text: ESTIMATING SYMPLECTIC CAPACITIES FROM. . . 5 (cid:107) · (cid:107) we use the Euclidean norm, not . Now assume t(cid:88)hat this attempt to build a no worse w-invariant solution to Clarke’s variational problem fails and we have | | mA + α v 2 < A + A . i m i i i Moreover, assume this attempt fails for all i = 1, . . . , m. Then summing such inequalities we obtain (cid:88) | | (2.2) α v 2 < mA, m i i (cid:32) (cid:33) (cid:88) (cid:88) and from the quadratic-arithmetic mean inequality 2 α | | ≤ | | m v α v 2 < mA. i m i m i i Note that we can ﬁll the polygonal loop composed of v , . . . , v by a 2-chain F of triangles 1 m T , . . . , T − and note that by the calibration property [10, Section I] the Riemannian area 1 m 2 (cid:32) (cid:33) of every such Ti is no less than its sym(cid:88)plectic action, hence for the area SF of the whole 2-chain F we have 2 α | | m v < mS . i F m i This looks like the inverse isoperimetric inequality. Moreover, we can rotate T , T , . . . , T − 2 3 m 2 into the same 2-plane one by one and see that this inequality is the opposite to the stan- dard isoperimetric inequality for m-gons in the plane, thus obtaining a contradiction. Therefore, for some i the attempt to produce no worse solution with the required symmetry must succeed. From the existence and uniqueness theorem for the ODE for this variational problem we see that in the smooth case this modiﬁcation of γ must in fact give the same loop, just because the segment γ remains at its place. i An alternative way to deduce (2.2): Note that the action of a loop is the sum of actions of its projections to the coordinate complex lines. At the same time the Euclidean square of a vector is the sum of the Euclidean squares of its projection to the same complex lines. Therefore it is suﬃcient to prove the opposite to (2.2) in the plane, where it again follows (cid:3) from the standard isoperimetry for m-gons. In view of the previous symmetry results we need to investigate the length of centrally ⊂ R symmetric curves on the boundary of K measured in the norm with unit ball K d. The following result is essentially [12, Theorems 13E and 13F]: ⊂ R ⊂ Lemma 2.4 (Schaﬀer, 1976). If K d is convex and centrally symmetric and γ ∂K is a centrally symmetric closed curve then the length of γ in the norm whose unit ball is − K is at least 4 + 4/d. For odd d the bound can be improved to 4 + 4/(d 1). The example in [12, Theorem 10E] shows that this bound is tight for even d, in the case if interest to us. ##LTLine## ##LTLine##

text: ESTIMATING

text: SYMPLECTIC

text: CAPACITIES

text: FROM.

text: we

text: use

text: the

text: Euclidean

text: norm,

text: not

text: Now

text: assume

text: this

text: attempt

text: to

text: build

text: a

text: no

text: worse

text: w-invariant

text: solution

text: to

text: Clarke’s

text: variational

text: problem

text: fails

text: and

text: we

text: have

text: mA

text: α

text: v

text: A

text: A

text: i

text: m

text: i

text: i

text: i

text: Moreover,

text: assume

text: this

text: attempt

text: fails

text: for

text: all

text: i

text: m.

text: Then

text: summing

text: such

text: inequalities

text: we

text: obtain

label: equation

text: α

text: v

text: mA,

text: m

text: i

text: i

text: and

text: from

text: the

text: quadratic-arithmetic

text: mean

text: inequality

text: α

text: m

text: v

text: α

text: v

text: mA.

text: i

text: m

text: i

text: m

text: i

text: i

text: Note

text: that

text: we

text: can

text: ﬁll

text: the

text: polygonal

text: loop

text: composed

text: of

text: v

text: v

text: by

text: a

text: F

text: of

text: triangles

text: m

text: T

text: T

text: and

text: note

text: that

text: by

text: the

text: calibration

text: property

text: Section

text: I]

text: the

text: Riemannian

text: area

text: m

text: of

text: every

text: such

text: Ti

text: is

text: no

text: less

text: than

text: its

text: action,

text: hence

text: for

text: the

text: area

text: SF

text: of

text: the

text: whole

text: F

text: we

text: have

text: α

text: m

text: v

text: mS

text: i

text: F

text: m

text: i

text: This

text: looks

text: like

text: the

text: inverse

text: isoperimetric

text: inequality.

text: Moreover,

text: we

text: can

text: rotate

text: T

text: T

text: T

text: m

text: into

text: the

text: same

text: one

text: by

text: one

text: and

text: see

text: that

text: this

text: inequality

text: is

text: the

text: opposite

text: to

text: the

text: stan-

text: dard

text: isoperimetric

text: inequality

text: for

text: m-gons

text: in

text: the

text: plane,

text: thus

text: obtaining

text: a

text: contradiction.

text: Therefore,

text: for

text: some

text: i

text: the

text: attempt

text: to

text: produce

text: no

text: worse

text: solution

text: with

text: the

text: required

text: symmetry

text: must

text: succeed.

text: From

text: the

text: existence

text: and

text: uniqueness

text: theorem

text: for

text: the

text: ODE

text: for

text: this

text: variational

text: problem

text: we

text: see

text: that

text: in

text: the

text: smooth

text: case

text: this

text: modiﬁcation

text: of

text: γ

text: must

text: in

text: fact

text: give

text: the

text: same

text: loop,

text: just

text: because

text: the

text: segment

text: γ

text: remains

text: at

text: its

text: place.

text: i

text: An

text: alternative

text: way

text: to

text: deduce

text: Note

text: that

text: the

text: action

text: of

text: a

text: loop

text: is

text: the

text: sum

text: of

text: actions

text: of

text: its

text: projections

text: to

text: the

text: coordinate

text: complex

text: lines.

text: At

text: the

text: same

text: time

text: the

text: Euclidean

text: square

text: of

text: a

text: vector

text: is

text: the

text: sum

text: of

text: the

text: Euclidean

text: squares

text: of

text: its

text: projection

text: to

text: the

text: same

text: complex

text: lines.

text: Therefore

text: it

text: is

text: suﬃcient

text: to

text: prove

text: the

text: opposite

text: to

text: in

text: the

text: plane,

text: where

text: it

text: again

text: follows

text: from

text: the

text: standard

text: isoperimetry

text: for

text: m-gons.

text: In

text: view

text: of

text: the

text: previous

text: symmetry

text: results

text: we

text: need

text: to

text: investigate

text: the

text: length

text: of

text: centrally

text: R

text: symmetric

text: curves

text: on

text: the

text: boundary

text: of

text: K

text: measured

text: in

text: the

text: norm

text: with

text: unit

text: ball

text: K

text: d.

text: The

text: following

text: result

text: is

text: essentially

text: Theorems

text: and

text: R

text: Lemma

text: (Schaﬀer,

text: If

text: K

text: d

text: is

text: convex

text: and

text: centrally

text: symmetric

text: and

text: γ

text: ∂K

text: is

text: a

text: centrally

text: symmetric

text: closed

text: curve

text: then

text: the

text: length

text: of

text: γ

text: in

text: the

text: norm

text: whose

text: unit

text: ball

text: is

text: K

text: is

text: at

text: least

text: For

text: odd

text: d

text: the

text: bound

text: can

text: be

text: improved

text: to

text: The

text: example

text: in

text: Theorem

text: shows

text: that

text: this

text: bound

text: is

text: tight

text: for

text: even

text: d,

text: in

text: the

text: case

text: if

text: interest

text: to

text: us.

