present

text: study

text: since

text: there

text: was

text: no

text: individual

text: data

text: available

text: for

text: the

text: period

text: during

text: which

text: they

text: were

text: part

text: of

text: former

text: Yugoslavia,

text: was

text: ignored

text: due

text: to

text: incomplete

text: records.

text: ##LTLine##


text: SURFACE RESPONSE ANALYSIS II 3 2. The model, parameters, and descriptive quantities We recall the ﬁnal second-order model found in [3] to provide a good ﬁt for the data and to have robust features for prediction and estimation: (cid:100) − − − × − [CO ] 2.376 = 0.00000123 + (710.85F l 30.64Ga 3.4501Li) 10 13 + 2 · · − · − + (37.34Ga Bu + 1.35Li Li 65.12Bu Bu − · − · × − 133.05Li F l 5.35Li Bu) 10 18. Throughout the paper, we will be using the the notation x = Liquid 1 Fuels (Li), x = Gas Fuels (Ga), x = Gas Flares (Fl), x = Bunker (Bu) 2 3 4 for the relevant attributable variables. With respect to these variables, the model becomes (cid:88) (cid:88) (cid:100) 4 4 − (1) [CO ] 2.376 = β + β x + β x x , 2 0 i i ij i j ≤ i=1 i j=1 with the corresponding ranks determined by the stepwise SAS procedure are given in Table 1, along with the coeﬃcients in the ﬁnal regression model. In matrix notation (where prime denotes transposition), (1) becomes (cid:48) · (cid:48) · · (2) Y = β + β X + X B X, 0 with the obvious identiﬁcations (cid:48) (cid:48) 1 X = (x , . . . , x ), β = (β , . . . , β ), B = B = β (i < j). 1 4 1 4 ij ji ij 2 Table 1. Ranking by statist(cid:2)ical rele(cid:3)vance for attributable variables and interactions. × − − Rank Variable β 10 18 F Value − × 1 Ga 30.635 105 197.22 2 Ga:Bu 37.3391 50.25 3 Li:Li 1.35565 47.74 − 4 Bu:Bu 65.115 31.49 × 5 Fl 710.848 105 26.98 − 6 Li:Fl 133.05 20.49 − 7 Li:Bu 5.3501 19.07 − × 8 Li 3.4501 105 11.57 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: SURFACE

text: RESPONSE

text: ANALYSIS

text: II

text: The

text: model,

text: parameters,

text: and

text: descriptive

text: quantities

text: We

text: recall

text: the

text: ﬁnal

text: second-order

text: model

text: found

text: in

text: to

text: provide

text: a

text: good

text: ﬁt

text: for

text: the

text: data

text: and

text: to

text: have

text: robust

text: features

text: for

text: prediction

text: and

text: estimation:

text: [CO

text: l

text: Bu

text: Li

text: Bu

text: F

text: l

text: Bu)

text: Throughout

text: the

text: paper,

text: we

text: will

text: be

text: using

text: the

text: the

text: notation

text: x

text: Liquid

text: Fuels

text: (Li),

text: x

text: Gas

text: Fuels

text: (Ga),

text: x

text: Gas

text: Flares

text: (Fl),

text: x

text: Bunker

text: (Bu)

text: for

text: the

text: relevant

text: attributable

text: variables.

text: With

text: respect

text: to

text: these

text: variables,

text: the

text: model

text: becomes

text: [CO

text: β

text: β

text: x

text: β

text: x

text: x

text: i

text: i

text: ij

text: i

text: j

text: i

text: with

text: the

text: corresponding

text: ranks

text: determined

text: by

text: the

text: stepwise

text: SAS

text: procedure

text: are

text: given

text: in

text: Table

text: along

text: with

text: the

text: coeﬃcients

text: in

text: the

text: ﬁnal

text: regression

text: model.

text: In

text: matrix

text: notation

text: (where

text: prime

text: denotes

text: transposition),

text: becomes

text: Y

text: β

text: β

text: X

text: X

text: B

text: X,

text: with

text: the

text: obvious

text: identiﬁcations

text: X

text: (x

text: x

text: β

text: (β

text: β

text: B

text: B

text: β

text: (i

text: j).

text: ij

text: ji

text: ij

text: Table

text: Ranking

text: by

text: for

text: attributable

text: variables

text: and

text: interactions.

