text: 6 TABLE I. Datasets used in the quantiﬁcation of the calibration uncertainty. CT data was obtained with the 80/140Sn kVp voltage combination of a Siemens Somatom Definition Flash CT scanner. Diﬀerent container phantoms for the inserts made of solid water (SW) or acrylic glass (PMMA) with diameters typical for head (H) or abdomen (A) were used. The listed parameters were tested individually for statistical signiﬁcance of their inﬂuence on the variation in α by analysis of variance (ANOVA) (p-values listed below the parameter name). Only the bone-tissue substitutes of the respective phantom were used for the regression, while additional inclusion of soft-tissue substitutes did not alter the resulting α in the four signiﬁcant digits shown. ID author inserts container phantom reconstruction α vendor model material diameter kernel slice p = 0.37 p = 0.50 p = 0.75 p = 0.44 p = 0.007* p = 0.32 p = 0.77 ∅ ≈ C1 [20] CIRS 062 SW 30 cm (A) D30 5 mm 1.465 ∅ C2 [20] CIRS 062 SW 18 cm (H) D30 5 mm 1.433 ∅ C3 [5] Gammex 467 PMMA 16 cm (H) D30 2 mm 1.442 ∅ C4 [21] Gammex 465 SW 33 cm (A) B10 3 mm 1.472 ∅ C5 [12] Gammex 467 SW 33 cm (A) n/a 3 mm 1.475 (cid:98) TABLE II. Selected published algorithms for image-based determination of relative electron density, n, and/or eﬀective atomic number, Z . M1 and M2 use the alpha-blending equation 1 and thus do not require a parameterization for the cross section eﬀ (parameterization in M2 only for Z ). M3-6 use parameterizations, which do not respect the condition of equation A3. Some eﬀ (cid:98) methods require the input of x-ray spectra (‘spec’), while others are based on calibration (‘cal’). (cid:98) ID reference n(cid:98)/Zeﬀ kind parameterization M1 [11] n (cid:98) cal - → M2 [5] n Z cal a(E) + b(E)Zm (only for Z ) →eﬀ (cid:80) eﬀ M3 [10] Zeﬀ n(cid:98) spec Z4F(E,Z) + G(E,Z) M4 [12] Zeﬀ → (cid:98) cal [3] M5 [13] Z n cal a Zk eﬀ → k k M6 [14] Z n spec [22] eﬀ B. Calibration uncertainty The blending parameters obtained from published data sets show very little variation, in spite of a variety of experimental conﬁgurations and CT protocols used (table I). With a mean and(cid:98)standard deviation of α = 1.457±0.017, the relative uncertainty in α is estimated as u(α)/α = 1.2%. Deﬁning ∆max = max |1 − σ(cid:98) | for a given tissue σl i l,i gr(cid:98)oup, an upper limit of the calibration uncertainty can be calculated using equation 5. For soft (bony) tissue with ∆max = 0.05(0.7), we get u (n(cid:98))/n(cid:98) < 0.06%(0.8%). σl calib Of the parameters listed in table I, only the phantom diameter (head vs. abdomen) showed a signiﬁcant impact on the variability in α (p = 0.007), with the head setup leading to smaller α-values than the abdomen setup. This is most likely explained by beam hardening eﬀects, as no beam hardening correction for bones was applied in the cited studies. IV. DISCUSSION A. Methodological uncertainty The methodological uncertainty, u (n(cid:98))/n(cid:98) < 0.15%, for tissue excl. thyroid and the error, δ(n(cid:98))/n(cid:98) ≈ 0.5% for meth thyroid are very similar to the ones of the ‘stoichiometric method’ in [13]. Their theoretical analysis resulted in a maximum absolute deviation of determined from reference electron densities of 0.2% for reference human tissues excl. thyroid and a positive deviation of 0.6% for thyroid (ﬁgure 6 in Bourque et al.[13]). Furthermore, the distribution of residuals in ﬁgure 1B is remarkably similar to the residuals observed in a ﬁtting step of the ratio of attenuation coeﬃcients to Z in ﬁgure 7 of Landry et al.[12]. These similarities are made plausible by the fact that these eﬀ approaches use a single energy-independent eﬀective atomic number, which requires a cross section parameterization of the type of equation A3 (appendix ). From this parameterization, equation 1 can be rigorously derived (appendix ). Every approach, in which an eﬀective atomic number is deﬁned (table II, M3-6) thus implicitly recognizes the validity of equation 1. It is natural that such an algorithm would include a ﬁtting step at some point with residuals to reference tissues that are (in the best case) of similar structure and magnitude as shown in ﬁgure 1B. This also means that the use of a parameterization not respecting the form of equation A3 to determine a unique Z (as in eﬀ M3-6) is inconsistent. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I.

text: Datasets

text: used

text: in

text: the

text: quantiﬁcation

text: of

text: the

text: calibration

text: uncertainty.

text: CT

text: data

text: was

text: obtained

text: with

text: the

text: kVp

text: voltage

text: combination

text: of

text: a

text: Siemens

text: Somatom

text: Definition

text: Flash

text: CT

text: scanner.

text: Diﬀerent

text: container

text: phantoms

text: for

text: the

text: inserts

text: made

text: of

text: solid

text: water

text: (SW)

text: or

text: acrylic

text: glass

text: (PMMA)

text: with

text: diameters

text: typical

text: for

text: head

text: (H)

text: or

text: abdomen

text: (A)

text: were

text: used.

text: The

text: listed

text: parameters

text: were

text: tested

text: individually

text: for

text: statistical

text: signiﬁcance

text: of

text: their

text: inﬂuence

text: on

text: the

text: variation

text: in

text: α

text: by

text: analysis

text: of

text: variance

text: (ANOVA)

text: (p-values

text: listed

text: below

text: the

text: parameter

text: name).

text: Only

text: the

text: bone-tissue

text: substitutes

text: of

text: the

text: respective

text: phantom

text: were

text: used

text: for

text: the

text: regression,

text: while

text: additional

text: inclusion

text: of

text: soft-tissue

text: substitutes

text: did

text: not

text: alter

text: the

text: resulting

text: α

text: in

text: the

text: four

text: signiﬁcant

text: digits

text: shown.

text: ID

text: author

text: inserts

text: container

text: phantom

text: reconstruction

text: α

text: vendor

text: model

text: material

text: diameter

text: kernel

text: slice

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: CIRS

text: SW

text: cm

text: (A)

text: mm

text: CIRS

text: SW

text: cm

text: (H)

text: mm

text: Gammex

text: PMMA

text: cm

text: (H)

text: mm

text: Gammex

text: SW

text: cm

text: (A)

text: mm

text: Gammex

text: SW

text: cm

text: (A)

text: mm

text: TABLE

text: II.

text: Selected

text: published

text: algorithms

text: for

text: image-based

text: determination

text: of

text: relative

text: electron

text: density,

text: n,

text: and/or

text: eﬀective

text: atomic

text: number,

text: Z

text: and

text: use

text: the

text: alpha-blending

text: equation

text: and

text: thus

text: do

text: not

text: require

text: a

text: parameterization

text: for

text: the

text: cross

text: section

text: eﬀ

text: (parameterization

text: in

text: only

text: for

text: Z

text: use

text: parameterizations,

text: which

text: do

text: not

text: respect

text: the

text: condition

text: of

text: equation

text: Some

text: eﬀ

text: methods

text: require

text: the

text: input

text: of

text: x-ray

text: spectra

text: (‘spec’),

text: while

text: others

text: are

text: based

text: on

text: calibration

text: (‘cal’).

text: ID

text: reference

text: kind

text: parameterization

text: n

text: cal

text: n

text: Z

text: cal

text: a(E)

text: b(E)Zm

text: (only

text: for

text: Z

text: →eﬀ

text: eﬀ

text: Zeﬀ

text: spec

text: G(E,Z)

text: Zeﬀ

text: cal

text: Z

text: n

text: cal

text: a

text: Zk

text: eﬀ

text: k

text: k

text: Z

text: n

text: spec

text: eﬀ

text: B.

text: Calibration

text: uncertainty

text: The

text: blending

text: parameters

text: obtained

text: from

text: published

text: data

text: sets

text: show

text: very

text: little

text: variation,

text: in

text: spite

text: of

text: a

text: variety

text: of

text: experimental

text: conﬁgurations

text: and

text: CT

text: protocols

text: used

text: (table

text: I).

text: With

text: a

text: mean

text: deviation

text: of

text: α

text: the

text: relative

text: uncertainty

text: in

text: α

text: is

text: estimated

text: as

text: u(α)/α

text: Deﬁning

text: ∆max

text: max

text: for

text: a

text: given

text: tissue

text: σl

text: i

text: l,i

text: an

text: upper

text: limit

text: of

text: the

text: calibration

text: uncertainty

text: can

text: be

text: calculated

text: using

text: equation

text: For

text: soft

text: (bony)

text: tissue

text: with

text: ∆max

text: we

text: get

text: u

text: σl

text: calib

text: Of

text: the

text: parameters

text: listed

text: in

text: table

text: I,

text: only

text: the

text: phantom

text: diameter

text: (head

text: vs.

text: abdomen)

text: showed

text: a

text: signiﬁcant

text: impact

text: on

text: the

text: variability

text: in

text: α

text: (p

text: with

text: the

text: head

text: setup

text: leading

text: to

text: smaller

text: α-values

text: than

text: the

text: abdomen

text: setup.

text: This

text: is

text: most

text: likely

text: explained

text: by

text: beam

text: hardening

text: eﬀects,

text: as

text: no

text: beam

text: hardening

text: correction

text: for

text: bones

text: was

text: applied

text: in

text: the

text: cited

text: studies.

text: IV.

text: DISCUSSION

text: A.

text: Methodological

text: uncertainty

text: The

text: methodological

text: uncertainty,

text: u

text: for

text: tissue

text: excl.

text: thyroid

text: and

text: the

text: error,

text: for

text: meth

text: thyroid

text: are

text: very

text: similar

text: to

text: the

text: ones

text: of

text: the

text: ‘stoichiometric

text: method’

text: in

text: Their

text: theoretical

text: analysis

text: resulted

text: in

text: a

text: maximum

text: absolute

text: deviation

text: of

text: determined

text: from

text: reference

text: electron

text: densities

text: of

text: for

text: reference

text: human

text: tissues

text: excl.

text: thyroid

text: and

text: a

text: positive

text: deviation

text: of

text: for

text: thyroid

text: (ﬁgure

text: in

text: Bourque

text: et

text: Furthermore,

text: the

text: distribution

text: of

text: residuals

text: in

text: ﬁgure

text: is

text: remarkably

text: similar

text: to

text: the

text: residuals

text: observed

text: in

text: a

text: ﬁtting

text: step

text: of

text: the

text: ratio

text: of

text: attenuation

text: coeﬃcients

text: to

text: Z

text: in

text: ﬁgure

text: of

text: Landry

text: et

text: These

text: similarities

text: are

text: made

text: plausible

text: by

text: the

text: fact

text: that

text: these

text: eﬀ

text: approaches

text: use

text: a

text: single

text: energy-independent

text: eﬀective

text: atomic

text: number,

text: which

text: requires

text: a

text: cross

text: section

text: parameterization

text: of

text: the

text: type

text: of

text: equation

text: (appendix

text: From

text: this

text: parameterization,

text: equation

text: can

text: be

text: rigorously

text: derived

text: (appendix

text: Every

text: approach,

text: in

text: which

text: an

text: eﬀective

text: atomic

text: number

text: is

text: deﬁned

text: (table

text: II,

text: thus

text: implicitly

text: recognizes

text: the

text: validity

text: of

text: equation

text: It

text: is

text: natural

text: that

text: such

text: an

text: algorithm

text: would

text: include

text: a

text: ﬁtting

text: step

text: at

text: some

text: point

text: with

text: residuals

text: to

text: reference

text: tissues

text: that

text: are

text: (in

text: the

text: best

text: case)