text: ##LTLine##

text: ##LTLine##


text: 34 LETEB 1δ 0δ ∗ C.O.P ES.ST.vA DS.tsoP.vA naeM.tsoP.vA C.O.P ES.ST.vA DS.tsoP.vA naeM.tsoP.vA v 1 41000.0 33600.0 57999.0 1 59100.0 58130.0 54200.2 1 43.0 94000.0 56210.0 19779.0 28.0 98200.0 72830.0 18469.1 89.0 53.0 84000.0 18210.0 21779.0 28.0 42300.0 00140.0 07469.1 59.0 43.0 94000.0 89210.0 24779.0 87.0 57300.0 92440.0 30769.1 29.0 LETEBR 1δ 0δ ∗ C.O.P ES.ST.vA DS.tsoP.vA naeM.tsoP.vA C.O.P ES.ST.vA DS.tsoP.vA naeM.tsoP.vA v 1 21000.0 34010.0 37999.0 1 58200.0 51020.0 72200.2 1 89.0 94000.0 14210.0 51899.0 79.0 75200.0 71130.0 61899.1 89.0 99.0 92000.0 35800.0 12999.0 89.0 88300.0 72830.0 93999.1 59.0 89.0 52000.0 05700.0 14999.0 89.0 13300.0 93530.0 36100.2 29.0 seulav detroper ehT .gnittes noisserger raenil eht ni sdohtem LETEBR dna LETEB lanoitidart gnisu dna rof seirammus roiretsoP :4 elbaT 1δ 0δ )ES.ST.vA( rorre dradnats seires emit egareva ,)DS.tsoP.vA( noitaived dradnats roiretsop egareva ,)naeM.tsoP.vA( naem roiretsop egareva era .1 = ∗ dna 2 = ∗ seulav dengised eht fo)C.O.P( egarevoc fo noitroporp dna 1δ 0δ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: freedom parameter, , with shape equal to 10 and rate equal to 5. We give a ﬂat Normal prior with ν mean zero and variance equal to 100 to the parameters and . For RBETEL estimation, the δ0 δ1 prior distribution of the indicator vector, , conditional on the probability parameter, , is shown s v in Section 4.3.4.1. The prior of the probability parameter, , given by (29) is a truncated Beta v distribution and we set the hyperparameters to be = 30 and = 15. We produce 50 000 MCMC α0 β0 , draws from the posterior distribution, and use the ﬁrst 20 000 draws to warm-up the chains and , the subsequent 30 000 draws to make inference. , The inferential results are summarized in Table 6, and Figure 7 provides the plots of the marginal posterior densities for the parameters anbd , respectively. Wbe can see that the RBETEL pos- δ0 δ1 terior mean estimates for the parameters, δ0RBETEL = 2.1482 and δ1RBETEL = 0.7512, are similar to the robust Frequentist estimates produced by M bestimation. The pobsterior mean estimates produced by the standard BETEL method are given by δ0BETEL = 2.2016 and δ1BETEL = 0.6055, which are similar to those estimated by the non-robust OLS method. We ﬁnd that the 95% credible intervals for produced by the two methods considered here have little overlap. This suggests that inference δ1 about the relationship between the variables estimated by these two methods are signiﬁcantly dif- ferent. The parametric Bayesian approach assuming a Student-t distribution for the error term shows some extent of robustness with respect to outliers. The posterior mean estimates produced by this parametric approach for and are 2 1114 and 0 7191, respectively, and they are both δ0 δ1 . . between the corresponding estimates produced by the standard BETEL and RBETEL methods. Notably, however, the RBETEL posteriors here are much sharper than are the posteriors that result from the parametric and the BETEL methodologies. The estimated regression lines with parameter values given by the posterior means are added to the data plot in Figure 8. We can see that the regression line estimated by the RBETEL method ﬁts the majority of the observations well. The red points marked by number 1, 2 and 3 in Figure 8, are the outliers identiﬁed by the RBETEL method, while the corresponding estimated probabilities that these points are ‘good’, i.e. ( = 1), are 0 053, 0 032 and 0 001, respectively. All other P r s . . . i observations have estimated probabilities that they are ‘good’ that are greater than 0 8. Compared . to the robust estimate produced by the RBETEL method, the regression line produced by the traditional BETEL method has an obviously lower slope due to the eﬀect of the leverage points. 36

text: freedom

text: parameter,

text: with

text: shape

text: equal

text: to

text: and

text: rate

text: equal

text: to

text: We

text: give

text: a

text: ﬂat

text: Normal

text: prior

text: with

text: ν

text: mean

text: zero

text: and

text: variance

text: equal

text: to

text: to

text: the

text: parameters

text: and

text: For

text: RBETEL

text: estimation,

text: the

text: δ0

text: δ1

text: prior

text: distribution

text: of

text: the

text: indicator

text: vector,

text: conditional

text: on

text: the

text: probability

text: parameter,

text: is

text: shown

text: s

text: v

text: in

text: Section

text: The

text: prior

text: of

text: the

text: probability

text: parameter,

text: given

text: by

text: is

text: a

text: truncated

text: Beta

text: v

text: distribution

text: and

text: we

text: set

text: the

text: hyperparameters

text: to

text: be

text: and

text: We

text: produce

text: MCMC

text: α0

text: β0

text: draws

text: from

text: the

text: posterior

text: distribution,

text: and

text: use

text: the

text: ﬁrst

text: draws

text: to

text: warm-up

text: the

text: chains

text: and

text: the

text: subsequent

text: draws

text: to

text: make

text: inference.

text: The

text: inferential

text: results

text: are

text: summarized

text: in

text: Table

text: and

text: Figure

text: provides

text: the

text: plots

text: of

text: the

text: marginal

text: posterior

text: densities

text: for

text: the

text: parameters

text: anbd

text: respectively.

text: Wbe

text: can

text: see

text: that

text: the

text: RBETEL

text: pos-

text: δ0

text: δ1

text: terior

text: mean

text: estimates

text: for

text: the

text: parameters,

text: and

text: are

text: similar

text: to

text: the

text: robust

text: Frequentist

text: estimates

text: produced

text: by

text: M

text: bestimation.

text: The

text: pobsterior

text: mean

text: estimates

text: produced

text: by

text: the

text: standard

text: BETEL

text: method

text: are

text: given

text: by

text: and

text: which

text: are

text: similar

text: to

text: those

text: estimated

text: by

text: the

text: non-robust

text: OLS

text: method.

text: We

text: ﬁnd

text: that

text: the

text: credible

text: intervals

text: for

text: produced

text: by

text: the

text: two

text: methods

text: considered

text: here

text: have

text: little

text: overlap.

text: This

text: suggests

text: that

text: inference

text: δ1

text: about

text: the

text: relationship

text: between

text: the

text: variables

text: estimated

text: by

text: these

text: two

text: methods

text: are

text: signiﬁcantly

text: dif-

text: ferent.

text: The

text: parametric

text: Bayesian

text: approach

text: assuming

text: a

text: Student-t

text: distribution

text: for

text: the

text: error

text: term

text: shows

text: some

text: extent

text: of

text: robustness

text: with

text: respect

text: to

text: outliers.

text: The

text: posterior

text: mean

text: estimates

text: produced

text: by

text: this

text: parametric

text: approach

text: for

text: and

text: are

text: and

text: respectively,

text: and

text: they

text: are

text: both

text: δ0

text: δ1

text: between

text: the

text: corresponding

text: estimates

text: produced

text: by

text: the

text: standard

text: BETEL

text: and

text: RBETEL

text: methods.

text: Notably,

text: however,

text: the

text: RBETEL

text: posteriors

text: here

text: are

text: much

text: sharper

text: than

text: are

text: the

text: posteriors

text: that

text: result

text: from

text: the

text: parametric

text: and

text: the

text: BETEL

text: methodologies.

text: The

text: estimated

text: regression

text: lines

text: with

text: parameter

text: values

text: given

text: by

text: the

text: posterior

text: means

text: are

text: added

text: to

text: the

text: data

text: plot

text: in

text: Figure

text: We

text: can

text: see

text: that

text: the

text: regression

text: line

text: estimated

text: by

text: the

text: RBETEL

text: method

text: ﬁts

text: the

text: majority

text: of

text: the

text: observations

text: well.

text: The

text: red

text: points

text: marked

text: by

text: number

text: and

text: in

text: Figure

text: are

text: the

text: outliers

text: identiﬁed

text: by

text: the

text: RBETEL

text: method,

text: while

text: the

text: corresponding

text: estimated

text: probabilities

text: that

text: these

text: points

text: are

text: ‘good’,

text: i.e.

text: are

text: and

text: respectively.

text: All

text: other

text: P

text: r

text: s

text: i

text: observations

text: have

text: estimated

text: probabilities

text: that

text: they

text: are

text: ‘good’

text: that

text: are

text: greater

text: than

text: Compared

text: to

text: the

text: robust

text: estimate

text: produced

text: by

text: the

text: RBETEL

text: method,

text: the

text: regression

text: line

text: produced

text: by

text: the

text: traditional

text: BETEL

text: method

text: has

text: an

text: obviously

text: lower

text: slope

text: due

text: to

text: the

text: eﬀect

text: of

text: the

text: leverage

text: points.


text: Research in Astron. Astrophys. Vol.0 (200x) No.0, 000–000 Research in http://www.raa-journal.org http://www.iop.org/journals/raa Astronomy and Astrophysics Effect of non-stationary accretion on spectral state transitions: An example of a persistent neutron star LMXB 4U 1636-536 H. Zhang1 and W.-F. Yu1 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China; hzhang@shao.ac.cn, zhanghuienator@gmail.com Abstract Observations of the black hole and neutron star X-ray binaries show that the luminosity of the hard-to-soft state transition is usually higher than that of the soft-to- hard state transition, indicating additional parameters other than the mass accretion rate is required to interpret spectral state transitions. It has been found in some individual black hole or neutron star soft X-ray transients that the luminosity corresponding to the hard-to- soft state transition is positively correlated with the peak luminosity of the following soft state. In this work, we report the discovery of the same correlation in a single persistent neutron star low mass X-ray binary (LMXB) 4U1636-536 based on the data from the All Sky Monitor(ASM) on board the RXTE, the Gas Slit Camera (GSC) on board the MAXI and the Burst Alert Telescope (BAT) on board the Swift. We also found such a positive correlation holds in the persistent neutron star LMXB in a luminosity range spanning by about a factor of four. Our results indicates that non-stationary accretion also plays an im- portant role in driving X-ray spectral state transitions in persistent accreting systems with small accretion ﬂares, which is much less dramatic compared with the bright outbursts seen in many Galactic LMXB transients. Key words: accretion, accretion disks — X-rays: binaries — stars: neutron 1 INTRODUCTION Black hole transients usually exhibit different spectral states during their outbursts, namely, the hard state, the soft state and the state between the two, which is called the intermediate state or the very high state (see detailed deﬁnitions in Remilliad & McClintock 2006; Done et al. 2007). Similar to black hole systems, according to the source location in the X-ray color-color diagram, neutron star low mass X-ray binaries also exhibit the hard state and the soft state (Hasinger & van der Klis 1989; Yu et al. 2003; Mun˜oz-Darias et al. 2014). In current popular theoretical models, X-ray spectral states are thought to be determined primarily by the mass accretion rate, and there is a critical threshold of the mass accretion rate – below or above which the accretion states, and therefore the spectral states, will change (e.g. Esin et al. 1997). However, X-ray monitoring observations show that the luminosity corresponding to the hard-to-soft state transition during the rising phase of the outbursts in LMXB tran- sients is usually higher than the luminosity corresponding to the soft-to-hard state transition (Miyamoto et al. 1995; Maccarone & Coppi 2003), and this is also common in neutron star soft X-ray transients (Bouchacourt et al. 1984; Yu et al. 2003; Yu & Dolence 2007). This hysteresis effect indicates that the source can stay in distinct spectral states at the same luminosity (or mass accretion rate). Moreover, the hysteresis effect is mainly driven by the large luminosity range in the hard-to-soft state transition. The luminosity corresponding to the hard-to-soft transition has been observed to vary by several times to ##LTLine## ##LTLine##