text: Rank

text: Variable

text: β

text: F

text: Value

text: Ga

text: Ga:Bu

text: Li:Li

text: Bu:Bu

text: Fl

text: Li:Fl

text: Li:Bu

text: Li

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: SURFACE RESPONSE ANALYSIS II 5 − (cid:88) § Let B 1 represent the inverse of the matrix B ([5, Appendix C]) 4 − − (cid:48) B 1 = λ 1V V , k k k k=1 and start from the model (2) (cid:48) · (cid:48) · · Y = β + β X + X B X. 0 In order to bring this expression to its normal form, we begin by shifting (cid:98) the variable X by a constant term 1 − · X = X + B 1 β. 2 (cid:98) (cid:98) (cid:98) (cid:98) Since B is a non-singular matrix, we obtain the model (cid:48) · − 1 (cid:48) · − · (cid:48) · · − (cid:48) · · − Y = β + β X β B 1 β + X B X β B B 1X, 0 4 − · I(cid:98) (cid:98) where we have used the property B 1 B = . Therefore, Y = β − 1β(cid:48) · B−1 · β + X(cid:48) · B · X, (cid:98) (cid:98) 0 4(cid:98) (cid:98) (cid:48) · · so we(cid:32)are now wo(cid:33)rking with the normal quadratic form X B X. Using (cid:88) (cid:88) (cid:48) · · (cid:88) (cid:88) ag(cid:98)ain (3), the quadra(cid:98)tic form X (cid:98) B X b(cid:98)ecomes (cid:98) 4 4 4 4 (cid:48) (cid:48) (cid:48) (cid:48) | (cid:48) · | X λ V V X = λ (X V )(V X) = λ V X 2 = λ z2, k k k k k k k k k k k=1 k=1 (cid:98) k=1 k=1 where we have introduced the canonical coordinates (cid:48) · (5) z := V X, k = 1, 2, 3, 4. k k To conclude, we have the canonical form of the model − − − × − (6) Y Y = ( 140.176z2 + 134.413z2 131.701z2 + 9.94612z2) 10 18, 0 1 2 3 4 with z given in (5). k To ﬁnd the stationary point of the model, deﬁned generically as the zero- gradient point, we must solve simultaneously for all k = 1, . . . , 4: ∂Y ⇒ (cid:48) (cid:48) · ⇒ · −1 = 0 β + 2X B = 0 B X = β, ∂x (cid:98) (cid:98) 2 k which is equivalent to · ⇒ B X = 0 X = 0, because B is non-degenerate. Together with (5), this gives the stationary   point as the origin of the z coordinates, z = z = z = z = 0. In the 1 2 3 4 original variables, the stationary point is found to be:  −   534271  − −1 − · 286155 (7) X = B 1 β = − , s 2 8294.32 − 82045.4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SURFACE

text: RESPONSE

text: ANALYSIS

text: II

text: Let

text: B

text: represent

text: the

text: inverse

text: of

text: the

text: matrix

text: B

text: Appendix

text: C])

text: B

text: λ

text: V

text: k

text: k

text: k

text: and

text: start

text: from

text: the

text: model

text: Y

text: β

text: β

text: X

text: X

text: B

text: X.

text: In

text: order

text: to

text: bring

text: this

text: expression

text: to

text: its

text: normal

text: form,

text: we

text: begin

text: by

text: shifting

text: the

text: variable

text: X

text: by

text: a

text: constant

text: term

text: X

text: X

text: B

text: β.

text: Since

text: B

text: is

text: a

text: non-singular

text: matrix,

text: we

text: obtain

text: the

text: model

text: Y

text: β

text: β

text: X

text: β

text: B

text: β

text: X

text: B

text: X

text: β

text: B

text: B

text: where

text: we

text: have

text: used

text: the

text: property

text: B

text: B

text: Therefore,

text: Y

text: β

text: β

text: B

text: X,

text: so

text: now

text: with

text: the

text: normal

text: quadratic

text: form

text: X

text: B

text: X.