text: of

text: similar

text: structure

text: and

text: magnitude

text: as

text: shown

text: in

text: ﬁgure

text: This

text: also

text: means

text: that

text: the

text: use

text: of

text: a

text: parameterization

text: not

text: respecting

text: the

text: form

text: of

text: equation

text: to

text: determine

text: a

text: unique

text: Z

text: (as

text: in

text: eﬀ

text: is

text: inconsistent.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Here, a hat on a variable’s symbol marks a quantity x normalized by the same quantity for water xw, i.e. x(cid:98) = (cid:82) x/xw. The index s signiﬁes spectral weighting in the form x = S(E)x(E) dE, where the detected spectrum S(E), (cid:82) s normalized such that S(E) dE = 1, contains the emitted x-ray spectrum and energy-dependent detector properties such as eﬃciency. 2. Cross section parameterization Many speciﬁc parameterizations of the photon absorption cross section, σ(E, Z), in terms of photon energy, E, and atomic number of the absorber, Z, have been proposed for the energy regime of medical imaging, i.e. E < 1 MeV [3, 22, 23]. Among these, an important general shape is given by σ(E, Z) = A(E) + B(E)C(Z) , (A3) corresponding to eq. 4.6 in Jackson and Hawkes[22]. The important feature of this parameterization is the separation of E- from Z-dependence in the second term. A power law, C(Z) = Zm , (A4) is often used in literature, where m ∈ [3, 4] if only the photoelectric eﬀect is considered. 3. Eﬀective atomic number (cid:80) From the attenuation sum rule for chemical compounds or mixtures, µ = µ , and equation A1, the energy- i i dependent cross section of a composite material with electron-density fractions, ν = n /n, results as i i (cid:88) σ(E) = ν σ(E, Z ) ≡ σ(E, Z ) . (A5) i i eﬀ i On the right-hand side, the eﬀective atomic number, Z , is deﬁned as the virtual decimal atomic number of a eﬀ hypothetical one-atomic element that would express the same cross section at a given photon energy as the compound in consideration, based on a given cross section parameterization, σ(E, Z). It thus reduces the number of variables to describe photon interaction with a mixture. A cross section parameterization in the form of equation A3 is a necessary condition for the deﬁnition of a single energy-independent eﬀective atomic number [22]. The deﬁnition A5 then translates to (cid:88) C(Z ) = ν C(Z ) . (A6) eﬀ i i i In order to solve for Z , C(Z) has to be invertible. Using equation A4 leads to the well-known and frequently used eﬀ equation (cid:16)(cid:88) (cid:17)1/m Z = ν Zm . (A7) eﬀ i i i Appendix B: Mathematical framework The alpha-blending equation 1 for electron-density assessment can be derived rigorously by requiring a universal solution for the electron density, meaning that the solution is valid for any single chemical element as well as any arbitrary compound (ﬁgure 3(a), proof in appendix ). Alternatively, the central equations can be derived from the given general functional dependency of the cross section on Z and E (ﬁgure 3(b), proof in appendix ). A cross section parameterization of the given form is a necessary requirement for the deﬁnition of a unique, energy-independent eﬀective atomic number (ﬁgure 3(c), Jackson and Hawkes[22]).

label: paragraph

text: Here,

text: a

text: hat

text: on

text: a

text: variable’s

text: symbol

text: marks

text: a

text: quantity

text: x

text: normalized

text: by

text: the

text: same

text: quantity

text: for

text: water

text: xw,

text: i.e.

text: x/xw.

text: The

text: index

text: s

text: signiﬁes

text: spectral

text: weighting

text: in

text: the

text: form

text: x

text: S(E)x(E)

text: dE,

text: where

text: the

text: detected

text: spectrum

text: S(E),

text: s

text: normalized

text: such

text: that

text: S(E)

text: dE

text: contains

text: the

text: emitted

text: x-ray

text: spectrum

text: and

text: energy-dependent

text: detector

text: properties

text: such

text: as

text: eﬃciency.

label: section

text: Cross

text: section

text: parameterization

text: Many

text: speciﬁc

text: parameterizations

text: of

text: the

text: photon

text: absorption

text: cross

text: section,

text: σ(E,

text: Z),

text: in

text: terms

text: of

text: photon

text: energy,

text: E,

text: and

text: atomic

text: number

text: of

text: the

text: absorber,

text: Z,

text: have

text: been

text: proposed

text: for

text: the

text: energy

text: regime

text: of

text: medical

text: imaging,

text: i.e.

text: E

text: MeV

text: Among

text: these,

text: an

text: important

text: general

text: shape

text: is

text: given

text: by

text: σ(E,

text: Z)

text: A(E)

text: B(E)C(Z)

text: corresponding

text: to

text: eq.

text: in

text: Jackson

text: and

text: The

text: important

text: feature

text: of

text: this

text: parameterization

text: is

text: the

text: separation

text: of

text: E-

text: from

text: Z-dependence

text: in

text: the

text: second

text: term.

text: A

text: power

text: law,

text: C(Z)

text: Zm

text: is

text: often

text: used

text: in

text: literature,

text: where

text: m

text: if

text: only

text: the

text: photoelectric

text: eﬀect

text: is

text: considered.

text: Eﬀective

text: atomic

text: number

text: From

text: the

text: attenuation

text: sum

text: rule

text: for

text: chemical

text: compounds

text: or

text: mixtures,

text: µ

text: µ

text: and

text: equation

text: the

text: energy-

text: i

text: i

text: dependent

text: cross

text: section

text: of

text: a

text: composite

text: material

text: with

text: electron-density

text: fractions,

text: ν

text: n

text: /n,

text: results

text: as

text: i

text: i

text: σ(E)

text: ν

text: σ(E,

text: Z

text: σ(E,

text: Z

text: i

text: i

text: eﬀ

text: i

text: On

text: the

text: right-hand

text: side,

text: the

text: eﬀective

text: atomic

text: number,

text: Z

text: is

text: deﬁned

text: as

text: the

text: virtual

text: decimal

text: atomic

text: number

text: of

text: a

text: eﬀ

text: hypothetical

text: one-atomic

text: element

text: that

text: would

text: express

text: the

text: same

text: cross

text: section

text: at

text: a

text: given

text: photon

text: energy

text: as

text: the

text: compound

text: in

text: consideration,

text: based

text: on

text: a

text: given

text: cross

text: section

text: parameterization,

text: σ(E,

text: Z).

text: It

text: thus

text: reduces

text: the

text: number

text: of

text: variables

text: to

text: describe

text: photon

text: interaction

text: with

text: a

text: mixture.

text: A

text: cross

text: section

text: parameterization

text: in

text: the

text: form

text: of

text: equation

text: is

text: a

text: necessary

text: condition

text: for

text: the

text: deﬁnition

text: of

text: a

text: single

text: energy-independent

text: eﬀective

text: atomic

text: number

text: The

text: deﬁnition

text: then

text: translates

text: to

text: C(Z

text: ν

text: C(Z

text: eﬀ

text: i

text: i

text: i

text: In

text: order

text: to

text: solve

text: for

text: Z

text: C(Z)

text: has

text: to

text: be

text: invertible.

text: Using

text: equation

text: leads

text: to

text: the

text: well-known

text: and

text: frequently

text: used

text: eﬀ

text: equation

text: Z

text: ν

text: Zm

text: eﬀ

text: i

text: i

text: i

text: Appendix

text: B:

text: Mathematical

text: framework

text: The

text: alpha-blending

text: equation

text: for

text: electron-density

text: assessment

text: can

text: be

text: derived

text: rigorously

text: by

text: requiring

text: a

text: universal

text: solution

text: for

text: the

text: electron

text: density,

text: meaning

text: that

text: the

text: solution

text: is

text: valid

text: for

text: any

text: single

text: chemical

text: element

text: as

text: well

text: as

text: any

text: arbitrary

text: compound

text: (ﬁgure

text: proof

text: in

text: appendix

text: Alternatively,

text: the

text: central

text: equations

text: can

text: be

text: derived

text: from

text: the

text: given

text: general

text: functional

text: dependency

text: of

text: the

text: cross

text: section

text: on

text: Z

text: and

text: E

text: (ﬁgure

text: proof

text: in

text: appendix

text: A

text: cross

text: section

text: parameterization

text: of

text: the

text: given

text: form

text: is

text: a

text: necessary

text: requirement

text: for

text: the

text: deﬁnition

text: of

text: a

text: unique,

text: energy-independent

text: eﬀective

text: atomic

text: number

text: (ﬁgure

text: Jackson

text: and


text: 4 Bernardini et al. 8&&( 8&&) 8&&9 8&%& 8&%% 8&%8 8&%: 8&%# 8&%! 7 7 7 7 7 7 7 7 7 01 2+34 %’ 5 2+34 %( 6 2+34 !5!"."#7#$1$!5!%!%$1$&0&’’4 !5!"."#7#(2(%!%$1$&0&’4’ 2M23J34D4 %) !"#$%&# !"!$%&# !"’$%&# !"($%&# -./ (cid:48) Fig. 1.— Optical lightcurve in i (triangles), R (squares), and V (circles) band from 2006 to 2015. Magnitudes include ﬂux from the nearby contaminating star. The dot-dashed line at MJD 57188.772 shows the X-ray trigger. The red points are 2015 June 8 (MJD 57181.5) (cid:48) and 9 (MJD 57182.5). No observation in i band was performed on 2015 June 8. In the insert, a zoom-in of the 2015 R band data compared with the 3σ BAT detections (blue crosses). (cid:48) (cid:48) Cackett 2014). bands, < ∆ i , ∆ R >= 0.5(< ∆ i > + < ∆ R >), where We ﬁt the orbital lightcurves (except the last two we use three points per bin. The decreasing-increasing points) with a constant plus double sinusoid function, trend is now clearer. We ﬁt the ﬁrst part of the lightcurve whose phases are free to vary so as to account for un- (up to MJD 55834.5) with a constant plus a linear func- equal minima and maxima, as frequently seen in qui- tion. An F-test gives a 4.4σ signiﬁcance for the inclusion escent LMXBs including V404 (Zurita et al. 2004). of the latter. Between MJD 53860 and 55834.5 we mea- ∼ ∼ ∼ On 2015 June 8 (MJD 57181.5, φ 0.11), and 9 sure a decrease of 0.02 mag/year. ∼ ∼ (MJD 57182.5, φ 0.25) the optical magnitude in all We detect an optical precursor to the ﬁrst X-ray ﬂare bands is signiﬁcantly brighter than the average quies- registered by BAT. Assuming that the two events are di- cent modulation level by 0.1–0.3 mag, and above the rectly linked and that the rise of the X-ray ﬂare is mono- (cid:46) ∼ low amplitude ﬂaring behavior (where ∆ mag 0.1). tonic, we estimate a delay in the rise of the X-ray (MJD ∼ We subtract the best ﬁtting model from the orbital 57188.5) compared to the optical (MJD 57181.5) emis- lightcurves, and show in Fig. 3 the residual lightcurves. sion of at least 7 days. It could be as long as 13 days, We note that the last two points of the lightcurves have considering that the optical ﬂux may have started ris- the highest residuals, that the residuals in the diﬀerent ing immediately after the pointing before the precursor ∼ bands look correlated on short timescales (days) and that (MJD 57175.6). The delay’s length is similar to those a long-term trend (years) seems present (ﬁrst a decay seen in the other four LMXBs that showed indication of and then a rise). this behavior. We have also found evidence of a long- We measure the statistical signiﬁcance of the correla- term trend. The last 2 points of the residual lightcurve (cid:48) tion using Spearman’s rank test on the R and i bands, before the outburst are signiﬁcantly above this quiescent where the S/N is higher compared to V. The Spearman’s downward trend. Consequently, we can constrain the coeﬃcient is ρ = 0.78, and the null hypothesis probability long-term rise to have started before MJD 57129.5 (2015 × − is P = 3.5 10 27, so the residuals are positively corre- April 17). lated. This and the small error bars on each data point, 3.3. Spectral Energy Distribution suggest that the observed variability is intrinsic to the To construct the Spectral Energy Distribution (SED) source and is likely accretion activity on timescales longer of the optical precursor, we ﬁrst measure the lower enve- than minutes (the time between two consecutive expo- lope of the modulation (see Zurita et al. 2004), which is (cid:48) sures). We combine the i and R band residuals. In the the donor contribution at each phase (Fig. 2). We com- (cid:48) (cid:48) bottom panel of Fig. 3 we show the average of R and i bine the 2015 June 8 and 9 images (in the i band only ,+* ✆ ✆ ✆ 111,,,000///...---,,,+++ *** (cid:0)✂☎ )))$$$ $$$""")))$$$ 1$."0& $1&0".&0 33221100//..--)),,++**))&&!! !!$$ ✁✄ ##LTFigure##

text: Bernardini

text: et

text: al.