text: Research

text: in

text: Astron.

text: Astrophys.

text: Research

text: in

text: Astronomy

text: and

text: Astrophysics

text: Effect

label: title

text: of

text: non-stationary

text: accretion

text: on

text: spectral

text: state

text: transitions:

text: An

text: example

text: of

text: a

text: persistent

text: neutron

text: star

text: LMXB

text: H.

label: author

text: and

text: W.-F.

text: Key

text: Laboratory

text: for

text: Research

text: in

text: Galaxies

text: and

text: Cosmology,

text: Shanghai

text: Astronomical

text: Observatory,

text: Chinese

text: Academy

text: of

text: Sciences,

text: Nandan

text: Road,

text: Shanghai

text: China;

text: hzhang@shao.ac.cn,

text: zhanghuienator@gmail.com

text: Abstract

text: Observations

label: abstract

text: of

text: the

text: black

text: hole

text: and

text: neutron

text: star

text: X-ray

text: binaries

text: show

text: that

text: the

text: luminosity

text: of

text: the

text: hard-to-soft

text: state

text: transition

text: is

text: usually

text: higher

text: than

text: that

text: of

text: the

text: soft-to-

text: hard

text: state

text: transition,

text: indicating

text: additional

text: parameters

text: other

text: than

text: the

text: mass

text: accretion

text: rate

text: is

text: required

text: to

text: interpret

text: spectral

text: state

text: transitions.

text: It

text: has

text: been

text: found

text: in

text: some

text: individual

text: black

text: hole

text: or

text: neutron

text: star

text: soft

text: X-ray

text: transients

text: that

text: the

text: luminosity

text: corresponding

text: to

text: the

text: hard-to-

text: soft

text: state

text: transition

text: is

text: positively

text: correlated

text: with

text: the

text: peak

text: luminosity

text: of

text: the

text: following

text: soft

text: state.

text: In

text: this

text: work,

text: we

text: report

text: the

text: discovery

text: of

text: the

text: same

text: correlation

text: in

text: a

text: single

text: persistent

text: neutron

text: star

text: low

text: mass

text: X-ray

text: binary

text: (LMXB)

text: based

text: on

text: the

text: data

text: from

text: the

text: All

text: Sky

text: Monitor(ASM)

text: on

text: board

text: the

text: RXTE,

text: the

text: Gas

text: Slit

text: Camera

text: (GSC)

text: on

text: board

text: the

text: MAXI

text: and

text: the

text: Burst

text: Alert

text: Telescope

text: (BAT)

text: on

text: board

text: the

text: Swift.

text: We

text: also

text: found

text: such

text: a

text: positive

text: correlation

text: holds

text: in

text: the

text: persistent

text: neutron

text: star

text: LMXB

text: in

text: a

text: luminosity

text: range

text: spanning

text: by

text: about

text: a

text: factor

text: of

text: four.

text: Our

text: results

text: indicates

text: that

text: non-stationary

text: accretion

text: also

text: plays

text: an

text: im-

text: portant

text: role

text: in

text: driving

text: X-ray

text: spectral

text: state

text: transitions

text: in

text: persistent

text: accreting

text: systems

text: with

text: small

text: accretion

text: ﬂares,

text: which

text: is

text: much

text: less

text: dramatic

text: compared

text: with

text: the

text: bright

text: outbursts

text: seen

text: in

text: many

text: Galactic

text: LMXB

text: transients.

text: Key

text: words:

text: accretion,

text: accretion

text: disks

text: X-rays:

text: binaries

text: stars:

text: neutron

text: INTRODUCTION

text: Black

text: hole

text: transients

text: usually

text: exhibit

text: different

text: spectral

text: states

text: during

text: their

text: outbursts,

text: namely,

text: the

text: hard

text: state,

text: the

text: soft

text: state

text: and

text: the

text: state

text: between

text: the

text: two,

text: which

text: is

text: called

text: the

text: intermediate

text: state

text: or

text: the

text: very

text: high

text: state

text: (see

text: detailed

text: deﬁnitions

text: in

text: Remilliad

text: McClintock

text: Done

text: et

text: al.

text: Similar

text: to

text: black

text: hole

text: systems,

text: according

text: to

text: the

text: source

text: location

text: in

text: the

text: X-ray

text: color-color

text: diagram,

text: neutron

text: star

text: low

text: mass

text: X-ray

text: binaries

text: also

text: exhibit

text: the

text: hard

text: state

text: and

text: the

text: soft

text: state

text: (Hasinger

text: van

text: der

text: Klis

text: Yu

text: et

text: al.

text: Mun˜oz-Darias

text: et

text: al.

text: In

text: current

text: popular

text: theoretical

text: models,

text: X-ray

text: spectral

text: states

text: are

text: thought

text: to

text: be

text: determined

text: primarily

text: by

text: the

text: mass

text: accretion

text: rate,

text: and

text: there

text: is

text: a

text: critical

text: threshold

text: of

text: the

text: mass

text: accretion

text: rate

text: below

text: or

text: above

text: which

text: the

text: accretion

text: states,

text: and

text: therefore

text: the

text: spectral

text: states,

text: will

text: change

text: (e.g.

text: Esin

text: et

text: al.

text: However,

text: X-ray

text: monitoring

text: observations

text: show

text: that

text: the

text: luminosity

text: corresponding

text: to

text: the

text: hard-to-soft

text: state

text: transition

text: during

text: the

text: rising

text: phase

text: of

text: the

text: outbursts

text: in

text: LMXB

text: tran-

text: sients

text: is

text: usually

text: higher

text: than

text: the

text: luminosity

text: corresponding

text: to

text: the

text: soft-to-hard

text: state

text: transition

text: (Miyamoto

text: et

text: al.

text: Maccarone

text: Coppi

text: and

text: this

text: is

text: also

text: common

text: in

text: neutron

text: star

text: soft

text: X-ray

text: transients

text: (Bouchacourt

text: et

text: al.

text: Yu

text: et

text: al.

text: Yu

text: Dolence

text: This

text: hysteresis

text: effect

text: indicates

text: that

text: the

text: source

text: can

text: stay

text: in

text: distinct

text: spectral

text: states

text: at

text: the

text: same

text: luminosity

text: (or

text: mass

text: accretion

text: rate).

text: Moreover,

text: the

text: hysteresis

text: effect

text: is

text: mainly

text: driven

text: by

text: the

text: large

text: luminosity

text: range

text: in

text: the

text: hard-to-soft

text: state

text: transition.

text: The

text: luminosity

text: corresponding

text: to

text: the

text: hard-to-soft

text: transition

text: has

text: been

text: observed

text: to

text: vary

text: by

text: several

text: times

text: to

text: ##LTLine##

text: ##LTLine##


text: 6 H. Zhang & W.-F. Yu 0.1 0.01 Hard-to-Soft State Transition Luminosity (L , 15-50 keV) Edd Fig.3 The same as Figure 2 with the unit in luminosity. 5 DISCUSSION The luminosity corresponding to the hard-to-soft state transition can vary by about a factor of 4 and the peak luminosity of the soft state can vary by about a factor of 3 during the ﬂares in 4U 1636-536. Though the outﬂows (wind or jet) will reduce the actual mass accretion rate accreted to the central com- pact object and the radiation efﬁciency may depends on the mass accretion rate for different accretion ﬂow (Narayan et al. 1998) , the variation of the X-ray ﬂux or luminosity can still roughly indicates the variation of the mass accretion rate. Our results therefore indicate that the hard-to-soft state transitions can occur at different mass accretion rates in a large range in 4U 1636-536. This is contrary to theoretical models based on stationary accretion which predicted the occurrence of the hard-to-soft state transitions when the mass accretion rate exceeds a critical mass accretion rate (e.g. Esin et al. 1997). The series of studies of the hard-to-soft spectral state transitions in black hole and neutron star X- ray binaries demonstrate that we are in need of non-stationary accretion models, which is characterized by the consideration of rate-of-change of the mass accretion rate, for the interpretation of spectral state )Vek 01-2 ro 21-2 , L( etatS tfoS fo ytisonimuL kaeP ddE ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: H.