text: Using

text: the

text: form

text: X

text: B

text: X

text: X

text: λ

text: V

text: V

text: X

text: λ

text: (X

text: V

text: )(V

text: X)

text: λ

text: V

text: X

text: λ

text: k

text: k

text: k

text: k

text: k

text: k

text: k

text: k

text: k

text: k

text: where

text: we

text: have

text: introduced

text: the

text: canonical

text: coordinates

text: z

text: V

text: X,

text: k

text: k

text: k

text: To

text: conclude,

text: we

text: have

text: the

text: canonical

text: form

text: of

text: the

text: model

text: Y

text: Y

text: with

text: z

text: given

text: in

text: k

text: To

text: ﬁnd

text: the

text: stationary

text: point

text: of

text: the

text: model,

text: deﬁned

text: generically

text: as

text: the

text: zero-

text: gradient

text: point,

text: we

text: must

text: solve

text: simultaneously

text: for

text: all

text: k

text: ∂Y

text: −1

text: β

text: B

text: B

text: X

text: β,

text: ∂x

text: k

text: which

text: is

text: equivalent

text: to

text: B

text: X

text: X

text: because

text: B

text: is

text: non-degenerate.

text: Together

text: with

text: this

text: gives

text: the

text: stationary

text: point

text: as

text: the

text: origin

text: of

text: the

text: z

text: coordinates,

text: z

text: z

text: z

text: z

text: In

text: the

text: original

text: variables,

text: the

text: stationary

text: point

text: is

text: found

text: to

text: be:

text: −1

text: X

text: B

text: β

text: s

text: ##LTLine##

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text: 6 IULIANA TEODORESCU AND CHRIS TSOKOS Figure 2. Conﬁdence regions for the elliptical case. up to an overall scale factor of 105. We repeat here the discussion regarding conﬁdence region types presented in [2]. In order to distinguish between various types of shapes the conﬁdence regions may have, we now specialize to a pair of variables (z , z ) from the i j normal quadratic form written in canonical variables, and impose the in- equality | − | ≤ (cid:12) (cid:12) Y (cid:12) Y0 M, (cid:12) M > 0, (cid:12) (cid:12) leading to ≤ λ z2 + λ z2 M, i i j j which deﬁnes the conﬁdence region centered at (0, 0). We ﬁnd the following cases, corresponding to classes of conic sections: If λ are either all positive or all negative, the point (0, 0) is a point of i,j minimum or of maximum, respectively. The inequality becomes | | | | ≤ ⇒ z2 z(cid:112)2 ≤ (cid:112) (8) λ z2 + λ z2 M i| | + |j | 1, i i j j M/ λ M/ λ i j | | | | which deﬁn(cid:115)es the interior of an el(cid:115)lipse of semiaxes M/ λ , M/ λ (see i j Figure 2, right panel). The conﬁdence region is given parametrically by: M M ≤ ≤ ∈ (9) z = | |r cos(θ), z = | |r sin(θ), 0 r 1, θ [0, 2π]. i j λ λ i j ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IULIANA

text: TEODORESCU

text: AND

text: CHRIS

text: TSOKOS

text: Figure

text: Conﬁdence

text: regions

text: for

text: the

text: elliptical

text: case.

text: up

text: to

text: an

text: overall

text: scale

text: factor

text: of

text: We

text: repeat

text: here

text: the

text: discussion

text: regarding

text: conﬁdence

text: region

text: types

text: presented

text: in

text: In

text: order

text: to

text: distinguish

text: between

text: various

text: types

text: of

text: shapes

text: the

text: conﬁdence

text: regions

text: may

text: have,

text: we

text: now

text: specialize

text: to

text: a

text: pair

text: of

text: variables

text: (z

text: z

text: from

text: the

text: i

text: j

text: normal

text: quadratic

text: form

text: written

text: in

text: canonical

text: variables,

text: and

text: impose

text: the

text: in-

text: equality

text: Y

text: M,

text: M

text: leading

text: to

text: λ

text: λ

text: M,

text: i

text: i

text: j

text: j

text: which

text: deﬁnes

text: the

text: conﬁdence

text: region

text: centered

text: at

text: We

text: ﬁnd

text: the

text: following

text: cases,

text: corresponding

text: to

text: classes

text: of

text: conic

text: sections:

text: If

text: λ

text: are

text: either

text: all

text: positive

text: or

text: all

text: negative,

text: the

text: point

text: is

text: a

text: point

text: of

text: i,j

text: minimum

text: or

text: of

text: maximum,

text: respectively.