font: default

label: caption

text: Fig.

text: Optical

text: lightcurve

text: in

text: i

text: (triangles),

text: R

text: (squares),

text: and

text: V

text: (circles)

text: band

text: from

text: to

text: Magnitudes

text: include

text: ﬂux

text: from

text: the

text: nearby

text: contaminating

text: star.

text: The

text: dot-dashed

text: line

text: at

text: MJD

text: shows

text: the

text: X-ray

text: trigger.

text: The

text: red

text: points

text: are

text: June

text: (MJD

text: and

text: (MJD

text: No

text: observation

text: in

text: i

text: band

text: was

text: performed

text: on

text: June

text: In

text: the

text: insert,

text: a

text: zoom-in

text: of

text: the

text: R

text: band

text: data

text: compared

text: with

text: the

text: 3σ

text: BAT

text: detections

text: (blue

text: crosses).

text: Cackett

text: bands,

text: i

text: R

text: i

text: R

text: where

text: We

text: ﬁt

text: the

text: orbital

text: lightcurves

text: (except

text: the

text: last

text: two

text: we

text: use

text: three

text: points

text: per

text: bin.

text: The

text: decreasing-increasing

text: points)

text: with

text: a

text: constant

text: plus

text: double

text: sinusoid

text: function,

text: trend

text: is

text: now

text: clearer.

text: We

text: ﬁt

text: the

text: ﬁrst

text: part

text: of

text: the

text: lightcurve

text: whose

text: phases

text: are

text: free

text: to

text: vary

text: so

text: as

text: to

text: account

text: for

text: un-

text: (up

text: to

text: MJD

text: with

text: a

text: constant

text: plus

text: a

text: linear

text: func-

text: equal

text: minima

text: and

text: maxima,

text: as

text: frequently

text: seen

text: in

text: qui-

text: tion.

text: An

text: F-test

text: gives

text: a

text: signiﬁcance

text: for

text: the

text: inclusion

text: escent

text: LMXBs

text: including

text: (Zurita

text: et

text: al.

text: of

text: the

text: latter.

text: Between

text: MJD

text: and

text: we

text: mea-

text: On

text: June

text: (MJD

text: φ

text: and

text: sure

text: a

text: decrease

text: of

text: mag/year.

text: (MJD

text: φ

text: the

text: optical

text: magnitude

text: in

text: all

text: We

text: detect

text: an

text: optical

text: precursor

text: to

text: the

text: ﬁrst

text: X-ray

text: ﬂare

text: bands

text: is

text: signiﬁcantly

text: brighter

text: than

text: the

text: average

text: quies-

text: registered

text: by

text: BAT.

text: Assuming

text: that

text: the

text: two

text: events

text: are

text: di-

text: cent

text: modulation

text: level

text: by

text: mag,

text: and

text: above

text: the

text: rectly

text: linked

text: and

text: that

text: the

text: rise

text: of

text: the

text: X-ray

text: ﬂare

text: is

text: mono-

text: low

text: amplitude

text: ﬂaring

text: behavior

text: (where

text: mag

text: tonic,

text: we

text: estimate

text: a

text: delay

text: in

text: the

text: rise

text: of

text: the

text: X-ray

text: (MJD

text: We

text: subtract

text: the

text: best

text: ﬁtting

text: model

text: from

text: the

text: orbital

text: compared

text: to

text: the

text: optical

text: (MJD

text: emis-

text: lightcurves,

text: and

text: show

text: in

text: Fig.

text: the

text: residual

text: lightcurves.

text: sion

text: of

text: at

text: least

text: days.

text: It

text: could

text: be

text: as

text: long

text: as

text: days,

text: We

text: note

text: that

text: the

text: last

text: two

text: points

text: of

text: the

text: lightcurves

text: have

text: considering

text: that

text: the

text: optical

text: ﬂux

text: may

text: have

text: started

text: ris-

text: the

text: highest

text: residuals,

text: that

text: the

text: residuals

text: in

text: the

text: diﬀerent

text: ing

text: immediately

text: after

text: the

text: pointing

text: before

text: the

text: precursor

text: bands

text: look

text: correlated

text: on

text: short

text: timescales

text: (days)

text: and

text: that

text: (MJD

text: The

text: delay’s

text: length

text: is

text: similar

text: to

text: those

text: a

text: long-term

text: trend

text: (years)

text: seems

text: present

text: (ﬁrst

text: a

text: decay

text: seen

text: in

text: the

text: other

text: four

text: LMXBs

text: that

text: showed

text: indication

text: of

text: and

text: then

text: a

text: rise).

text: this

text: behavior.

text: We

text: have

text: also

text: found

text: evidence

text: of

text: a

text: long-

text: We

text: measure

text: the

text: statistical

text: signiﬁcance

text: of

text: the

text: correla-

text: term

text: trend.

text: The

text: last

text: points

text: of

text: the

text: residual

text: lightcurve

text: tion

text: using

text: Spearman’s

text: rank

text: test

text: on

text: the

text: R

text: and

text: i

text: bands,

text: before

text: the

text: outburst

text: are

text: signiﬁcantly

text: above

text: this

text: quiescent

text: where

text: the

text: S/N

text: is

text: higher

text: compared

text: to

text: V.

text: The

text: Spearman’s

text: downward

text: trend.

text: Consequently,

text: we

text: can

text: constrain

text: the

text: coeﬃcient

text: is

text: ρ

text: and

text: the

text: hypothesis

text: probability

text: long-term

text: rise

text: to

text: have

text: started

text: before

text: MJD

text: is

text: P

text: so

text: the

text: residuals

text: are

text: positively

text: corre-

text: April

text: lated.

text: This

text: and

text: the

text: small

text: error

text: bars

text: on

text: each

text: data

text: point,

text: Spectral

text: Energy

text: Distribution

text: suggest

text: that

text: the

text: observed

text: variability

text: is

text: intrinsic

text: to

text: the

text: To

text: construct

text: the

text: Spectral

text: Energy

text: Distribution

text: (SED)

text: source

text: and

text: is

text: likely

text: accretion

text: activity

text: on

text: timescales

text: longer

text: of

text: the

text: optical

text: precursor,

text: we

text: ﬁrst

text: measure

text: the

text: lower

text: enve-

text: than

text: minutes

text: (the

text: time

text: between

text: two

text: consecutive

text: expo-

text: lope

text: of

text: the

text: modulation

text: (see

text: Zurita

text: et

text: al.

text: which

text: is

text: sures).

text: We

text: combine

text: the

text: i

text: and

text: R

text: band

text: residuals.

text: In

text: the

text: the

text: donor

text: contribution

text: at

text: each

text: phase

text: (Fig.

text: We

text: com-

text: bottom

text: panel

text: of

text: Fig.

text: we

text: show

text: the

text: average

text: of

text: R

text: and

text: i

text: bine

text: the

text: June

text: and

text: images

text: (in

text: the

text: i

text: band

text: only

text: ##LTFigure##


text: 3 by an appropriate gating which would reduce the car- rier concentration[12]. As both ARPES and Hall mea- surements give similar carrier concentration, the atmo- spheric contamination plays a minor role and is not the main source of doping, which is likely to be due to the interface. The Josephson devices are fabricated by conven- tional e-beam lithography. The graphene junctions are patterned by oxygen plasma. Then, the contacts are deposited, see Fig. 1(d,e). They consist of a 5 nm inter- facial layer of titanium, a 80 nm thick aluminium layer and a 3 nm thick gold layer. The gaps L between the electrodes ranges from 200 nm to 600 nm. The width is ﬁxed at W = 4 µm. A total of twelve junctions have been deﬁned, with two orientations with respect to the substrate. The samples have been thermally anchored to the 3He pot of a 3He cryostat (Heliox VL Oxford Instruments) and four probes electrical measurements have been performed. The cryostat is equipped with a room temperature electromagnetic interference ﬁlter stage followed by low pass RC ﬁlters anchored at 1.5 Fig. 2 (a) Temperature dependence of the resistance R(T) for K and by copper powder ﬁlters stage anchored at the ≈ four junctions of lengths L 200,300,400 and 600 nm, normal- sample stage[25]. Standard measurements of current- ized by their resistance at T = 1.2 K. The inset represents the voltage V (I) characteristics as a function of tempera- resistance at T = 1.2 K of the same four junctions, plotted as a function of their width as ﬁlled squares with the same colour ture and magnetic ﬁeld along with R(T) curves with a ≈ code. Two additional junctions of width L 200 nm are also bias current of about 5 nA have been performed. In ad- plotted as open squares. (b) V(I) characteristics of the four junc- dition, conductance spectra dI/dV (V ) have been mea- tions presented in panel (a), at the base temperature of T = 280 sured by superimposing a low amplitude (a few nA) mK. sinusoidal signal with frequency of about 30 Hz to a triangular slow ramp, with a frequency of about 1 mHz, and by reading the response from the sample by using voltage below the Ti/Al gap can be attributed to local- the lock-in technique. ized subgap states (see below). The mean free path of ∼ 3 Results graphene on hydrogenated SiC is estimated to be (cid:96) 50 nm from room temperature measurements and the car- The temperature dependence of the resistance of rier density and mobility are only weakly modiﬁed from four junctions is presented in Fig. 2(a). There is a kink room temperature to liquid Helium temperature[26,24]. at about 1 K monitoring the transition to superconduc- Thus even the smallest of the junctions should be dif- ≈ ≈ tivity of the Ti/Al contacts. The junction with L 600 fusive. Estimating the aluminium gap ∆ 100 µeV (cid:112) nm (green curve), appears to be going insulating. Not from conductance dI/dV (V ) measurements (see Fig. ≈ even in the smallest junctions with L 200 nm (black 4(a)), we obtain a superconducting coherence length ≈ ≈ curve) and L 300 nm (red curve) is the transition to ξ0 = ¯hD/∆ 300–400 nm, where D≤is the diﬀusion the superconducting state complete above 280 mK. The constant. Hence our junctions with L 300 nm are in precursory eﬀects of the dissipationless conductance in the crossover between the short and the long junction these junctions will be discussed elsewhere. Here we limit. A reduction of the Josephson critical current is show that, no matter that some residual resistance is expected in the long junction limit, as conﬁrmed by the present above 280 mK, these junctions attain phase co- experiment. ≈ herence and Josephson conduction. Fig. 2(b) shows the For the L 200 nm junction, the product eI R ≈ c N corresponding V (I) characteristics taken at T = 280 /∆ 0.09 is smaller than the theoretical estimate[27] ≈ mK for the four junctions presented in panel (a). The eI R / ∆ 0.66 derived for junctions in which the c N ∼ V (I) characteristic is practically ohmic for the junction Thouless energy is ¯hD/L2 ∆. This is the upper limit ≈ ≈ ≈ with L 600 nm. A Josephson critical current I 40 of the diﬀerence, since it is calculated for I at T 0.3 c≈ c and 10 nA can be read oﬀ for the junctions with L 200 TAl (TAl is the critical temperature of the electrodes). c c ≈ nm and 300 nm respectively. The small steps at ﬁnite The measured critical current density J = I /W 10 c c ##LTFigure## ##LTLine## ##LTLine##

text: by

text: an

text: appropriate

text: gating

text: which

text: would

text: reduce

text: the

text: car-

text: rier

text: As

text: both

text: ARPES

text: and

text: Hall

text: mea-

text: surements

text: give

text: similar

text: carrier

text: concentration,

text: the

text: atmo-

text: spheric

text: contamination

text: plays

text: a

text: minor

text: role

text: and

text: is

text: not

text: the

text: main

text: source

text: of

text: doping,

text: which

text: is

text: likely

text: to

text: be

text: due

text: to

text: the

text: interface.