text: Zhang

text: W.-F.

text: Yu

font: default

text: Hard-to-Soft

text: State

text: Transition

text: Luminosity

text: (L

text: keV)

text: Edd

text: The

text: same

text: as

text: Figure

text: with

text: the

text: unit

text: in

text: luminosity.

label: section

text: DISCUSSION

text: The

text: luminosity

text: corresponding

text: to

text: the

text: hard-to-soft

text: state

text: transition

text: can

text: vary

text: by

text: about

text: a

text: factor

text: of

text: and

text: the

text: peak

text: luminosity

text: of

text: the

text: soft

text: state

text: can

text: vary

text: by

text: about

text: a

text: factor

text: of

text: during

text: the

text: ﬂares

text: in

text: Though

text: the

text: outﬂows

text: (wind

text: or

text: jet)

text: will

text: reduce

text: the

text: actual

text: mass

text: accretion

text: rate

text: accreted

text: to

text: the

text: central

text: com-

text: pact

text: object

text: and

text: the

text: radiation

text: efﬁciency

text: may

text: depends

text: on

text: the

text: mass

text: accretion

text: rate

text: for

text: different

text: accretion

text: ﬂow

text: (Narayan

text: et

text: al.

text: the

text: variation

text: of

text: the

text: X-ray

text: ﬂux

text: or

text: luminosity

text: can

text: still

text: roughly

text: indicates

text: the

text: variation

text: of

text: the

text: mass

text: accretion

text: rate.

text: Our

text: results

text: therefore

text: indicate

text: that

text: the

text: hard-to-soft

text: state

text: transitions

text: can

text: occur

text: at

text: different

text: mass

text: accretion

text: rates

text: in

text: a

text: large

text: range

text: in

text: This

text: is

text: contrary

text: to

text: theoretical

text: models

text: based

text: on

text: stationary

text: accretion

text: which

text: predicted

text: the

text: occurrence

text: of

text: the

text: hard-to-soft

text: state

text: transitions

text: when

text: the

text: mass

text: accretion

text: rate

text: exceeds

text: a

text: critical

text: mass

text: accretion

text: rate

text: (e.g.

text: Esin

text: et

text: al.

text: The

text: series

text: of

text: studies

text: of

text: the

text: hard-to-soft

text: spectral

text: state

text: transitions

text: in

text: black

text: hole

text: and

text: neutron

text: star

text: X-

text: ray

text: binaries

text: demonstrate

text: that

text: we

text: are

text: in

text: need

text: of

text: non-stationary

text: accretion

text: models,

text: which

text: is

text: characterized

text: by

text: the

text: consideration

text: of

text: rate-of-change

text: of

text: the

text: mass

text: accretion

text: rate,

text: for

text: the

text: interpretation

text: of

text: spectral

text: state

text: )Vek

text: ro

text: L(

text: etatS

text: tfoS

text: fo

text: ytisonimuL

text: kaeP

text: ddE

text: ##LTFigure##

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text: TILING ENUMERATION OF DOUBLY-INTRUDED HALVED HEXAGONS 13 Theorem 2.9. With the same notations in Theorem 2.8, the number of tilings of the R(2)-type region is given by (2) (2) M(R (a, b)) M(R (a, b)) M(R(2) (a, b)) = x+y,0,z 0,2y,z x,y,z (2) M(R (a, b)) y,0,z − − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 3, b + 2y + z 2, y) − − T(1, 2a + b + 2y + z 1, y) V(2a + 3, b + 2y + z 2, y) = K(a , . . . , a (cid:100) − (cid:101), a (cid:100) − (cid:101) + x + y + b (cid:98) (cid:99), b (cid:98) (cid:99)− , . . . , b ) 1 2 m 1 2 m 1 +1 2 n+1 2 n+1 1 1 × 2 2 2 2 K(0, a , . . . , a (cid:98) (cid:99)− , a (cid:98) (cid:99) + x + y + b (cid:100) − (cid:101) , b (cid:100) − (cid:101), . . . , b , z) 1 2 m+1 1 2 m+1 2 n 1 +1 2 n 1 1 − 2 2 2 2 × (2a 1)!! H (2 e +2 o ) H (2 o +2 e +2z) − 2 a b 2 a b (2a + 2y 1)!! H (2 e +2 o +2y) H (2 o +2 e +2y + 2z) 2 a b 2 a b × H(2a + b + 2y + z) H(b + y + z) H(2a + b + y + z) H(b + z) − − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 3, b + 2y + z 2, y) (2.19) − − . T(1, 2a + b + 2y + z 1, y) V(2a + 3, b + 2y + z 2, y) We are also investigate the weighted versions of the above ‘reversing’ regions with the verti- (i) cal lozenges along the west side weighted by 1/2. Denote the weighted version of R (a, b) by x,y,z (i) RW (a, b), for i = 1, 2. These newly deﬁned regions are illustrated in Figures 2.6(c) and (d), x,y,z respectively, and their tiling numbers are given by simple product formulas as below. Theorem 2.10. For non-negative integers x, y, z and sequences of non-negative integers a = (a , a , . . . , a ) 1 2 m and b = (b , b , . . . , b ), we have 1 2 n (1) (1) M(RW (a, b)) M(RW (a, b)) M(RW (1) (a, b)) = x+y,0,z 0,2y,z x,y,z (1) M(RW (a, b)) y,0,z − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 1, b + 2y + z, y) − T(1, 2a + b + 2y + z 1, y) V(2a + 1, b + 2y + z, y) − (cid:48) = 2a1 2y Q (a , . . . , a (cid:100) − (cid:101), a (cid:100) − (cid:101) + x + y + b (cid:98) (cid:99), b (cid:98) (cid:99)− , . . . , b ) 1 2 m 1 2 m 1 +1 2 n+1 2 n+1 1 1 × (cid:48) 2 2 2 2 Q (0, a , . . . , a (cid:98) (cid:99)− , a (cid:98) (cid:99) + x + y + b (cid:100) − (cid:101) , b (cid:100) − (cid:101), . . . , b , z) 1 2 m+1 1 2 m+1 2 n 1 +1 2 n 1 1 − 2 2 2 2 × (2a + 2y 1)!! H (2 e +2 o +1) H (2 o +2 e +2z + 1) − 2 a b 2 a b (2a 1)!! H (2 e +2 o +2y + 1) H (2 o +2 e +2y + 2z + 1) 2 a b 2 a b × H(2a + b + 2y + z) H(b + y + z) H(2a + b + y + z) H(b + z) − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 1, b + 2y + z, y) (2.20) − . T(1, 2a + b + 2y + z 1, y V(2a + 1, b + 2y + z, y) Theorem 2.11. With the same notations in Theorem 2.10, the number of tilings of the RW (2)-type region is given by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: TILING

text: ENUMERATION

text: OF

text: DOUBLY-INTRUDED

text: HALVED

text: HEXAGONS

text: Theorem

text: With

text: the

text: same

text: notations

text: in

text: Theorem

text: the

text: number

text: of

text: tilings

text: of

text: the

text: region

text: is

text: given

text: by

text: M(R

text: (a,

text: b))

text: M(R

text: (a,

text: b))

text: (a,

text: b))

text: x,y,z

text: M(R

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text: b))

text: T(x

text: b

text: z

text: y)

text: b

text: z

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text: b

text: z

text: y)

text: b

text: z

text: y)

text: K(a

text: a

text: a

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text: y

text: b

text: b

text: b

text: m

text: m

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text: M(RW

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text: Theorem

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text: the

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text: of

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text: RW

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text: given

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text: TILING ENUMERATION OF DOUBLY-INTRUDED HALVED HEXAGONS 27 x+a1 b2 x+a2+b2+b4 2y 2y+2a2+ +2a1+b1 2b3 2a1 a2 b3 b2 z a2 b4 b2 b1 z 2a1 b2 2b4 + 2b2 + +b2 2a2 +2a1 +z+ +z 2y 2y x+a2+b3 x+a1+b1 (a) (b) x+a2+b2 x+a1+b2 2y+2a1+b1 2y+2a2+b1 2a1 b1 a2 b1 a2 b2 b2 2a1 +2b2 2b2 +2a2 2a1+ 2y 2y+ x+a1 b1 x+a2 b1 (c) (d) Figure 4.3. (a)–(b): Eliminating triangles of side length 0 from the b-fern. (c)–(d): Region reduction when z = 0. Q Q (2) (0, a , . . . , a , x + b , . . . , b ) and (a , . . . , a + x, b , . . . , b , z) if m is odd and n is even, Q 1 m n 1 Q 1 m n 1 (3) (0, a , . . . , a + x, b , . . . , b ) and (a , . . . , a , x + b , . . . , b , z) if m is even and n is odd, Q 1 m n 1 Q 1 m n 1 (4) (0, a , . . . , a , x, b , . . . , b ) and (a , . . . , a + x + b , . . . , b , z) if m, n are odd. 1 m n 1 1 m n 1 This means that (4.1) is implied by Lemma 2.3. The veriﬁcation for (4.2) in this case can be treated in the same manner. If b + n = 0, then we have the right fern empty. This case was already considered in [24]. For induction step, we assume that x, y, b + n are all positive and that the our tiling formulas (4.1) and (4.2) hold respectively for any H(1)-type and R(1)-type regions with the sum of the y-, z-, b- and n-parameters strictly less than y + z + b + n. Before obtaining the recurrences for the H(1)-type and R(1)-type regions, we have two important notices as follows. 3b+2a+y 2b+z+y 1b+y 2a+2b+y 1b+y 4b+2b+2a+z+y 1b+2a+y 2b+z+y ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TILING

text: ENUMERATION

text: OF

text: DOUBLY-INTRUDED

text: HALVED

text: HEXAGONS

text: z

text: z

text: +z+

text: +z

text: (a)

text: (b)

text: (c)

text: (d)

text: Figure

text: (a)–(b):

text: Eliminating

text: triangles

text: of

text: side

text: length

text: from

text: the

text: b-fern.