text: The

text: inequality

text: becomes

text: λ

text: λ

text: M

text: i|

text: |j

text: i

text: i

text: j

text: j

text: M/

text: λ

text: M/

text: λ

text: i

text: j

text: which

text: the

text: interior

text: of

text: an

text: of

text: semiaxes

text: M/

text: λ

text: M/

text: λ

text: (see

text: i

text: j

text: Figure

text: right

text: panel).

text: The

text: conﬁdence

text: region

text: is

text: given

text: parametrically

text: by:

text: M

text: M

text: z

text: |r

text: cos(θ),

text: z

text: |r

text: sin(θ),

text: r

text: θ

text: i

text: j

text: λ

text: λ

text: i

text: j

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: τ-COMPLETE FACTORIZATION IN COMMUTATIVE RINGS WITH ZERO-DIVISORS CHRISTOPHER PARK MOONEY Abstract. Much work has been done on generalized factorization techniques in integral domains, namely τ-factorization. There has also been substantial progress made in investi- gating factorization in commutative rings with zero-divisors. There are many ways authors have decided to study factorization when zero-divisors present. This paper focuses on the method τ-complete factorizations developed by D.D. Anderson and A. Frazier. There is a natural way to extend τ-complete factorization to commutative rings with zero-divisors. τ-complete factorization is a natural way to think of reﬁning factorizations into smaller pieces until one simply cannot reﬁne any further. We see that this notion translates well into the case of commutative rings with zero-divisors and there is an interesting relationship between the τ-complete ﬁnite factorization properties and the original τ-ﬁnite factorization properties in rings with zero-divisors developed by the author in 2012. 2010 AMS Subject Classiﬁcation: 13A05, 13E99, 13F15 1. Introduction There has been a substantial amount of research done on the factorization properties of commutative rings, especially domains. Unique factorization domains (UFDs) are well un- derstood and have been studied extensively over the years. More recently, many authors have studied rings which satisfy various weakenings of the UFD conditions. These factorization properties of domains have been extended in several ways to rings with zero-divisors. Tradi- tionally, in the domain case, authors have studied prime or irreducible factorizations. More recently, research has been done on generalizing the types of factorizations that have been studied to include things like co-maximal factorizations or using (cid:63)-operations to generalize factorization. Of particular interest to the current article is the 2011 work of D.D. Anderson and A. Frazier. This is a survey article, [2], on the study of factorization in domains in which they introduce τ-factorization. The use of τ-factorization yields a beautiful synthesis of many of these generalizations of factorizations studied in the integral domain case. The goal then has been to extend this powerful approach of τ-factorization to the case of a commutative ring with zero-divisors. There have been several unique ways of studying factorization in commutative rings with zero-divisors, so this has led to many approaches to extending τ- factorization. In [19], the author used the methods established by D.D. Anderson and S. Valdes-Leon in [4] to extend many of the τ-factorization deﬁnitions to work also in rings with zero-divisors. In [20], the author investigated extending τ-factorization using the notion of U-factorizations developed ﬁrst by C.R. Fletcher in [15, 16] and then studied extensively by M. Axtell, N. Baeth, and J. Stickles in [8, 9]. Date: May 26, 2020. Key words and phrases. factorization, zero-divisors, commutative rings, complete factorization. 1 ##LTLine##

text: τ-COMPLETE

text: FACTORIZATION

text: IN

text: COMMUTATIVE

text: RINGS

text: WITH

text: ZERO-DIVISORS

text: CHRISTOPHER

text: PARK

text: MOONEY

text: Abstract.

text: Much

text: work

text: has

text: been

text: done

text: on

text: generalized

text: factorization

text: techniques

text: in

text: integral

text: domains,

text: namely

text: τ-factorization.

text: There

text: has

text: also

text: been

text: substantial

text: progress

text: made

text: in

text: investi-

text: gating

text: factorization

text: in

text: commutative

text: rings

text: with

text: zero-divisors.

text: There

text: are

text: many

text: ways

text: authors

text: have

text: decided

text: to

text: study

text: factorization

text: when

text: zero-divisors

text: present.

text: This

text: paper

text: focuses

text: on

text: the

text: method

text: τ-complete

text: factorizations

text: developed

text: by

text: D.D.

text: Anderson

text: and

text: A.

text: Frazier.

text: There

text: is

text: a

text: natural

text: way

text: to

text: extend

text: τ-complete

text: factorization

text: to

text: commutative

text: rings

text: with

text: zero-divisors.