text: The

text: Josephson

text: devices

text: are

text: fabricated

text: by

text: conven-

text: tional

text: e-beam

text: lithography.

text: The

text: graphene

text: junctions

text: are

text: patterned

text: by

text: oxygen

text: plasma.

text: Then,

text: the

text: contacts

text: are

text: deposited,

text: see

text: Fig.

text: They

text: consist

text: of

text: a

text: nm

text: inter-

text: facial

text: layer

text: of

text: titanium,

text: a

text: nm

text: thick

text: aluminium

text: layer

text: and

text: a

text: nm

text: thick

text: gold

text: layer.

text: The

text: gaps

text: L

text: between

text: the

text: electrodes

text: ranges

text: from

text: nm

text: to

text: nm.

text: The

text: width

text: is

text: ﬁxed

text: at

text: W

text: µm.

text: A

text: total

text: of

text: twelve

text: junctions

text: have

text: been

text: deﬁned,

text: with

text: two

text: orientations

text: with

text: respect

text: to

text: the

text: substrate.

text: The

text: samples

text: have

text: been

text: thermally

text: anchored

text: to

text: the

text: pot

text: of

text: a

text: cryostat

text: (Heliox

text: VL

text: Oxford

text: Instruments)

text: and

text: four

text: probes

text: electrical

text: measurements

text: have

text: been

text: performed.

text: The

text: cryostat

text: is

text: equipped

text: with

text: a

text: room

text: temperature

text: electromagnetic

text: interference

text: ﬁlter

text: stage

text: followed

text: by

text: low

text: pass

text: RC

text: ﬁlters

text: anchored

text: at

text: Fig.

text: (a)

text: Temperature

text: dependence

text: of

text: the

text: resistance

text: R(T)

text: for

text: K

text: and

text: by

text: copper

text: powder

text: ﬁlters

text: stage

text: anchored

text: at

text: the

text: four

text: junctions

text: of

text: lengths

text: L

text: and

text: nm,

text: normal-

text: sample

text: Standard

text: measurements

text: of

text: current-

text: ized

text: by

text: their

text: resistance

text: at

text: T

text: K.

text: The

text: inset

text: represents

text: the

text: voltage

text: V

text: (I)

text: characteristics

text: as

text: a

text: function

text: of

text: tempera-

text: resistance

text: at

text: T

text: K

text: of

text: the

text: same

text: four

text: junctions,

text: plotted

text: as

text: a

text: function

text: of

text: their

text: width

text: as

text: ﬁlled

text: squares

text: with

text: the

text: same

text: colour

text: ture

text: and

text: magnetic

text: ﬁeld

text: along

text: with

text: R(T)

text: curves

text: with

text: a

text: code.

text: Two

text: additional

text: junctions

text: of

text: width

text: L

text: nm

text: are

text: also

text: bias

text: current

text: of

text: about

text: have

text: been

text: performed.

text: In

text: ad-

text: plotted

text: as

text: open

text: squares.

text: (b)

text: V(I)

text: characteristics

text: of

text: the

text: four

text: junc-

text: dition,

text: conductance

text: spectra

text: dI/dV

text: (V

text: have

text: been

text: mea-

text: tions

text: presented

text: in

text: panel

text: (a),

text: at

text: the

text: base

text: temperature

text: of

text: T

text: sured

text: by

text: superimposing

text: a

text: low

text: amplitude

text: (a

text: few

text: nA)

text: mK.

text: sinusoidal

text: signal

text: with

text: frequency

text: of

text: about

text: Hz

text: to

text: a

text: triangular

text: slow

text: ramp,

text: with

text: a

text: frequency

text: of

text: about

text: mHz,

text: and

text: by

text: reading

text: the

text: response

text: from

text: the

text: sample

text: by

text: using

text: voltage

text: below

text: the

text: Ti/Al

text: gap

text: can

text: be

text: attributed

text: to

text: local-

text: the

text: lock-in

text: technique.

text: ized

text: subgap

text: states

text: (see

text: below).

text: The

text: mean

text: free

text: path

text: of

text: Results

text: graphene

text: on

text: hydrogenated

text: SiC

text: is

text: estimated

text: to

text: be

text: nm

text: from

text: room

text: temperature

text: measurements

text: and

text: the

text: car-

text: The

text: temperature

text: dependence

text: of

text: the

text: resistance

text: of

text: rier

text: density

text: and

text: mobility

text: are

text: only

text: weakly

text: modiﬁed

text: from

text: four

text: junctions

text: is

text: presented

text: in

text: Fig.

text: There

text: is

text: a

text: kink

text: room

text: temperature

text: to

text: liquid

text: Helium

text: at

text: about

text: K

text: monitoring

text: the

text: transition

text: to

text: superconduc-

text: Thus

text: even

text: the

text: smallest

text: of

text: the

text: junctions

text: should

text: be

text: dif-

text: tivity

text: of

text: the

text: Ti/Al

text: contacts.

text: The

text: junction

text: with

text: L

text: fusive.

text: Estimating

text: the

text: aluminium

text: gap

text: µeV

text: nm

text: (green

text: curve),

text: appears

text: to

text: be

text: going

text: insulating.

text: Not

text: from

text: conductance

text: dI/dV

text: (V

text: measurements

text: (see

text: Fig.

text: even

text: in

text: the

text: smallest

text: junctions

text: with

text: L

text: nm

text: (black

text: we

text: obtain

text: a

text: superconducting

text: coherence

text: length

text: curve)

text: and

text: L

text: nm

text: (red

text: curve)

text: is

text: the

text: transition

text: to

text: ξ0

text: ¯hD/∆

text: nm,

text: where

text: D≤is

text: the

text: diﬀusion

text: the

text: superconducting

text: state

text: complete

text: above

text: mK.

text: The

text: constant.

text: Hence

text: our

text: junctions

text: with

text: L

text: nm

text: are

text: in

text: precursory

text: eﬀects

text: of

text: the

text: dissipationless

text: conductance

text: in

text: the

text: crossover

text: between

text: the

text: short

text: and

text: the

text: long

text: junction

text: these

text: junctions

text: will

text: be

text: discussed

text: elsewhere.

text: Here

text: we

text: limit.

text: A

text: reduction

text: of

text: the

text: Josephson

text: critical

text: current

text: is

text: show

text: that,

text: no

text: matter

text: that

text: some

text: residual

text: resistance

text: is

text: expected

text: in

text: the

text: long

text: junction

text: limit,

text: as

text: conﬁrmed

text: by

text: the

text: present

text: above

text: mK,

text: these

text: junctions

text: attain

text: phase

text: co-

text: experiment.

text: herence

text: and

text: Josephson

text: conduction.

text: Fig.

text: shows

text: the

text: For

text: the

text: L

text: nm

text: junction,

text: the

text: product

text: eI

text: R

text: c

text: N

text: corresponding

text: V

text: (I)

text: characteristics

text: taken

text: at

text: T

text: is

text: smaller

text: than

text: the

text: theoretical

text: mK

text: for

text: the

text: four

text: junctions

text: presented

text: in

text: panel

text: (a).

text: The

text: eI

text: R

text: derived

text: for

text: junctions

text: in

text: which

text: the

text: c

text: N

text: V

text: (I)

text: characteristic

text: is

text: practically

text: ohmic

text: for

text: the

text: junction

text: Thouless

text: energy

text: is

text: This

text: is

text: the

text: upper

text: limit

text: with

text: L

text: nm.

text: A

text: Josephson

text: critical

text: current

text: I

text: of

text: the

text: diﬀerence,

text: since

text: it

text: is

text: calculated

text: for

text: I

text: at

text: T

text: c≈

text: c

text: and

text: can

text: be

text: read

text: oﬀ

text: for

text: the

text: junctions

text: with

text: L

text: TAl

text: (TAl

text: is

text: the

text: critical

text: temperature

text: of

text: the

text: electrodes).

text: c

text: c

text: nm

text: and

text: nm

text: respectively.

text: The

text: small

text: steps

text: at

text: ﬁnite

text: The

text: measured

text: critical

text: current

text: density

text: J

text: I

text: /W

text: c

text: c

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: INTERPRETING THE MONADIC SECOND ORDER THEORY OF ONE SUCCESSOR IN EXPANSIONS OF THE REAL LINE PHILIPP HIERONYMI AND ERIK WALSBERG Abstract. We give suﬃcient conditions for a ﬁrst order expansion of the real line to deﬁne the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties deﬁned by Shelah, such as NIP or even NTP2. We use thisRto deduce the ﬁrst general results about deﬁnable sets in NTP2 expansions of ( ,<,+). The goal of this paper is to bring together the study of combinatorial tameness properties of ﬁrst order structures initiated by Morley and Shelah (neostability) R and the study of geometric tameness properties of expansions of the real line ( , <) B championed by Miller (tame geometry, see [Mil05]). Let be the two-sorted (ﬁrst P N N ∈ P N N order) structure ( ( ), , , +1) where ( ) is the power set of and +1 is the suc- N cessor function on . The theory of this structure is essentially the monadic second N B order theory of ( , +1). While Bu¨chi showed in his landmark paper [Bu¨c62] that B admits quantiﬁer elimination in a suitable language and its theory is decidable, obviously does not enjoy any Shelah-style combinatorial tameness properties, such as NIP or NTP (see e.g. Simon [Sim15] for deﬁnitions). Therefore any structure 2 B that deﬁnes an isomorphic copy of , can not satisfy any of those properties, and has to be considered complicated or wild in this framework of combinatorial tame- B ness. Here we study the consequences of the non-deﬁnability of a copy of in an R expansion of ( , <) on the geometric tameness of deﬁnable sets in this expansion. Our results are new even when the assumption “does not deﬁne an isomorphic B copy of ” is replaced by one of the stronger assumptions “has NTP ” or even 2 “has NIP”. Therefore these are arguably the very ﬁrst general results about NTP R 2 expansions of ( , <, +). Throughout deﬁnable will always mean deﬁnable with parameters. We will say that B B a structure deﬁnes if it deﬁnes an isomorphic copy of . The main technical result of this paper is as follows. R R ≺ ⊆ R Theorem A. Let = ( , <, D, ) where D is dense in some open interval ≺ R B and is an order on D with order type ω. Then deﬁnes . Date: May 28, 2020. 2010 Mathematics Subject Classiﬁcation. Primary 03C45 Secondary 03C64, 03D05, 28A80, 54F45. Key words and phrases. NTP2, neostability, tame geometry, Cantor set, expansions of the real line, monadic second order theory of one successor, Bu¨chi, metric dimensions, monotonicity theorem. The ﬁrst author was partially supported by NSF grant DMS-1300402. The second author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG. 1 ##LTLine##

text: INTERPRETING

text: THE

text: MONADIC

text: SECOND

text: ORDER

text: THEORY

text: OF

text: ONE

text: SUCCESSOR

text: IN

text: EXPANSIONS

text: OF

text: THE

text: REAL

text: LINE

text: PHILIPP

text: HIERONYMI

text: AND

text: ERIK

text: WALSBERG

text: Abstract.

text: We

label: abstract

text: give

text: suﬃcient

text: conditions

text: for

text: a

text: ﬁrst

text: order

text: expansion

text: of

text: the

text: real

text: line

text: to

text: deﬁne

text: the

text: standard

text: model

text: of

text: the

text: monadic

text: second

text: order

text: theory

text: of

text: one

text: successor.