text: (c)–(d):

text: Region

text: reduction

text: when

text: z

text: Q

label: list

text: Q

text: a

text: a

text: x

text: b

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text: (a

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text: b

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text: m

text: is

text: odd

text: and

text: n

text: is

text: even,

text: Q

text: m

text: n

text: Q

text: m

text: n

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text: a

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text: b

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text: z)

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text: m

text: is

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text: and

text: n

text: is

text: odd,

text: Q

text: m

text: n

text: Q

text: m

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text: b

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text: b

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text: if

text: m,

text: n

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text: odd.

text: m

text: n

text: m

text: n

text: This

text: means

text: that

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text: implied

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text: Lemma

text: The

text: veriﬁcation

text: for

text: in

text: this

text: case

text: can

text: be

text: treated

text: in

text: the

text: same

text: manner.

text: If

text: b

text: n

text: then

text: we

text: have

text: the

text: right

text: fern

text: empty.

text: This

text: case

text: was

text: already

text: considered

text: in

text: For

text: induction

text: step,

text: we

text: assume

text: that

text: x,

text: y,

text: b

text: n

text: are

text: all

text: positive

text: and

text: that

text: the

text: our

text: tiling

text: formulas

text: and

text: hold

text: respectively

text: for

text: any

text: and

text: regions

text: with

text: the

text: sum

text: of

text: the

text: y-,

text: z-,

text: b-

text: and

text: n-parameters

text: strictly

text: less

text: than

text: y

text: z

text: b

text: n.

text: Before

text: obtaining

text: the

text: recurrences

text: for

text: the

text: and

text: regions,

text: we

text: have

text: two

text: important

text: notices

text: as

text: follows.

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text: 4 TRI LAI c c a a b b b b a a c c (a) (b) x x j-1 j-1 k a+k k a+k a+k-j+1 a+k-j+1 a a x+k x+k (c) (d) P Figure 2.1. (a) Halved hexagon (with defect) . (b) The weighted halved hexagon P(cid:48) 4,7,3 (with defect) . (c)–(d) The regions in Rohatgi’s paper [30]. 4,7,3 x + a + a 2 4 x + a 2 y y + + a a1 1+ + 2 2a a3 3 a3 a2 a1 z a4 a3 a2 a1 z a2 2 + 2a4 z + + y 2a2 + z + y x + a + a 1 3 (a) x + a + a 1 3 (b) Figure 2.2. Halved hexagons with an array of triangles removed from the non- staircase boundary that were enumerated in [23]. a + a + a + z + y 3 2 1 a + a + a + a + z + y 4 3 2 1 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TRI

text: LAI

text: c

text: c

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text: P

text: Figure

text: (a)

text: Halved

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text: defect)

text: (b)

text: The

text: weighted

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text: (c)–(d)

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text: paper

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text: 34 TRI LAI We now only need to show that the expressions on the right-hand sides of (4.22) and (4.23), denoted by f (a, b) and g (a, b), respectively, satisfy the same recurrence. It is equivalent to show that x,y,z x,y,z f − (a, b) · f − (a, b+1) f − (a, b) · f − − (a, b+1) x,y,z 1 x,y 2,z x+1,y 2,z x 1,y,z 1 (4.26) + + = 1. f (a, b) f − − (a, b+1) f (a, b) f − − (a, b+1) x,y,z x,y 2,z 1 x,y,z x,y 2,z 1 and that g − (a, b) · g − (a, b+1) g − (a, b) · g − − (a, b+1) x,y,z 1 x,y 2,z x+1,y 2,z x 1,y,z 1 (4.27) + + = 1. g (a, b) g − − (a, b+1) g (a, b) g − − (a, b+1) x,y,z x,y 2,z 1 x,y,z x,y 2,z 1 Performing a simpliﬁcation using Lemmas 3.5 and 3.4, (4.26) and (4.26) are both reduced to the following true identity (2x + 2a + b + 2y + z)(a + b + 2y + z) x(x + a) (4.28) + = 1. (x + 2a + b + 2y + z)(x + a + b + 2y + z) (x + 2a + b + 2y + z)(x + a + b + 2y + z) (cid:3) This ﬁnishes the proof. The combined proofs of Theorems 2.5 and 2.9, Theorems 2.6 and 2.10, Theorems 2.7 and 2.11, Theorems 2.13 and 2.17, Theorems 2.14 and 2.18, and Theorems 2.15 and 2.19 are similar and omitted. We conclude this section by presenting the proof of Theorem 2.20. The proof of Theorem 2.21 is essentially the same and is also omitted. Proof of Theorem 2.20. Apply the cutting procedure in Ciucu’s Factorization Theorem (Lemma 3.3) (1) to the dual graph G of the symmetric hexagon with three ferns removed S (a; b), we have x,y,z − − (4.29) M(S(1) (a; b)) = 2y+z+a+b a1 M(G+) M(G ), x,y,z − as there are exactly 2y + 2z + 2a + 2b 2a vertices of G lying on the vertical symmetry axis. 1 We ﬁrst consider the case when a , x, y are all even. It is easy to see that the right component graph 1 G+ is congruent to the dual graph of the left subregion restricted by the bold contour in Figure 4.6(a) (for the case x = y = z = 2, a = 4, a = 2, b = b = 2; the lozenges with shaded core are weighted 1 2 1 2 by 1/2). This region is exactly the weighted region W (2) (a1 , a , . . . , a ; b). The right component x,y,z 2 2 m 2 2 graph corresponds to the dual graph of the right subregion. By removing several forced lozenges on the top and the bottom of this region, we get the region H(2) (a1 , a , . . . , a ; b). This means that (cid:16) (cid:16)x2,y2,z 2 2 m(cid:17)(cid:17) we get (cid:16) (cid:16) (cid:17)(cid:17) − a M(S(1) (a; b)) = 2y+z+a+b a1 M H(2) 1 , a , . . . , a ; b x,y,z x,y,z 2 2 m 2 2 × a (2) 1 (4.30) M W , a , . . . , a ; b . x,y,z 2 2 m 2 2 (cid:18) (cid:19) If a is even, but x and y are odd, then as shown in Figur(cid:16)e 4.6(b), we get (cid:17) 1 − (cid:18) a(cid:16) (cid:17)(cid:19) M(Sx(1,y),z(a; b)) = 2y+z+a+b a1 M H(x2−)1,y+1,z 21 , a2, . . . , am; b 2 2 × a (2) 1 (4.31) M Wx−1,y+1,z 2 , a2, . . . , am; b . 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: TRI

text: LAI

text: We

text: now

text: only

text: need

text: to

text: show

text: that

text: the

text: expressions

text: on

text: the

text: right-hand

text: sides

text: of

text: and

text: denoted

text: by

text: f

text: (a,

text: b)

text: and

text: g

text: (a,

text: b),

text: respectively,

text: satisfy

text: the

text: same

text: recurrence.

text: It

text: is

text: equivalent

text: to

text: show

text: that

text: x,y,z

text: x,y,z

text: f

text: (a,

text: b)

text: f

text: (a,

text: f

text: (a,

text: b)

text: f

text: (a,

text: x,y,z

text: x,y

text: x

text: f

text: (a,

text: b)

text: f

text: (a,

text: f

text: (a,

text: b)

text: f

text: (a,

text: x,y,z

text: x,y

text: x,y,z

text: x,y

text: and

text: that

text: g

text: (a,

text: b)

text: g

text: (a,

text: g

text: (a,

text: b)

text: g

text: (a,

text: x,y,z

text: x,y

text: x

text: g

text: (a,

text: b)

text: g

text: (a,

text: g

text: (a,

text: b)

text: g

text: (a,

text: x,y,z

text: x,y

text: x,y,z

text: x,y

text: Performing

text: a

text: simpliﬁcation

text: using

text: Lemmas

text: and

text: and

text: are

text: both

text: reduced

text: to

text: the

text: following

text: true

text: identity

text: b

text: z)(a

text: b

text: z)

text: x(x

text: a)

text: (x

text: b

text: z)(x

text: a

text: b

text: z)

text: (x

text: b

text: z)(x

text: a

text: b

text: z)

text: This

text: ﬁnishes

text: the

text: proof.