text: τ-complete

text: factorization

text: is

text: a

text: natural

text: way

text: to

text: think

text: of

text: reﬁning

text: factorizations

text: into

text: smaller

text: pieces

text: until

text: one

text: simply

text: cannot

text: reﬁne

text: any

text: further.

text: We

text: see

text: that

text: this

text: notion

text: translates

text: well

text: into

text: the

text: case

text: of

text: commutative

text: rings

text: with

text: zero-divisors

text: and

text: there

text: is

text: an

text: interesting

text: relationship

text: between

text: the

text: τ-complete

text: ﬁnite

text: factorization

text: properties

text: and

text: the

text: original

text: τ-ﬁnite

text: factorization

text: properties

text: in

text: rings

text: with

text: zero-divisors

text: developed

text: by

text: the

text: author

text: in

text: AMS

text: Subject

text: Classiﬁcation:

text: Introduction

text: There

text: has

text: been

text: a

text: substantial

text: amount

text: of

text: research

text: done

text: on

text: the

text: factorization

text: properties

text: of

text: commutative

text: rings,

text: especially

text: domains.

text: Unique

text: factorization

text: domains

text: (UFDs)

text: are

text: well

text: un-

text: derstood

text: and

text: have

text: been

text: studied

text: extensively

text: over

text: the

text: years.

text: More

text: recently,

text: many

text: authors

text: have

text: studied

text: rings

text: which

text: satisfy

text: various

text: weakenings

text: of

text: the

text: UFD

text: conditions.

text: These

text: factorization

text: properties

text: of

text: domains

text: have

text: been

text: extended

text: in

text: several

text: ways

text: to

text: rings

text: with

text: zero-divisors.

text: Tradi-

text: tionally,

text: in

text: the

text: domain

text: case,

text: authors

text: have

text: studied

text: prime

text: or

text: irreducible

text: factorizations.

text: More

text: recently,

text: research

text: has

text: been

text: done

text: on

text: generalizing

text: the

text: types

text: of

text: factorizations

text: that

text: have

text: been

text: studied

text: to

text: include

text: things

text: like

text: co-maximal

text: factorizations

text: or

text: using

text: to

text: generalize

text: factorization.

text: Of

text: particular

text: interest

text: to

text: the

text: current

text: article

text: is

text: the

text: work

text: of

text: D.D.

text: Anderson

text: and

text: A.

text: Frazier.

text: This

text: is

text: a

text: survey

text: article,

text: on

text: the

text: study

text: of

text: factorization

text: in

text: domains

text: in

text: which

text: they

text: introduce

text: τ-factorization.

text: The

text: use

text: of

text: τ-factorization

text: yields

text: a

text: beautiful

text: synthesis

text: of

text: many

text: of

text: these

text: generalizations

text: of

text: factorizations

text: studied

text: in

text: the

text: integral

text: domain

text: case.

text: The

text: goal

text: then

text: has

text: been

text: to

text: extend

text: this

text: powerful

text: approach

text: of

text: τ-factorization

text: to

text: the

text: case

text: of

text: a

text: commutative

text: ring

text: with

text: zero-divisors.

text: There

text: have

text: been

text: several

text: unique

text: ways

text: of

text: studying

text: factorization

text: in

text: commutative

text: rings

text: with

text: zero-divisors,

text: so

text: this

text: has

text: led

text: to

text: many

text: approaches

text: to

text: extending

text: τ-

text: factorization.

text: In

text: the

text: author

text: used

text: the

text: methods

text: established

text: by

text: D.D.

text: Anderson

text: and

text: S.

text: Valdes-Leon

text: in

text: to

text: extend

text: many

text: of

text: the

text: τ-factorization

text: deﬁnitions

text: to

text: work

text: also

text: in

text: rings

text: with

text: zero-divisors.

text: In

text: the

text: author

text: investigated

text: extending

text: τ-factorization

text: using

text: the

text: notion

text: of

text: U-factorizations

text: developed

text: ﬁrst

text: by

text: C.R.

text: Fletcher

text: in

text: and

text: then

text: studied

text: extensively

text: by

text: M.

text: Axtell,

text: N.

text: Baeth,

text: and

text: J.

text: Stickles

text: in

text: Date:

text: May

text: Key

text: words

text: and

text: phrases.

text: factorization,

text: zero-divisors,

text: commutative

text: rings,

text: complete

text: factorization.

text: ##LTLine##