text: Such

text: an

text: expansion

text: does

text: not

text: satisfy

text: any

text: of

text: the

text: combinatorial

text: tameness

text: properties

text: deﬁned

text: by

text: Shelah,

text: such

text: as

text: NIP

text: or

text: even

text: We

text: use

text: thisRto

text: deduce

text: the

text: ﬁrst

text: general

text: results

text: about

text: deﬁnable

text: sets

text: in

text: expansions

text: of

text: The

text: goal

text: of

text: this

text: paper

text: is

text: to

text: bring

text: together

text: the

text: study

text: of

text: combinatorial

text: tameness

text: properties

text: of

text: ﬁrst

text: order

text: structures

text: initiated

text: by

text: Morley

text: and

text: Shelah

text: (neostability)

text: R

text: and

text: the

text: study

text: of

text: geometric

text: tameness

text: properties

text: of

text: expansions

text: of

text: the

text: real

text: line

text: B

text: championed

text: by

text: Miller

text: (tame

text: geometry,

text: see

text: Let

text: be

text: the

text: two-sorted

text: (ﬁrst

text: P

text: N

text: N

text: P

text: N

text: N

text: order)

text: structure

text: where

text: is

text: the

text: power

text: set

text: of

text: and

text: is

text: the

text: suc-

text: N

text: cessor

text: function

text: on

text: The

text: theory

text: of

text: this

text: structure

text: is

text: essentially

text: the

text: monadic

text: second

text: N

text: B

text: order

text: theory

text: of

text: While

text: Bu¨chi

text: showed

text: in

text: his

text: landmark

text: paper

text: that

text: B

text: admits

text: quantiﬁer

text: elimination

text: in

text: a

text: suitable

text: language

text: and

text: its

text: theory

text: is

text: decidable,

text: obviously

text: does

text: not

text: enjoy

text: any

text: Shelah-style

text: combinatorial

text: tameness

text: properties,

text: such

text: as

text: NIP

text: or

text: NTP

text: (see

text: e.g.

text: Simon

text: for

text: deﬁnitions).

text: Therefore

text: any

text: structure

text: B

text: that

text: deﬁnes

text: an

text: isomorphic

text: copy

text: of

text: can

text: not

text: satisfy

text: any

text: of

text: those

text: properties,

text: and

text: has

text: to

text: be

text: considered

text: complicated

text: or

text: wild

text: in

text: this

text: framework

text: of

text: combinatorial

text: tame-

text: B

text: ness.

text: Here

text: we

text: study

text: the

text: consequences

text: of

text: the

text: non-deﬁnability

text: of

text: a

text: copy

text: of

text: in

text: an

text: R

text: expansion

text: of

text: on

text: the

text: geometric

text: tameness

text: of

text: deﬁnable

text: sets

text: in

text: this

text: expansion.

text: Our

text: results

text: are

text: new

text: even

text: when

text: the

text: assumption

text: “does

text: not

text: deﬁne

text: an

text: isomorphic

text: B

text: copy

text: of

text: is

text: replaced

text: by

text: one

text: of

text: the

text: stronger

text: assumptions

text: “has

text: NTP

text: or

text: even

text: “has

text: NIP”.

text: Therefore

text: these

text: are

text: arguably

text: the

text: very

text: ﬁrst

text: general

text: results

text: about

text: NTP

text: R

text: expansions

text: of

text: Throughout

text: deﬁnable

text: will

text: always

text: mean

text: deﬁnable

text: with

text: parameters.

text: We

text: will

text: say

text: that

text: B

text: B

text: a

text: structure

text: deﬁnes

text: if

text: it

text: deﬁnes

text: an

text: isomorphic

text: copy

text: of

text: The

text: main

text: technical

text: result

text: of

text: this

text: paper

text: is

text: as

text: follows.

text: R

text: R

text: R

text: Theorem

text: A.

text: Let

text: D,

text: where

text: D

text: is

text: dense

text: in

text: some

text: open

text: interval

text: R

text: B

text: and

text: is

text: an

text: order

text: on

text: D

text: with

text: order

text: type

text: ω.

text: Then

text: deﬁnes

text: Date:

text: May

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Primary

text: Secondary

text: Key

text: words

text: and

text: phrases.

text: neostability,

text: tame

text: geometry,

text: Cantor

text: set,

text: expansions

text: of

text: the

text: real

text: line,

text: monadic

text: second

text: order

text: theory

text: of

text: one

text: successor,

text: Bu¨chi,

text: metric

text: dimensions,

text: monotonicity

text: theorem.

text: The

text: ﬁrst

text: author

text: was

text: partially

text: supported

text: by

text: NSF

text: grant

text: The

text: second

text: author

text: was

text: supported

text: by

text: the

text: European

text: Research

text: Council

text: under

text: the

text: European

text: Union’s

text: Seventh

text: Framework

text: Programme

text: ERC

text: Grant

text: agreement

text: no.

text: MODAG.

text: ##LTLine##


text: 2 PHILIPP HIERONYMI AND ERIK WALSBERG R Heuristically, an expansion of ( , <) can only satisfy any kind of Shelah-style tame- ness if there is no dense set that can be deﬁnably ordered with order type ω. We envision Theorem A to be a starting point for a forthcoming study of the connec- tions between the two notions of tameness described above. We will now state the main consequences of Theorem A established in this paper. R We ﬁrst consider expansions of ( , <, +) that deﬁne a Cantor set. A Cantor R set is a subset of that is nonempty, compact, and has neither isolated nor inte- rior points. Such expansions were ﬁrst studied by Friedman, Miller, Kurdyka and Speisseger [FKMS10]. R B Theorem B. An expansion of ( , <, +) that deﬁnes a Cantor set, deﬁnes . R Heuristically, an expansion of ( , <, +) can only satisfy any kind of Shelah-style tameness if it does not deﬁne a Cantor set. We do not know whether the con- verse implication of Theorem B is true as well. However, by Boigelot, Rassart and B R Wolper [BRW98, Theorem 5], deﬁnes an isomorphic copy of ( , <, +, C) where C is the usual Cantor set constructed by repeatedly deleting middle-thirds of a line R B segment. Therefore, if an expansion of ( , <, +) deﬁnes , it also deﬁnes an iso- R morphic copy of an expansion of ( , <, +) that deﬁnes a Cantor set. Also note that R Theorem B fails for expansions of ( , <) by Dolich, Miller and Steinhorn [DMS10, 7.1]. This result is also signiﬁcant as a complement to [Hie15b]. There a Cantor set K R · is constructed such that every ( , <, +, , K)-deﬁnable set is Borel. The key idea is R · B to ﬁnd a Cantor set K such that ( , <, +, , K) deﬁnes a model of and essentially B nothing else, and to use known results on to prove a quantiﬁer elimination result R · B for ( , <, +, , K). Theorem B arguably shows that one must use results about to prove model theoretic tameness results for expansions which deﬁne Cantor sets. R · By [Hie11, Theorem A] an expansion of ( , <, +, ) that deﬁnes a discrete set ⊆ R → R Z D and a function f : Dk whose image is somewhere dense, also deﬁnes . The theory of such an expansion is clearly undecidable, and moreover deﬁnes every real projective set in the sense of descriptive set theory (see e.g. Kechris [Kec95, 37.6]). Therefore such structures are maximally wild from the point of view of ﬁrst R order model theory. In contrast, there are expansions of ( , <, +) that deﬁne a discrete set D ⊆ R and a function f : Dk → R whose image is some√where dense, R Z Z but whose theory is decidable. For example, by [Hie15a] ( , <, +, , 2 ) is such B a structure. All known examples of such structures deﬁne . We show that this is necessairly the case. R ⊆ R Theorem C. Let R be an expansion of ( , <, +) that deﬁnes a discrete set D k → R B and a map f : D whose image is somewhere dense. Then R deﬁnes . R Heuristically, an expansion of ( , <, +) that deﬁnably collapses a discrete set, can not satisfy any kind of Shelah-style tameness. The converse implication of Theo- rem C is not true. The expansion of the real ﬁeld by the Cantor set K constructed B √ in [Hie15b] deﬁnes and does not satisfy the assumptions of Theorem C. However, B R Z Z deﬁnes an isomorphic copy of ( , <, +, , 2 ) by [Hie15a, Theorem C]. For that ##LTLine## ##LTLine##

text: PHILIPP

text: HIERONYMI

text: AND

text: ERIK

text: WALSBERG

text: R

text: Heuristically,

text: an

text: expansion

text: of

text: can

text: only

text: satisfy

text: any

text: kind

text: of

text: Shelah-style

text: tame-

text: ness

text: if

text: there

text: is

text: no

text: dense

text: set

text: that

text: can

text: be

text: deﬁnably

text: ordered

text: with

text: order

text: type

text: ω.

text: We

text: envision

text: Theorem

text: A

text: to

text: be

text: a

text: starting

text: point

text: for

text: a

text: forthcoming

text: study

text: of

text: the

text: connec-

text: tions

text: between

text: the

text: two

text: notions

text: of

text: tameness

text: described

text: above.

text: We

text: will

text: now

text: state

text: the

text: main

text: consequences

text: of

text: Theorem

text: A

text: established

text: in

text: this

text: paper.

text: R

text: We

text: ﬁrst

text: consider

text: expansions

text: of

text: that

text: deﬁne

text: a

text: Cantor

text: set.

text: A

text: Cantor

text: R

text: set

text: is

text: a

text: subset

text: of

text: that

text: is

text: nonempty,

text: compact,

text: and

text: has

text: neither

text: isolated

text: nor

text: inte-

text: rior

text: points.

text: Such

text: expansions

text: were

text: ﬁrst

text: studied

text: by

text: Friedman,

text: Miller,

text: Kurdyka

text: and

text: Speisseger

text: R

text: B

text: Theorem

text: B.

text: An

text: expansion

text: of

text: that

text: deﬁnes

text: a

text: Cantor

text: set,

text: deﬁnes

text: R

text: Heuristically,

text: an

text: expansion

text: of

text: can

text: only

text: satisfy

text: any

text: kind

text: of

text: Shelah-style

text: tameness

text: if

text: it

text: does

text: not

text: deﬁne

text: a

text: Cantor

text: set.

text: We

text: do

text: not

text: know

text: whether

text: the

text: con-

text: verse

text: implication

text: of

text: Theorem

text: B

text: is

text: true

text: as

text: well.

text: However,

text: by

text: Boigelot,

text: Rassart

text: and

text: B

text: R

text: Wolper

text: Theorem

text: deﬁnes

text: an

text: isomorphic

text: copy

text: of

text: C)

text: where

text: C

text: is

text: the

text: usual

text: Cantor

text: set

text: constructed

text: by

text: repeatedly

text: deleting

text: middle-thirds

text: of

text: a

text: line

text: R

text: B

text: segment.

text: Therefore,

text: if

text: an

text: expansion

text: of

text: deﬁnes

text: it

text: also

text: deﬁnes

text: an

text: iso-

text: R

text: morphic

text: copy

text: of

text: an

text: expansion

text: of

text: that

text: deﬁnes

text: a

text: Cantor

text: set.

text: Also

text: note

text: that

text: R

text: Theorem

text: B

text: fails

text: for

text: expansions

text: of

text: by

text: Dolich,

text: Miller

text: and

text: Steinhorn

text: This

text: result

text: is

text: also

text: signiﬁcant

text: as

text: a

text: complement

text: to

text: There

text: a

text: Cantor

text: set

text: K

text: R

text: is

text: constructed

text: such

text: that

text: every

text: K)-deﬁnable

text: set

text: is

text: Borel.

text: The

text: key

text: idea

text: is

text: R

text: B

text: to

text: ﬁnd

text: a

text: Cantor

text: set

text: K

text: such

text: that

text: K)

text: deﬁnes

text: a

text: model

text: of

text: and

text: essentially

text: B

text: nothing

text: else,

text: and

text: to

text: use

text: known

text: results

text: on

text: to

text: prove

text: a

text: quantiﬁer

text: elimination

text: result

text: R

text: B

text: for

text: K).

text: Theorem

text: B

text: arguably

text: shows

text: that

text: one

text: must

text: use

text: results

text: about

text: to

text: prove

text: model

text: theoretic

text: tameness

text: results

text: for

text: expansions

text: which

text: deﬁne

text: Cantor

text: sets.

text: R

text: By

text: Theorem

text: A]

text: an

text: expansion

text: of

text: that

text: deﬁnes

text: a

text: discrete

text: set

text: R

text: R

text: Z

text: D

text: and

text: a

text: function

text: f

text: Dk

text: whose

text: image

text: is

text: somewhere

text: dense,

text: also

text: deﬁnes

text: The

text: theory

text: of

text: such

text: an

text: expansion

text: is

text: clearly

text: undecidable,

text: and

text: moreover

text: deﬁnes

text: every

text: real

text: projective

text: set

text: in

text: the

text: sense

text: of

text: descriptive

text: set

text: theory

text: (see

text: e.g.