text: The

text: combined

text: proofs

text: of

text: Theorems

text: and

text: Theorems

text: and

text: Theorems

text: and

text: Theorems

text: and

text: Theorems

text: and

text: and

text: Theorems

text: and

text: are

text: similar

text: and

text: omitted.

text: We

text: conclude

text: this

text: section

text: by

text: presenting

text: the

text: proof

text: of

text: Theorem

text: The

text: proof

text: of

text: Theorem

text: is

text: essentially

text: the

text: same

text: and

text: is

text: also

text: omitted.

text: Proof

text: of

text: Theorem

text: Apply

text: the

text: cutting

text: procedure

text: in

text: Ciucu’s

text: Factorization

text: Theorem

text: (Lemma

text: to

text: the

text: dual

text: graph

text: G

text: of

text: the

text: symmetric

text: hexagon

text: with

text: three

text: ferns

text: removed

text: S

text: (a;

text: b),

text: we

text: have

text: x,y,z

text: (a;

text: b))

text: M(G+)

text: M(G

text: x,y,z

text: as

text: there

text: are

text: exactly

text: vertices

text: of

text: G

text: lying

text: on

text: the

text: vertical

text: symmetry

text: axis.

text: We

text: ﬁrst

text: consider

text: the

text: case

text: when

text: a

text: x,

text: y

text: are

text: all

text: even.

text: It

text: is

text: easy

text: to

text: see

text: that

text: the

text: right

text: component

text: graph

text: G+

text: is

text: congruent

text: to

text: the

text: dual

text: graph

text: of

text: the

text: left

text: subregion

text: restricted

text: by

text: the

text: bold

text: contour

text: in

text: Figure

text: (for

text: the

text: case

text: x

text: y

text: z

text: a

text: a

text: b

text: b

text: the

text: lozenges

text: with

text: shaded

text: core

text: are

text: weighted

text: by

text: This

text: region

text: is

text: exactly

text: the

text: weighted

text: region

text: W

text: a

text: a

text: b).

text: The

text: right

text: component

text: x,y,z

text: m

text: graph

text: corresponds

text: to

text: the

text: dual

text: graph

text: of

text: the

text: right

text: subregion.

text: By

text: removing

text: several

text: forced

text: lozenges

text: on

text: the

text: top

text: and

text: the

text: bottom

text: of

text: this

text: region,

text: we

text: get

text: the

text: region

text: a

text: a

text: b).

text: This

text: means

text: that

text: we

text: get

text: a

text: (a;

text: b))

text: M

text: a

text: a

text: b

text: x,y,z

text: x,y,z

text: m

text: a

text: M

text: W

text: a

text: a

text: b

text: x,y,z

text: m

text: If

text: a

text: is

text: even,

text: but

text: x

text: and

text: y

text: are

text: odd,

text: then

text: as

text: shown

text: in

text: we

text: get

text: b))

text: M

text: am;

text: b

text: a

text: M

text: am;

text: b

text: ##LTLine##

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text: 36 TRI LAI (cid:18) (cid:19) − (cid:18) a (cid:18)+ 1 (cid:19)(cid:19) M(Sx(1,y),z(a; b)) = 2y+z+a+b a1 M N(x4−)1,y+1,z( 1 2 , a2, . . . , am; b) 2 2 − × a 1 (1) 1 (4.33) M Nx+1,y−1,z 2 , a2, . . . , am; b . 2 2 (cid:3) 5. Several open questions The ﬁrst equality signs in the formulas of Theorems 2.4–2.19 show some factorizations for the numbers tilings of halved hexagons. It is interesting to ﬁnd a direct explanation for these factorizations. As shown in Theorems 2.20 and 2.21, only eight over sixteen halved hexagons (in particular, the H(2)-, W (1)-, R(2), RW (1)-, N(1)-, N(4)-, NR(1)-, NR(4)-types regions) are really halves of some symmetric hexagons with three ferns removed. How about the other eight? Is there any regions whose halves are corresponding to these remaining regions? References [1] G. E. Andrews, Plane partitions (III): The weak Macdonald conjecture, Invent. Math. 53 (1979), 193–225. [2] M. Ciucu, Enumeration of perfect matchings in graphs with reﬂective symmetry, J. Combin. Theory Ser. A 77 (1997), 67–97. [3] M. Ciucu, Plane partition I: A generalization of MacMahon’s formula, Memoirs of Amer. Math. Soc., 178 (2005), no. 839, 107–144. [4] M. Ciucu, Another dual of MacMahon’s theorem on plane partitions, arXiv:1509.06421. [5] M. Ciucu and C. Krattenthaler, Enumeration of lozenge tilings of hexagons with cut oﬀ corners, J. Combin. Theory Ser. A, 100 (2002), 201–231. [6] M. Ciucu and T. Lai, Lozenge tilings of doubly-intruded hexagons, Preprint http://arxiv.org/abs/1712.08024. [7] M. Ciucu and C. Krattenthaler, A dual of MacMahon’s theorem on plane partitions, Proc. Natl. Acad. Sci. USA 110 (2013), 4518–4523. [8] H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998), 137–165. [9] C. Dodgson, Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetical Values, Proceedings of the Royal Society of London 15 (1866–1867), 150–155. [10] N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings (Part I), J. Algebraic Combin. 1 (1992), 111–132. [11] N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings (Part II), J. Algebraic Combin. 1 (1992), 219–234. [12] I. M. Gessel and X. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985), 300–32. [13] C. Krattenthaler, Advanced determinant calculus, S´eminaire Lotharingien Combin. 42 (“ The Andrews Festschrift”): paper B42q, 1999. [14] C. Krattenthaler, A. J. Guttmann, and X. G. Viennot, Vicious walkers, friendly walkers and Young tableaux II: with a wall, J. Phys. A: Math. Gen. 33 (2000), 8835–8866. [15] C. Koutschan, M. Kauers and D. Zeilberger, A proof of George Andrews’ and David Robbins’ q-TSPP-conjecture, Proc. Natl. Acad. Sci. USA 108 (2011), 2196–2199. [16] E. H. Kuo, Applications of Graphical Condensation for Enumerating Matchings and Tilings, Theor. Comput. Sci. 319 (2004), 29–57. [17] G. Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Combin. Theory Ser. A 68 (1994), 115–151. [18] W. Jockusch and J. Propp, Antisymmetric monotone triangles and domino tilings of quartered Aztec diamonds, Unpublished work. [19] T. Lai, Enumeration of tilings of quartered Aztec rectangles, Electron. J. Combin. 21 (4), #P4.46. [20] T. Lai, A new proof for the number of lozenge tilings of quartered hexagons, Discrete Math 338 (2015), 1866–1872. [21] T. Lai, A q-enumeration of lozenge tilings of a hexagon with three dents, Adv. Applied Math 82 (2017), 23–57. [22] T. Lai, A q-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary, European J. Combin. 64 (2017), 66–87. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TRI

text: LAI

text: a

text: b))

text: M

text: am;

text: b)

text: a

text: M

text: am;

text: b

text: Several

text: open

text: questions

text: The

text: ﬁrst

text: equality

text: signs

text: in

text: the

text: formulas

text: of

text: Theorems

text: show

text: some

text: factorizations

text: for

text: the

text: numbers

text: tilings

text: of

text: halved

text: hexagons.

text: It

text: is

text: interesting

text: to

text: ﬁnd

text: a

text: direct

text: explanation

text: for

text: these

text: factorizations.

text: As

text: shown

text: in

text: Theorems

text: and

text: only

text: eight

text: over

text: sixteen

text: halved

text: hexagons

text: (in

text: particular,

text: the

text: W

text: RW

text: regions)

text: are

text: really

text: halves

text: of

text: some

text: symmetric

text: hexagons

text: with

text: three

text: ferns

text: removed.

text: How

text: about

text: the

text: other

text: eight?

text: Is

text: there

text: any

text: regions

text: whose

text: halves

text: are

text: corresponding

text: to

text: these

text: remaining

text: regions?

text: References

text: G.

text: E.

text: Andrews,

text: Plane

text: partitions

text: (III):

text: The

text: weak

text: Macdonald

text: conjecture,

text: Invent.

text: Math.

text: M.

text: Ciucu,

text: Enumeration

text: of

text: perfect

text: matchings

text: in

text: graphs

text: with

text: reﬂective

text: symmetry,

text: J.

text: Combin.

text: Theory

text: Ser.

text: A

text: M.

text: Ciucu,

text: Plane

text: partition

text: I:

text: A

text: generalization

text: of

text: MacMahon’s

text: formula,

text: Memoirs

text: of

text: Amer.

text: Math.

text: Soc.,

text: no.

text: M.

text: Ciucu,

text: Another

text: dual

text: of

text: MacMahon’s

text: theorem

text: on

text: plane

text: partitions,

text: M.

text: Ciucu

text: and

text: C.