text: Kechris

text: Therefore

text: such

text: structures

text: are

text: maximally

text: wild

text: from

text: the

text: point

text: of

text: view

text: of

text: ﬁrst

text: R

text: order

text: model

text: theory.

text: In

text: contrast,

text: there

text: are

text: expansions

text: of

text: that

text: deﬁne

text: a

text: discrete

text: set

text: D

text: R

text: and

text: a

text: function

text: f

text: Dk

text: R

text: whose

text: image

text: is

text: some√where

text: dense,

text: R

text: Z

text: Z

text: but

text: whose

text: theory

text: is

text: decidable.

text: For

text: example,

text: by

text: is

text: such

text: B

text: a

text: structure.

text: All

text: known

text: examples

text: of

text: such

text: structures

text: deﬁne

text: We

text: show

text: that

text: this

text: is

text: necessairly

text: the

text: case.

text: R

text: R

text: Theorem

text: C.

text: Let

text: R

text: be

text: an

text: expansion

text: of

text: that

text: deﬁnes

text: a

text: discrete

text: set

text: D

text: k

text: R

text: B

text: and

text: a

text: map

text: f

text: D

text: whose

text: image

text: is

text: somewhere

text: dense.

text: Then

text: R

text: deﬁnes

text: R

text: Heuristically,

text: an

text: expansion

text: of

text: that

text: deﬁnably

text: collapses

text: a

text: discrete

text: set,

text: can

text: not

text: satisfy

text: any

text: kind

text: of

text: Shelah-style

text: tameness.

text: The

text: converse

text: implication

text: of

text: Theo-

text: rem

text: C

text: is

text: not

text: true.

text: The

text: expansion

text: of

text: the

text: real

text: ﬁeld

text: by

text: the

text: Cantor

text: set

text: K

text: constructed

text: B

text: in

text: deﬁnes

text: and

text: does

text: not

text: satisfy

text: the

text: assumptions

text: of

text: Theorem

text: C.

text: However,

text: B

text: R

text: Z

text: Z

text: deﬁnes

text: an

text: isomorphic

text: copy

text: of

text: by

text: Theorem

text: C].

text: For

text: that

text: ##LTLine##

text: ##LTLine##


text: N R INTERPRETING THE MSO THEORY OF ( ,+1) IN EXPANSIONS OF ( ,<) 11 subinterval of I on which f is either strictly increasing or strictly decreasing. It 0 suﬃces to ﬁnd an open subinterval of I on which f is a homeomorphism. We 0 assume that f is nonconstant and ﬁnd a subinterval on which f is a homeomor- phism. Let P be the image of f. Then P is an interval. As f is nonconstant P has → nonempty interior. Let g : P I be given by letting g(p) be the minimal element − 0 of f 1(p). This deﬁnition makes sense by continuity of f. Then g is an inverse of f. It suﬃces to ﬁnd an open subinterval of P on which g is continuous. We suppose towards a contradiction that there is no such subinterval. Applying Lemma 4.1 we ⊆ ﬁx an open subinterval J P and an (cid:15) > 0 such that g has oscillation at least 2(cid:15) ∈ − at every point in J. Thus for every p J and δ > 0 there is a p δ < q < p + δ | − | ≥ such that f(p) f(q) (cid:15). As I is bounded the image of the restriction of g to J ∈ 0 is bounded. We let p J be such that { ∈ } − sup g(q) : q J g(p) < (cid:15). { } → → ∞ Let x ∈N be a sequence of elements of J such that x p as n and | −n n | ≥ n − ≥ g(x ) g(p) (cid:15) for every n. This implies that g(x ) < g(p) and so g(p) g(x ) (cid:15) n n n for every n. We declare y = g(x ) for all n. As I is bounded there is a subsequence { } n n 0 of y ∈N which converges to a point y. As the closure of I is contained in I n n ∈ 0 we have y I. After restricting to a subsequence if necessary we suppose that { } ∈ y ∈N converges to y I. By deﬁnition of g we have f(y ) = x for all n. As f n n n n is continuous we have: f(y) = lim f(y ) = lim x = p. →∞ n →∞ n n n − ≥ − ≥ As g(p) y (cid:15) for all n we have g(p) y (cid:15). Thus g(p) > y. This is a n − (cid:3) contradiction as f(y) = p and g(p) is the minimum of f 1(p). The theorem below is an easy consequence of the previous proposition. ⊆ R → R Theorem 4.3. Let I be an open interval and let f : I be a continuous ⊆ deﬁnable function. There is a deﬁnable open U I dense in I such that f is strictly increasing, strictly decreasing, or constant on each connected component of U. Proof. We deﬁne several subsets of I. ⊆ ∈ − (1) Let U I be the set of p I such that f is strictly increasing on (p δ, p+δ) 1 for some δ > 0. ⊆ ∈ − (2) Let U I be the set of p I such that f is strictly decreasing on (p 2 δ, p + δ) for some δ > 0. ∈ − (3) Let U be the set of p I such that f is constant on (p δ, p + δ) for some 3 δ > 0. ∈ { } Each U is open for i 1, 2, 3 by deﬁnition. Proposition 4.2 implies that every i open subinterval of I contains an open interval on which f is strictly increasing, ∪ ∪ strictly decreasing or constant. Thus U U U is dense in I. No connected com- 1 2 3 ∈ { } ponent of U intersects any connected component of U for distinct i, j 1, 2, 3 . i ∪ ∪ j Therefore the connected components of U U U are exactly the connected com- 1 2 3 ponents of U , U and U . It follows that f is strictly increasing, strictly decreasing 1 2 3 ∪ ∪ (cid:3) or constant on each connected component of U U U . 1 2 3

text: N

text: R

text: INTERPRETING

text: THE

text: MSO

text: THEORY

text: OF

text: IN

text: EXPANSIONS

text: OF

text: subinterval

text: of

text: I

text: on

text: which

text: f

text: is

text: either

text: strictly

text: increasing

text: or

text: strictly

text: decreasing.

text: It

text: suﬃces

text: to

text: ﬁnd

text: an

text: open

text: subinterval

text: of

text: I

text: on

text: which

text: f

text: is

text: a

text: homeomorphism.

text: We

text: assume

text: that

text: f

text: is

text: nonconstant

text: and

text: ﬁnd

text: a

text: subinterval

text: on

text: which

text: f

text: is

text: a

text: homeomor-

text: phism.

text: Let

text: P

text: be

text: the

text: image

text: of

text: f.

text: Then

text: P

text: is

text: an

text: interval.

text: As

text: f

text: is

text: nonconstant

text: P

text: has

text: nonempty

text: interior.

text: Let

text: g

text: P

text: I

text: be

text: given

text: by

text: letting

text: g(p)

text: be

text: the

text: minimal

text: element

text: of

text: f

text: This

text: deﬁnition

text: makes

text: sense

text: by

text: continuity

text: of

text: f.

text: Then

text: g

text: is

text: an

text: inverse

text: of

text: f.

text: It

text: suﬃces

text: to

text: ﬁnd

text: an

text: open

text: subinterval

text: of

text: P

text: on

text: which

text: g

text: is

text: continuous.

text: We

text: suppose

text: towards

text: a

text: contradiction

text: that

text: there

text: is

text: no

text: such

text: subinterval.

text: Applying

text: Lemma

text: we

text: ﬁx

text: an

text: open

text: subinterval

text: J

text: P

text: and

text: an

text: such

text: that

text: g

text: has

text: oscillation

text: at

text: least

text: at

text: every

text: point

text: in

text: J.

text: Thus

text: for

text: every

text: p

text: J

text: and

text: δ

text: there

text: is

text: a

text: p

text: δ

text: q

text: p

text: δ

text: such

text: that

text: f(p)

text: f(q)

text: As

text: I

text: is

text: bounded

text: the

text: image

text: of

text: the

text: restriction

text: of

text: g

text: to

text: J

text: is

text: bounded.

text: We

text: let

text: p

text: J

text: be

text: such

text: that

text: sup

text: g(q)

text: q

text: J

text: g(p)

text: Let

text: x

text: ∈N

text: be

text: a

text: sequence

text: of

text: elements

text: of

text: J

text: such

text: that

text: x

text: p

text: as

text: n

text: and

text: −n

text: n

text: n

text: g(x

text: g(p)

text: for

text: every

text: n.

text: This

text: implies

text: that

text: g(x

text: g(p)

text: and

text: so

text: g(p)

text: g(x

text: n

text: n

text: n

text: for

text: every

text: n.

text: We

text: declare

text: y

text: g(x

text: for

text: all

text: n.

text: As

text: I

text: is

text: bounded

text: there

text: is

text: a

text: subsequence

text: n

text: n

text: of

text: y

text: ∈N

text: which

text: converges

text: to

text: a

text: point

text: y.

text: As

text: the

text: closure

text: of

text: I

text: is

text: contained

text: in

text: I

text: n

text: n

text: we

text: have

text: y

text: I.

text: After

text: restricting

text: to

text: a

text: subsequence

text: if

text: necessary

text: we

text: suppose

text: that

text: y

text: ∈N

text: converges

text: to

text: y

text: I.

text: By

text: deﬁnition

text: of

text: g

text: we

text: have

text: f(y

text: x

text: for

text: all

text: n.

text: As

text: f

text: n

text: n

text: n

text: n

text: is

text: continuous

text: we

text: have:

text: f(y)

text: lim

text: f(y

text: lim

text: x

text: p.

text: n

text: n

text: n

text: n

text: As

text: g(p)

text: y

text: for

text: all

text: n

text: we

text: have

text: g(p)

text: y

text: Thus

text: g(p)

text: y.

text: This

text: is

text: a

text: n

text: contradiction

text: as

text: f(y)

text: p

text: and

text: g(p)

text: is

text: the

text: minimum

text: of

text: f

text: The

text: theorem

text: below

text: is

text: an

text: easy

text: consequence

text: of

text: the

text: previous

text: proposition.

text: R

text: R

text: Theorem

text: Let

text: I

text: be

text: an

text: open

text: interval

text: and

text: let

text: f

text: I

text: be

text: a

text: continuous

text: deﬁnable

text: function.

text: There

text: is

text: a

text: deﬁnable

text: open

text: U

text: I

text: dense

text: in

text: I

text: such

text: that

text: f

text: is

text: strictly

text: increasing,

text: strictly

text: decreasing,

text: or

text: constant

text: on

text: each

text: connected

text: component

text: of

text: U.

text: Proof.

text: We

text: deﬁne

text: several

text: subsets

text: of

text: I.

text: Let

text: U

text: I

text: be

text: the

text: set

text: of

text: p

text: I

text: such

text: that

text: f

text: is

text: strictly

text: increasing

text: on

text: (p

text: δ,

text: p+δ)

text: for

text: some

text: δ

text: Let

text: U

text: I

text: be

text: the

text: set

text: of

text: p

text: I

text: such

text: that

text: f

text: is

text: strictly

text: decreasing

text: on

text: (p

text: δ,

text: p

text: δ)

text: for

text: some

text: δ

text: Let

text: U

text: be

text: the

text: set

text: of

text: p

text: I

text: such

text: that

text: f

text: is

text: constant

text: on

text: (p

text: δ,

text: p

text: δ)

text: for

text: some

text: δ

text: Each

text: U

text: is

text: open

text: for

text: i

text: by

text: deﬁnition.

text: Proposition

text: implies

text: that

text: every

text: i

text: open

text: subinterval

text: of

text: I

text: contains

text: an

text: open

text: interval

text: on

text: which

text: f

text: is

text: strictly

text: increasing,

text: strictly

text: decreasing

text: or

text: constant.

text: Thus

text: U

text: U

text: U

text: is

text: dense

text: in

text: I.

text: No

text: connected

text: com-

text: ponent

text: of

text: U

text: intersects

text: any

text: connected

text: component

text: of

text: U

text: for

text: distinct

text: i,

text: j

text: i

text: j

text: Therefore

text: the

text: connected

text: components

text: of

text: U

text: U

text: U

text: are

text: exactly

text: the

text: connected

text: com-

text: ponents

text: of

text: U

text: U

text: and

text: U

text: It

text: follows

text: that

text: f

text: is

text: strictly

text: increasing,

text: strictly

text: decreasing

text: or

text: constant

text: on

text: each

text: connected

text: component

text: of

text: U

text: U

text: U


text: 6 PHILIPP HIERONYMI AND ERIK WALSBERG (cid:54) Proof. With loss of generality we may assume a < b. Suppose that l(a, e) = l(b, e) (cid:54) ∈ ≺ or r(a, e) = r(b, e). Then there is d D(cid:22) such that a < d < b. Since (E, ) ≺e has order type ω, we can assume that e is -minimal in E(cid:31) with this property. e0 Together with l(a, e ) = l(b, e ) = 0 and r(a, e ) = r(b, e ) = 1, this gives 0 0 0 0 (1) l(a, p(e)) = l(b, p(e)) and r(a, p(e)) = r(b, p(e)). ≺ (cid:22) Thus p(e) d e. But then (cid:54) (cid:54) (2) l(a, e) = l(b, e) and r(a, e) = r(b, e). ∩ ∩ Since F E(cid:22) = F E(cid:22) , we get that either l(a, p(e)) = l(a, e) and l(b, p(e)) = a e b e l(b, e), or r(a, p(e)) = r(a, e) and r(b, p(e)) = r(b, e). This contradicts (1) and (cid:3) (2). ⊆ Lemma 1.3. For X E(cid:31) that is neither ﬁnite nor co-ﬁnite, there is unique ∈ e0 a C such that F = X. a Proof. The uniqueness follows immediately from Lemma 1.2 and the density of D. It is left to show the existence of such an a. We will construct a nested sequence { } ∈ I ∈N of non-empty closed subintervals of [0, 1] such that for every c int(I ) n n n (I) I = [l(c, e ), r(c, e )], n n n and if n > 0, (cid:54) ∈ (II) l(c, e − ) = l(c, e ) if and only if e X, n 1 n n (III) l(c, e − ) = l(c, e ) or r(c, e − ) = r(c, e ). n 1 n n 1 n Let I = [0, 1]. Since e = 0, I satisﬁes (I). Now suppose that we have already 0 0 0 ∈ constructed an interval In−1 satisfying (I)-(III). By (I) there are d1, d2 D(cid:22)en−1 such that I − = [d , d ]. n 1 1 2 ∈ ≤ ∩ First suppose that e X. Let d be the -maximal element of D(cid:22) (d , d ). n 3 en 1 2 Since E is a splitting set, such that a d exists. By (I) for I − , we necessar- ≺ ∈ 3 n 1 ily have d d . If c (d , d ), then it is easy to see that l(c, e ) = d . For 2 3 (cid:54) 3 2 ≤ n 3 such c, l(c, e − ) = l(c, e ). Since d was chosen to be the -maximal element ∩ n 1 n 3 ∈ of D(cid:22) (d , d ), we also get that r(c, e ) = r(c, e − ) = d for all c (d , d ). en 1 2 n n 1 2 3 2 Therefore set I = [d , d ]. n 3 2 ∈ ≤ ∩ Now consider the case that e / X. Let d be the -minimal element of D(cid:22) n 3 ≺ en (d , d ). Since E is a splitting set, such that a d exists, and d d by (I) for I − . 1 ∈2 3 1 3 n 1 If c (d , d ), then it is easy to see that r(c, e ) = d . Since d was chosen to be ≤ 1 3 ∩ n 3 3 the -minimal element of D(cid:22) (d , d ), we also get that l(c, e − ) = l(c, e ) = d en 1 2 n 1 n 1 for such c. Therefore set I = [d , d ]. (cid:84) n (cid:84) 1 3 It follows directly from the construction that I satisﬁes (I)-(III). By completeness R (cid:54) ∅ ∈ n ∈ of , ∈N I = . Let a ∈N I . In order to show that F = X and a C, n n n n a

text: PHILIPP

text: HIERONYMI

text: AND

text: ERIK

text: WALSBERG

text: Proof.

text: With

text: loss

text: of

text: generality

text: we

text: may

text: assume

text: a

text: b.

text: Suppose

text: that

text: l(a,

text: e)

text: l(b,

text: e)

text: or

text: r(a,

text: e)

text: r(b,

text: e).

text: Then

text: there

text: is

text: d

text: such

text: that

text: a

text: d

text: b.

text: Since

text: (E,

text: ≺e

text: has

text: order

text: type

text: ω,

text: we

text: can

text: assume

text: that

text: e

text: is

text: -minimal

text: in

text: with

text: this

text: property.

text: Together

text: with

text: l(a,

text: e

text: l(b,

text: e

text: and

text: r(a,

text: e

text: r(b,

text: e

text: this

text: gives

label: equation

text: l(a,

text: p(e))

text: l(b,

text: p(e))

text: and

text: r(a,

text: p(e))

text: r(b,

text: p(e)).

text: Thus

text: p(e)

text: d

text: e.

text: But

text: then

text: l(a,

text: e)

text: l(b,

text: e)

text: and

text: r(a,

text: e)

text: r(b,

text: e).

text: Since

text: F

text: F

text: we

text: get

text: that

text: either

text: l(a,

text: p(e))

text: l(a,

text: e)

text: and

text: l(b,

text: p(e))

text: a

text: e

text: b

text: e

text: l(b,

text: e),

text: or

text: r(a,

text: p(e))

text: r(a,

text: e)

text: and

text: r(b,

text: p(e))

text: r(b,

text: e).

text: This

text: contradicts

text: and

text: Lemma

text: For

text: X

text: that

text: is

text: neither

text: ﬁnite

text: nor

text: co-ﬁnite,

text: there

text: is

text: unique

text: a

text: C

text: such

text: that

text: F

text: X.

text: a

text: Proof.

text: The

text: uniqueness

text: follows

text: immediately

text: from

text: Lemma

text: and

text: the

text: density

text: of

text: D.

text: It

text: is

text: left

text: to

text: show

text: the

text: existence

text: of

text: such

text: an

text: a.

text: We

text: will

text: construct

text: a

text: nested

text: sequence

text: I

text: ∈N

text: of

text: non-empty

text: closed

text: subintervals

text: of

text: such

text: that

text: for

text: every

text: c

text: int(I

text: n

text: n

text: n

text: (I)

text: I

text: [l(c,

text: e

text: r(c,

text: e

text: n

text: n

text: n

text: and

text: if

text: n

text: (II)

text: l(c,

text: e

text: l(c,

text: e

text: if

text: and

text: only

text: if

text: e

text: X,

text: n

text: n

text: n

text: (III)

text: l(c,

text: e

text: l(c,

text: e

text: or

text: r(c,

text: e

text: r(c,

text: e

text: n

text: n

text: n

text: n

text: Let

text: I

text: Since

text: e

text: I

text: satisﬁes

text: (I).

text: Now

text: suppose

text: that

text: we

text: have

text: already

text: constructed

text: an

text: interval

text: satisfying

text: (I)-(III).

text: By

text: (I)

text: there

text: are

text: such

text: that

text: I

text: [d

text: d

text: n

text: First

text: suppose

text: that

text: e

text: X.

text: Let

text: d

text: be

text: the

text: -maximal

text: element

text: of

text: (d

text: d

text: n

text: en

text: Since

text: E

text: is

text: a

text: splitting

text: set,

text: such

text: that

text: a

text: d

text: exists.

text: By

text: (I)

text: for

text: I

text: we

text: necessar-

text: n

text: ily

text: have

text: d

text: d

text: If

text: c

text: (d

text: d

text: then

text: it

text: is

text: easy

text: to

text: see

text: that

text: l(c,

text: e

text: d

text: For

text: n

text: such

text: c,

text: l(c,

text: e

text: l(c,

text: e

text: Since

text: d

text: was

text: chosen

text: to

text: be

text: the

text: -maximal

text: element

text: n

text: n

text: of

text: (d

text: d

text: we

text: also

text: get

text: that

text: r(c,

text: e

text: r(c,

text: e

text: d

text: for

text: all

text: c

text: (d

text: d

text: en

text: n

text: n

text: Therefore

text: set

text: I

text: [d

text: d

text: n

text: Now

text: consider

text: the

text: case

text: that

text: e

text: X.

text: Let

text: d

text: be

text: the

text: -minimal

text: element

text: of

text: n

text: en

text: (d

text: d

text: Since

text: E

text: is

text: a

text: splitting

text: set,

text: such

text: that

text: a

text: d

text: exists,

text: and

text: d

text: d

text: by

text: (I)

text: for

text: I

text: ∈2

text: n

text: If

text: c

text: (d

text: d

text: then

text: it

text: is

text: easy

text: to

text: see

text: that

text: r(c,

text: e

text: d

text: Since

text: d

text: was

text: chosen

text: to

text: be

text: n

text: the

text: -minimal

text: element

text: of

text: (d

text: d

text: we

text: also

text: get

text: that

text: l(c,

text: e

text: l(c,

text: e

text: d

text: en

text: n

text: n

text: for

text: such

text: c.

text: Therefore

text: set

text: I

text: [d

text: d

text: n

text: It

text: follows

text: directly

text: from

text: the

text: construction

text: that

text: I

text: satisﬁes

text: (I)-(III).

text: By

text: completeness

text: R

text: n

text: of

text: ∈N

text: I

text: Let

text: a

text: ∈N

text: I

text: In

text: order

text: to

text: show

text: that

text: F

text: X

text: and

text: a

text: C,

text: n

text: n

text: n

text: n

text: a


text: CRUNCHING MORTALITY AND LIFE INSURANCE PORTFOLIOS 17 decades. In Austria we get a life expectancy of roughly 84.9 years for males and 86.9 for females born in 2014, i.e. signiﬁcantly above oﬃcial ﬁgures. For 60-year-old males and females we get expected future life times of 23.8 years and 26.1 years, respectively. Using estimated values of death probabilities, weights and risk factor variances, as well as population forecasts (which are usually freely available at oﬃcial statistical bureaus), our annuity model can be used to derive conﬁdence bands for absolute numbers of deaths due to certain causes setting Y = 1. Using diﬀerent MCMC parameter samples and using i the parameter δ as described above, parameter risk and uncertainty in forecasts can be incorporated, respectively. In light of the previous section, life tables can be projected into the future and, thus, it is straightforward to derive best estimate liabilities (BEL) of annuities and life insurance contracts. The possibility that death probabilities diﬀer from an expected curve, i.e. estimated parameters do no longer reﬂect the best estimate and have to be changed, contributes to mortality or longevity risk, when risk is measured over a one year time horizon as in Solvency II and the duration of in-force insurance contracts exceeds this time horizon. In our model, this risk can be captured by considering various MCMC samples (θˆh) , yielding h=1,...,m distributions of BELs. For example, taking D(T, T + t) as the discount curve from time T + t back to T and choosing an MCMC sample θˆh of parameters to calculate death probabilities qh (T + t) and survival probabilities ph (T ) at age a with gender g, the BEL for a term a+t,g a,g life insurance contract which pays 1 unit at the end of the year of death within the contract term of d years is given by (cid:0) (cid:1) (cid:88) d · AT θˆh = D(T, T + 1)qh (T ) + D(T, T + t + 1) ph (T )qh (T + t) . (4.3) a,g a,g t a,g a+t,g t=1 In a next step, this approach can be used as a building block for (partial) internal models to calculate basic solvency capital requirements (BSCR) for biometric underwriting risk under Solvency II, as illustrated in the following example. ∈ N Consider an insurance portfolio at time 0 with n whole life insurance policies with lump sum payments C > 0, for i = 1, . . . , n, upon death at the end of the year. Assume that i all assets are invested in an EU government bond (risk free under the standard model of the − Solvency II directive) with maturity 1, nominal A and coupon rate c > 1. Furthermore, 0 assume that we are only considering mortality risk and ignore proﬁt sharing, lapse, costs, reinsurance, deferred taxes, other assets and other liabilities, as well as the risk margin. Note that in this case, basic own funds, denoted by BOF , are given by market value of assets t minus BEL at time t, respectively. Then, the BSCR at time 0 is given by the 99.5% quantile of the change in basic own funds over the period [0, 1], denoted by ∆BOF , which can be 1 (cid:0) (cid:1) (cid:88) (cid:0) (cid:1) derived by, see (4.3), n − − − ∆BOF = BOF D(0, 1(cid:18))BOF = A 1 D(0, 1)(1 + c) C A0 θˆ(cid:19) 1 0 (cid:88) (cid:88) 1 0 (cid:0) (cid:1) (cid:88) (cid:88) (cid:0) i (cid:0)a,g(cid:1)(cid:1) i=1 D(0, 1) m n n Nih − + C A1 θˆh + C 1 A1 θˆh . m i a+1,g i a+1,g h=1 i=1 i=1 j=1 ##LTLine##

text: CRUNCHING

text: MORTALITY

text: AND

text: LIFE

text: INSURANCE

text: PORTFOLIOS

text: decades.