text: Krattenthaler,

text: Enumeration

text: of

text: lozenge

text: tilings

text: of

text: hexagons

text: with

text: cut

text: oﬀ

text: corners,

text: J.

text: Combin.

text: Theory

text: Ser.

text: A,

text: M.

text: Ciucu

text: and

text: T.

text: Lai,

text: Lozenge

text: tilings

text: of

text: doubly-intruded

text: hexagons,

text: Preprint

text: M.

text: Ciucu

text: and

text: C.

text: Krattenthaler,

text: A

text: dual

text: of

text: MacMahon’s

text: theorem

text: on

text: plane

text: partitions,

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: H.

text: Cohn,

text: M.

text: Larsen,

text: and

text: J.

text: Propp,

text: The

text: shape

text: of

text: a

text: typical

text: boxed

text: plane

text: partition,

text: New

text: York

text: J.

text: Math.

text: C.

text: Dodgson,

text: Condensation

text: of

text: Determinants,

text: Being

text: a

text: New

text: and

text: Brief

text: Method

text: for

text: Computing

text: their

text: Arithmetical

text: Values,

text: Proceedings

text: of

text: the

text: Royal

text: Society

text: of

text: London

text: N.

text: Elkies,

text: G.

text: Kuperberg,

text: M.

text: Larsen,

text: and

text: J.

text: Propp,

text: Alternating-sign

text: matrices

text: and

text: domino

text: tilings

text: (Part

text: I),

text: J.

text: Algebraic

text: Combin.

text: N.

text: Elkies,

text: G.

text: Kuperberg,

text: M.

text: Larsen,

text: and

text: J.

text: Propp,

text: Alternating-sign

text: matrices

text: and

text: domino

text: tilings

text: (Part

text: II),

text: J.

text: Algebraic

text: Combin.

text: I.

text: M.

text: Gessel

text: and

text: X.

text: Viennot,

text: Binomial

text: determinants,

text: paths,

text: and

text: hook

text: length

text: formulae,

text: Adv.

text: Math.

text: C.

text: Krattenthaler,

text: Advanced

text: determinant

text: calculus,

text: S´eminaire

text: Lotharingien

text: Combin.

text: The

text: Andrews

text: Festschrift”):

text: paper

text: C.

text: Krattenthaler,

text: A.

text: J.

text: Guttmann,

text: and

text: X.

text: G.

text: Viennot,

text: Vicious

text: walkers,

text: friendly

text: walkers

text: and

text: Young

text: tableaux

text: II:

text: with

text: a

text: wall,

text: J.

text: Phys.

text: A:

text: Math.

text: Gen.

text: C.

text: Koutschan,

text: M.

text: Kauers

text: and

text: D.

text: Zeilberger,

text: A

text: proof

text: of

text: George

text: Andrews’

text: and

text: David

text: Robbins’

text: q-TSPP-conjecture,

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: E.

text: H.

text: Kuo,

text: Applications

text: of

text: Graphical

text: Condensation

text: for

text: Enumerating

text: Matchings

text: and

text: Tilings,

text: Theor.

text: Comput.

text: Sci.

text: G.

text: Kuperberg,

text: Symmetries

text: of

text: plane

text: partitions

text: and

text: the

text: permanent-determinant

text: method,

text: J.

text: Combin.

text: Theory

text: Ser.

text: A

text: W.

text: Jockusch

text: and

text: J.

text: Propp,

text: Antisymmetric

text: monotone

text: triangles

text: and

text: domino

text: tilings

text: of

text: quartered

text: Aztec

text: diamonds,

text: Unpublished

text: work.

text: T.

text: Lai,

text: Enumeration

text: of

text: tilings

text: of

text: quartered

text: Aztec

text: rectangles,

text: Electron.

text: J.

text: Combin.

text: T.

text: Lai,

text: A

text: new

text: proof

text: for

text: the

text: number

text: of

text: lozenge

text: tilings

text: of

text: quartered

text: hexagons,

text: Discrete

text: Math

text: T.

text: Lai,

text: A

text: q-enumeration

text: of

text: lozenge

text: tilings

text: of

text: a

text: hexagon

text: with

text: three

text: dents,

text: Adv.

text: Applied

text: Math

text: T.

text: Lai,

text: A

text: q-enumeration

text: of

text: lozenge

text: tilings

text: of

text: a

text: hexagon

text: with

text: four

text: adjacent

text: triangles

text: removed

text: from

text: the

text: boundary,

text: European

text: J.

text: Combin.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 TRI LAI Theorem 2.5. For non-negative integers x, y, z and sequences of non-negative integers a = (a , a , . . . , a ) 1 2 m and b = (b , b , . . . , b ) 1 2 n (2) (2) M(H (a, b)) M(H (a, b)) M(H(2) (a, b)) = x+y,0,z 0,2y,z x,y,z (2) M(H (a, b)) y,0,z − − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 3, b + 2y + z 2, y) − − T(1, 2a + b + 2y + z 1, y) V(2a + 3, b + 2y + z 2, y) = K(0, a , . . . , a (cid:98) (cid:99)− , a (cid:98) (cid:99) + x + y + b (cid:98) (cid:99), b (cid:98) (cid:99)− , . . . , b ) 1 2 m+1 1 2 m+1 2 n+1 2 n+1 1 1 × 2 2 2 2 K(a , . . . , a(cid:100) − (cid:101), a(cid:100) − (cid:101) + x + y + b(cid:100) − (cid:101) , b(cid:100) − (cid:101), . . . , b , z) 1 m 1 m 1 +1 n 1 +1 n 1 1 − 2 2 2 2 × (2a 1)!! H (2 o +2 o ) H (2 e +2 e +2z) − 2 a b 2 a b (2a + 2y 1)!! H (2 o +2 o +2y) H (2 e +2 e +2y + 2z) 2 a b 2 a b × H(2a + b + 2y + z) H(b + y + z) H(2a + b + y + z) H(b + z) − − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 3, b + 2y + z 2, y) (2.15) − − . T(1, 2a + b + 2y + z 1, y) V(2a + 3, b + 2y + z 2, y) Similar to the case of the halved hexagons, we would like to enumerate tilings of the weighted version W (1) (a; b) and W (2) (a; b) of the above two H(1)- and H(2)-type regions by assigning to x,y,z x,y,z each vertical lozenge along the west side a weight 1/2 (see the lozenges with shaded cores in Figures 2.5 (c) and (d), respectively). The weighted numbers of tilings of these two newly deﬁned regions are also given by closed-form products. Theorem 2.6. For non-negative integers x, y, z and sequences of non-negative integers a = (a , a , . . . , a ) 1 2 m and b = (b , b , . . . , b ), we have 1 2 n (1) (1) M(W (a, b)) M(W (a, b)) M(W (1) (a, b)) = x+y,0,z 0,2y,z x,y,z (1) M(W (a, b)) y,0,z − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 1, b + 2y + z, y) − T(1, 2a + b + 2y + z 1, y) V(2a + 1, b + 2y + z, y) − (cid:48) = 2 2y+a1 Q (0, a , . . . , a (cid:98) (cid:99)− , a (cid:98) (cid:99) + x + y + b (cid:98) (cid:99), b (cid:98) (cid:99)− , . . . , b ) 1 2 m+1 1 2 m+1 2 n+1 2 n+1 1 1 × (cid:48) 2 2 2 2 Q (a , . . . , a (cid:100) − (cid:101), a (cid:100) − (cid:101) + x + y + b (cid:100) − (cid:101) , b (cid:100) − (cid:101), . . . , b , z) 1 2 m 1 2 m 1 +1 2 n 1 +1 2 n 1 1 − 2 2 2 2 × (2a + 2y 1)!! H (2 o +2 o +1) H (2 e +2 e +2z + 1) − 2 a b 2 a b (2a 1)!! H (2 o +2 o +2y + 1) H (2 e +2 e +2y + 2z + 1) 2 a b 2 a b × H(2a + b + 2y + z) H(b + y + z) H(2a + b + y + z) H(b + z) − × T(x + 1, 2a + b + 2y + z 1, y) V(2x + 2a + 1, b + 2y + z, y) (2.16) − . T(1, 2a + b + 2y + z 1, y) V(2a + 1, b + 2y + z, y) Theorem 2.7. With the same notations in Theorem 2.6, the number of tilings of the W (2)-type region is given by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: TRI

text: LAI

text: Theorem

text: For

text: non-negative

text: integers

text: x,

text: y,

text: z

text: and

text: sequences

text: of

text: non-negative

text: integers

text: a

text: (a

text: a

text: a

text: m

text: and

text: b

text: (b

text: b

text: b

text: n

text: M(H

text: (a,

text: b))

text: M(H

text: (a,

text: b))

text: (a,

text: b))

text: x,y,z

text: M(H

text: (a,

text: b))

text: T(x

text: b

text: z

text: y)

text: b

text: z

text: y)

text: b

text: z

text: y)

text: b

text: z

text: y)

text: a

text: a

text: a

text: x

text: y

text: b

text: b

text: b

text: K(a

text: x

text: y

text: b

text: z)

text: m

text: m

text: n

text: n

text: H

text: o

text: o

text: H

text: e

text: e

text: a

text: b

text: a

text: b

text: H

text: o

text: o

text: H

text: e

text: e

text: a

text: b

text: a

text: b

text: b

text: z)

text: H(b

text: y

text: z)

text: b

text: y

text: z)

text: H(b

text: z)

text: T(x

text: b

text: z

text: y)

text: b

text: z

text: y)

text: b

text: z

text: y)

text: b

text: z

text: y)

text: Similar

text: to

text: the

text: case

text: of

text: the

text: halved

text: hexagons,

text: we

text: would

text: like

text: to

text: enumerate

text: tilings

text: of

text: the

text: weighted

text: version

text: W

text: (a;

text: b)

text: and

text: W

text: (a;

text: b)

text: of

text: the

text: above

text: two

text: and

text: regions

text: by

text: assigning

text: to

text: x,y,z

text: x,y,z

text: each

text: vertical

text: lozenge

text: along

text: the

text: west

text: side

text: a

text: weight

text: (see

text: the

text: lozenges

text: with

text: shaded

text: cores

text: in

text: Figures

text: (c)

text: and

text: (d),

text: respectively).

text: The

text: weighted

text: numbers

text: of

text: tilings

text: of

text: these

text: two

text: newly

text: deﬁned

text: regions

text: are

text: also

text: given

text: by

text: closed-form

text: products.

text: Theorem

text: For

text: non-negative

text: integers

text: x,

text: y,

text: z

text: and

text: sequences

text: of

text: non-negative

text: integers

text: a

text: (a

text: a

text: a

text: m

text: and

text: b

text: (b

text: b

text: b

text: we

text: have

text: n

text: M(W

text: (a,

text: b))

text: M(W

text: (a,

text: b))

text: M(W

text: (a,

text: b))

text: x,y,z

text: M(W

text: (a,

text: b))

text: T(x

text: b

text: z

text: y)

text: b

text: z,

text: y)

text: b

text: z

text: y)

text: b

text: z,

text: y)

text: Q

text: a

text: a

text: a

text: x

text: y

text: b

text: b

text: b

text: Q

text: (a

text: a

text: a

text: x

text: y

text: b

text: b

text: b

text: z)

text: m

text: m

text: n

text: n

text: H

text: o

text: o

text: H

text: e

text: e

text: a

text: b

text: a

text: b

text: H

text: o

text: o

text: H

text: e

text: e

text: a

text: b

text: a

text: b

text: b

text: z)

text: H(b

text: y

text: z)

text: b

text: y

text: z)

text: H(b

text: z)

text: T(x

text: b

text: z

text: y)

text: b

text: z,

text: y)

text: b

text: z

text: y)

text: b

text: z,

text: y)

text: Theorem

text: With

text: the

text: same

text: notations

text: in

text: Theorem

text: the

text: number

text: of

text: tilings

text: of

text: the

text: W

text: region

text: is

text: given

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: 2.4 Taxes analysis PolicySpace contains four alternative ways to distribute the taxes collected. They are implemented using two parameters: Alternative0 and FPM distribution. The default conﬁguration has both of them set as true. Alternative0 refers to the fact that municipalities are autonomous and receive taxes according to the rules in place. When false, this parameter implies that all of the munici- palities of a given ACP are considered as one for ﬁscal distribution purposes. This hypothesis is supported by the fact that larger municipalities are more ef- ﬁcient in their spending [16] and receive a larger proportion of taxes per capita [14]. Further, less fragmentation of municipalities leads to more productivity [2]. FPM distribution in turn sets the rules of FPM (a municipal tax transfer) in eﬀect or not. The combination of the four alternatives and the criteria for distribution along with the allocated percentage of taxes can be seen from 1. Options Alt0:True Alt0:False Alt0:True Alt0:False FPM:True FPM:True FPM:False FPM:False Taxes Mun. State FPM State FPM Mun. State Consumption .1875 .8125 1 1 1 Labor .765 .235 .765 .235 1 1 Transaction 1 1 1 1 Firms .765 .235 .765 .235 1 1 Property 1 1 1 1 Criteria Locally Equally FPM Equally FPM Locally Equally Table 1. Alternative distributions of collected taxes among municipalities. Depending on the set of rules, taxes are distributed equally among all municipalities, locally, so that a municipality receives only what was collected within the own municipality, according to FPM transfer rules, or a combination of them. 2.5 Running PolicySpace As the idea of the paper is also to present the platform, we describe the basics of running PolicySpace GitHub.com/BAFurtado/PolicySpace. Once installed, you can run the web version with the command: python main.py web and open the model in a browser: http://127.0.0.1:5000/. 1. Run type: four alternatives are available: (a) The ﬁrst: ‘run’ is the running of a single model, for a single ACP, a single conﬁguration and plotting and saving the results. (b) The option ‘sensitivity’ automatically varies any of the parameters. If they are booleans, only the name of the parameter will suﬃce. However, when the parameter is a quantity, the modeler needs to set the ﬁrst value, the last value, and the number of times the parameter is going to ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Taxes

text: analysis

text: PolicySpace

text: contains

text: four

text: alternative

text: ways

text: to

text: distribute

text: the

text: taxes

text: collected.

text: They

text: are

text: implemented

text: using

text: two

text: parameters:

text: and

text: FPM

text: distribution.

text: The

text: default

text: conﬁguration

text: has

text: both

text: of

text: them

text: set

text: as

text: true.

text: refers

text: to

text: the

text: fact

text: that

text: municipalities

text: are

text: autonomous

text: and

text: receive

text: taxes

text: according

text: to

text: the

text: rules

text: in

text: place.

text: When

text: false,

text: this

text: parameter

text: implies

text: that

text: all

text: of

text: the

text: munici-

text: palities

text: of

text: a

text: given

text: ACP

text: are

text: considered

text: as

text: one

text: for

text: ﬁscal

text: distribution

text: purposes.

text: This

text: hypothesis

text: is

text: supported

text: by

text: the

text: fact

text: that

text: larger

text: municipalities

text: are

text: more

text: ef-

text: ﬁcient

text: in

text: their

text: spending

text: and

text: receive

text: a

text: larger

text: proportion

text: of

text: taxes

text: per

text: capita

text: Further,

text: less

text: fragmentation

text: of

text: municipalities

text: leads

text: to

text: more

text: productivity

text: FPM

text: distribution

text: in

text: turn

text: sets

text: the

text: rules

text: of

text: FPM

text: (a

text: municipal

text: tax

text: transfer)

text: in

text: eﬀect

text: or

text: not.

text: The

text: combination

text: of

text: the

text: four

text: alternatives

text: and

text: the

text: criteria

text: for

text: distribution

text: along

text: with

text: the

text: allocated

text: percentage

text: of

text: taxes

text: can

text: be

text: seen

text: from

text: Options

text: FPM:True

text: FPM:True

text: FPM:False

text: FPM:False

text: Taxes

text: Mun.

text: State

text: FPM

text: State

text: FPM

text: Mun.

text: State

text: Consumption

text: Labor

text: Transaction

text: Firms

text: Property

text: Criteria

text: Locally

text: Equally

text: FPM

text: Equally

text: FPM

text: Locally

text: Equally

text: Table

text: Alternative

text: distributions

text: of

text: collected

text: taxes

text: among

text: municipalities.

text: Depending

text: on

text: the

text: set

text: of

text: rules,

text: taxes

text: are

text: distributed

text: equally

text: among

text: all

text: municipalities,

text: locally,

text: so

text: that

text: a

text: municipality

text: receives

text: only

text: what

text: was

text: collected

text: within

text: the

text: own

text: municipality,

text: according

text: to

text: FPM

text: transfer

text: rules,

text: or

text: a

text: combination

text: of

text: them.

text: Running

text: PolicySpace

text: As

text: the

text: idea

text: of

text: the

text: paper

text: is

text: also

text: to

text: present

text: the

text: platform,

text: we

text: describe

text: the

text: basics

text: of

text: running

text: PolicySpace

text: GitHub.com/BAFurtado/PolicySpace.

text: Once

text: installed,

text: you

text: can

text: run

text: the

text: web

text: version

text: with

text: the

text: command:

text: python

text: main.py

text: web

text: and

text: open

text: the

text: model

text: in

text: a

text: browser:

text: Run

text: type:

text: four

text: alternatives

text: are

text: available:

text: (a)

text: The

text: ﬁrst:

text: ‘run’

text: is

text: the

text: running

text: of

text: a

text: single

text: model,

text: for

text: a

text: single

text: ACP,

text: a

text: single

text: conﬁguration

text: and

text: plotting

text: and

text: saving

text: the

text: results.

text: (b)

text: The

text: option

text: ‘sensitivity’

text: automatically

text: varies

text: any

text: of

text: the

text: parameters.

text: If

text: they

text: are

text: booleans,

text: only

text: the

text: name

text: of

text: the

text: parameter

text: will

text: suﬃce.

text: However,

text: when

text: the

text: parameter

text: is

text: a

text: quantity,

text: the

text: modeler

text: needs

text: to

text: set

text: the

text: ﬁrst

text: value,

text: the

text: last

text: value,

text: and

text: the

text: number

text: of

text: times

text: the

text: parameter

text: is

text: going

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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