text: In

text: Austria

text: we

text: get

text: a

text: life

text: expectancy

text: of

text: roughly

text: years

text: for

text: males

text: and

text: for

text: females

text: born

text: in

text: i.e.

text: signiﬁcantly

text: above

text: oﬃcial

text: ﬁgures.

text: For

text: males

text: and

text: females

text: we

text: get

text: expected

text: future

text: life

text: times

text: of

text: years

text: and

text: years,

text: respectively.

text: Using

text: estimated

text: values

text: of

text: death

text: probabilities,

text: weights

text: and

text: risk

text: factor

text: variances,

text: as

text: well

text: as

text: population

text: forecasts

text: (which

text: are

text: usually

text: freely

text: available

text: at

text: oﬃcial

text: statistical

text: bureaus),

text: our

text: annuity

text: model

text: can

text: be

text: used

text: to

text: derive

text: conﬁdence

text: bands

text: for

text: absolute

text: numbers

text: of

text: deaths

text: due

text: to

text: certain

text: causes

text: setting

text: Y

text: Using

text: diﬀerent

text: MCMC

text: parameter

text: samples

text: and

text: using

text: i

text: the

text: parameter

text: δ

text: as

text: described

text: above,

text: parameter

text: risk

text: and

text: uncertainty

text: in

text: forecasts

text: can

text: be

text: incorporated,

text: respectively.

text: In

text: light

text: of

text: the

text: previous

text: section,

text: life

text: tables

text: can

text: be

text: projected

text: into

text: the

text: future

text: and,

text: thus,

text: it

text: is

text: straightforward

text: to

text: derive

text: best

text: estimate

text: liabilities

text: (BEL)

text: of

text: annuities

text: and

text: life

text: insurance

text: contracts.

text: The

text: possibility

text: that

text: death

text: probabilities

text: diﬀer

text: from

text: an

text: expected

text: curve,

text: i.e.

text: estimated

text: parameters

text: do

text: no

text: longer

text: reﬂect

text: the

text: best

text: estimate

text: and

text: have

text: to

text: be

text: changed,

text: contributes

text: to

text: mortality

text: or

text: longevity

text: risk,

text: when

text: risk

text: is

text: measured

text: over

text: a

text: one

text: year

text: time

text: horizon

text: as

text: in

text: Solvency

text: II

text: and

text: the

text: duration

text: of

text: in-force

text: insurance

text: contracts

text: exceeds

text: this

text: time

text: horizon.

text: In

text: our

text: model,

text: this

text: risk

text: can

text: be

text: captured

text: by

text: considering

text: various

text: MCMC

text: samples

text: (θˆh)

text: yielding

text: distributions

text: of

text: BELs.

text: For

text: example,

text: taking

text: D(T,

text: T

text: t)

text: as

text: the

text: discount

text: curve

text: from

text: time

text: T

text: t

text: back

text: to

text: T

text: and

text: choosing

text: an

text: MCMC

text: sample

text: θˆh

text: of

text: parameters

text: to

text: calculate

text: death

text: probabilities

text: qh

text: (T

text: t)

text: and

text: survival

text: probabilities

text: ph

text: (T

text: at

text: age

text: a

text: with

text: gender

text: g,

text: the

text: BEL

text: for

text: a

text: term

text: a+t,g

text: a,g

text: life

text: insurance

text: contract

text: which

text: pays

text: unit

text: at

text: the

text: end

text: of

text: the

text: year

text: of

text: death

text: within

text: the

text: contract

text: term

text: of

text: d

text: years

text: is

text: given

text: by

text: d

text: AT

text: θˆh

text: D(T,

text: T

text: (T

text: D(T,

text: T

text: t

text: ph

text: (T

text: )qh

text: (T

text: t)

text: a,g

text: a,g

text: t

text: a,g

text: a+t,g

text: In

text: a

text: next

text: step,

text: this

text: approach

text: can

text: be

text: used

text: as

text: a

text: building

text: block

text: for

text: (partial)

text: internal

text: models

text: to

text: calculate

text: basic

text: solvency

text: capital

text: requirements

text: (BSCR)

text: for

text: biometric

text: underwriting

text: risk

text: under

text: Solvency

text: II,

text: as

text: illustrated

text: in

text: the

text: following

text: example.

text: N

text: Consider

text: an

text: insurance

text: portfolio

text: at

text: time

text: with

text: n

text: whole

text: life

text: insurance

text: policies

text: with

text: lump

text: sum

text: payments

text: C

text: for

text: i

text: n,

text: upon

text: death

text: at

text: the

text: end

text: of

text: the

text: year.

text: Assume

text: that

text: i

text: all

text: assets

text: are

text: invested

text: in

text: an

text: EU

text: government

text: bond

text: (risk

text: free

text: under

text: the

text: standard

text: model

text: of

text: the

text: Solvency

text: II

text: directive)

text: with

text: maturity

text: nominal

text: A

text: and

text: coupon

text: rate

text: c

text: Furthermore,

text: assume

text: that

text: we

text: are

text: only

text: considering

text: mortality

text: risk

text: and

text: ignore

text: proﬁt

text: sharing,

text: lapse,

text: costs,

text: reinsurance,

text: deferred

text: taxes,

text: other

text: assets

text: and

text: other

text: liabilities,

text: as

text: well

text: as

text: the

text: risk

text: margin.

text: Note

text: that

text: in

text: this

text: case,

text: basic

text: own

text: funds,

text: denoted

text: by

text: BOF

text: are

text: given

text: by

text: market

text: value

text: of

text: assets

text: t

text: minus

text: BEL

text: at

text: time

text: t,

text: respectively.

text: Then,

text: the

text: BSCR

text: at

text: time

text: is

text: given

text: by

text: the

text: quantile

text: of

text: the

text: change

text: in

text: basic

text: own

text: funds

text: over

text: the

text: period

text: denoted

text: by

text: ∆BOF

text: which

text: can

text: be

text: derived

text: by,

text: see

text: n

text: ∆BOF

text: BOF

text: A

text: c)

text: C

text: i

text: m

text: n

text: n

text: Nih

text: C

text: θˆh

text: C

text: θˆh

text: m

text: i

text: i

text: ##LTLine##


text: CRUNCHING MORTALITY AND LIFE INSURANCE PORTFOLIOS 9 infectious neoplasm endocrine mental nervous 1987 1990 1993 1996 1999 2002 2005 2008 2011 year circulatory respiratory digestive external genitourinary 1987 1990 1993 1996 1999 2002 2005 2008 2011 year Figure 4.1. Estimated risk factor realisations. Table 4.1. Estimated weights for all death causes with ﬁve and 95 percent quantiles in brackets. male female 60 to 64 years 80 to 84 years 60 to 64 years 80 to 84 years (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 2011 2031 (quant.) 2011 2031 (quant.) 2011 2031 (quant.) 2011 2031 (quant.) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) neop. 0.499 0.547 (cid:0)0.561(cid:1) 0.324 0.378 (cid:0)0.392(cid:1) 0.592 0.648 (cid:0)0.662(cid:1) 0.263 0.303 (cid:0)0.319(cid:1) 0.531 0.364 0.629 0.288 circ. 0.228 0.116 (cid:0)0.123(cid:1) 0.325 0.173 (cid:0)0.181(cid:1) 0.140 0.060 (cid:0)0.065(cid:1) 0.342 0.149 (cid:0)0.158(cid:1) 0.109 0.164 0.055 0.140 ext. 0.056 0.062 (cid:0)0.073(cid:1) 0.026 0.028 (cid:0)0.033(cid:1) 0.072 0.069 (cid:0)0.078(cid:1) 0.100 0.126 (cid:0)0.139(cid:1) 0.053 0.024 0.060 0.113 resp. 0.051 0.036 (cid:0)0.040(cid:1) 0.106 0.092 (cid:0)0.101(cid:1) 0.038 0.037 (cid:0)0.043(cid:1) 0.051 0.068 (cid:0)0.074(cid:1) 0.032 0.083 0.032 0.061 endo. 0.044 0.062 (cid:0)0.070(cid:1) 0.047 0.077 (cid:0)0.084(cid:1) 0.036 0.051 (cid:0)0.060(cid:1) 0.054 0.080 (cid:0)0.089(cid:1) 0.055 0.070 0.043 0.071 dig. 0.041 0.036 (cid:0)0.040(cid:1) 0.027 0.020 (cid:0)0.023(cid:1) 0.035 0.032 (cid:0)0.038(cid:1) 0.024 0.023 (cid:0)0.027(cid:1) 0.031 0.018 0.026 0.020 nerv. 0.029 0.052 (cid:0)0.061(cid:1) 0.045 0.061 (cid:0)0.068(cid:1) 0.031 0.024 (cid:0)0.029(cid:1) 0.034 0.023 (cid:0)0.027(cid:1) 0.045 0.055 0.020 0.020 idio. 0.018 0.028 (cid:0)0.034(cid:1) 0.015 0.018 (cid:0)0.020(cid:1) 0.022 0.023 (cid:0)0.028(cid:1) 0.023 0.024 (cid:0)0.027(cid:1) 0.023 0.016 0.019 0.022 inf. 0.014 0.025 (cid:0)0.033(cid:1) 0.015 0.022 (cid:0)0.027(cid:1) 0.014 0.020 (cid:0)0.027(cid:1) 0.017 0.024 (cid:0)0.028(cid:1) 0.020 0.019 0.015 0.020 ment. 0.013 0.027 0.036 0.041 0.105 0.130 0.012 0.032 0.046 0.062 0.155 0.188 0.019 0.078 0.021 0.118 geni. 0.008 0.008 0.010 0.028 0.025 0.028 0.009 0.005 0.006 0.029 0.026 0.028 0.006 0.023 0.004 0.023 uncertainty in forecasted weights is reﬂected by wide conﬁdence intervals (values in brackets) for the risk factor of mental and behavioural disorders. These conﬁdence intervals are derived noitasilaer FR noitasilaer FR 1.1 9.0 7.0 1.1 9.0 7.0 ##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##

text: CRUNCHING

text: MORTALITY

text: AND

text: LIFE

text: INSURANCE

text: PORTFOLIOS

text: infectious

text: neoplasm

text: endocrine

text: mental

text: nervous

text: year

text: circulatory

text: respiratory

text: digestive

text: external

text: genitourinary

text: year

text: Figure

text: Estimated

text: risk

text: factor

text: realisations.

text: Table

text: Estimated

text: weights

text: for

text: all

text: death

text: causes

text: with

text: ﬁve

text: and

text: percent

text: quantiles

text: in

text: brackets.

text: male

label: table

text: female

text: to

text: years

text: to

text: years

text: to

text: years

text: to

text: years

text: (quant.)

text: (quant.)

text: (quant.)

text: (quant.)

text: neop.

text: circ.

text: ext.

text: resp.

text: endo.

text: dig.

text: nerv.

text: idio.

text: inf.

text: ment.

text: geni.

text: uncertainty

text: in

text: forecasted

text: weights

text: is

text: reﬂected

text: by

text: wide

text: conﬁdence

text: intervals

text: (values

text: in

text: brackets)

text: for

text: the

text: risk

text: factor

text: of

text: mental

text: and

text: behavioural

text: disorders.

text: These

text: conﬁdence

text: intervals

text: are

text: derived

text: noitasilaer

text: FR

text: noitasilaer

text: FR

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