text: be divided, separated by colons, as such: python main.py sensitivity ALPHA:.04:.94:7. (c) The ‘distributions’ option tests and plots the four automatic alternatives of taxes distributions together. (d) Finally, the ‘acps’ type runs the model for each one of the 46 ACPs available in the model, at least once for each one. 2. Number of runs per conﬁg: on top of any of the runs selected in the ﬁrst option, the modeler may set the number of runs for each conﬁguration in order to consider stochasticity. For example, if the modeler is running a sensitivity analysis on ALPHA, for seven diﬀerent values, then she may choose to run 4 times for each value. That means that PolicySpace will run 28 times for this example. The output is organized as such that the plots and ﬁles will manage averages and folders. − 3. Number of cores to use: is a typically operational decision. The default ( 1) is to run on all cores of your machine. 4. Here we discuss the main parameters brieﬂy: (a) ALPHA: is an exponent applied to the years of qualiﬁcation of employees in order to qualify productivity. (b) ALTERNATIVE0 takes ‘false’ or ‘true’ and determines whether municipal- ities function as they are presently (true) or as if they are together, as a single unit for ﬁscal purposes. (c) BETA: is the parameter of propensity to consume for the families. A high beta implies in more consumption of the family and less money saved for the real estate market. (d) FPM DISTRIBUTION: FPM is a municipal transfer tax in Brazil. FPM as ‘true’, the default, means that taxes follow FPM distribution rules. (e) HOUSE VACANCY: is the parameter used when generating houses that de- termines the amount of houses that will exceed the number of families in a given simulation conﬁguration. (f) LABOR MARKET: is the frequency the ﬁrm makes decisions on the labor market. (g) MARKUP: is the percentage the ﬁrms use when they decide to raise prices. (h) MEMBERS PER FAMILY: is also a parameter used to construct the agents. (i) PCT DISTANCE HIRING: is the percentage of the sample of candidates in a given month that will be selected via the distance to the ﬁrm criteria. The remaining of the sample is chosen by qualiﬁcation. (j) PERCENTAGE ACTUAL POP: is the size of actual population to be included in the model. (k) PERCENTAGE CHECK NEW LOCATION: is the sample of families at each given month that decide to enter the real estate market. (l) PROCESSING ACPS: is the choice of ACP to be simulated. (m) SIZE MARKET: is the number of ﬁrms that families check when deciding on where to make purchases. (n) STICKY PRICES: is the factor that controls the frequency in which the ﬁrms evaluates whether to raise prices. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: be

text: divided,

text: separated

text: by

text: colons,

text: as

text: such:

text: python

text: main.py

text: sensitivity

text: (c)

text: The

text: ‘distributions’

text: option

text: tests

text: and

text: plots

text: the

text: four

text: automatic

text: alternatives

text: of

text: taxes

text: distributions

text: together.

text: (d)

text: Finally,

text: the

text: ‘acps’

text: type

text: runs

text: the

text: model

text: for

text: each

text: one

text: of

text: the

text: ACPs

text: available

text: in

text: the

text: model,

text: at

text: least

text: once

text: for

text: each

text: one.

text: Number

text: of

text: runs

text: per

text: conﬁg:

text: on

text: top

text: of

text: any

text: of

text: the

text: runs

text: selected

text: in

text: the

text: ﬁrst

text: option,

text: the

text: modeler

text: may

text: set

text: the

text: number

text: of

text: runs

text: for

text: each

text: conﬁguration

text: in

text: order

text: to

text: consider

text: stochasticity.

text: For

text: example,

text: if

text: the

text: modeler

text: is

text: running

text: a

text: sensitivity

text: analysis

text: on

text: ALPHA,

text: for

text: seven

text: diﬀerent

text: values,

text: then

text: she

text: may

text: choose

text: to

text: run

text: times

text: for

text: each

text: value.

text: That

text: means

text: that

text: PolicySpace

text: will

text: run

text: times

text: for

text: this

text: example.

text: The

text: output

text: is

text: organized

text: as

text: such

text: that

text: the

text: plots

text: and

text: ﬁles

text: will

text: manage

text: averages

text: and

text: folders.

text: Number

text: of

text: cores

text: to

text: use:

text: is

text: a

text: typically

text: operational

text: decision.

text: The

text: default

text: is

text: to

text: run

text: on

text: all

text: cores

text: of

text: your

text: machine.

text: Here

text: we

text: discuss

text: the

text: main

text: parameters

text: brieﬂy:

text: (a)

text: ALPHA:

text: is

text: an

text: exponent

text: applied

text: to

text: the

text: years

text: of

text: qualiﬁcation

text: of

text: employees

text: in

text: order

text: to

text: qualify

text: productivity.

text: (b)

text: takes

text: ‘false’

text: or

text: ‘true’

text: and

text: determines

text: whether

text: municipal-

text: ities

text: function

text: as

text: they

text: are

text: presently

text: (true)

text: or

text: as

text: if

text: they

text: are

text: together,

text: as

text: a

text: single

text: unit

text: for

text: ﬁscal

text: purposes.

text: (c)

text: BETA:

text: is

text: the

text: parameter

text: of

text: propensity

text: to

text: consume

text: for

text: the

text: families.

text: A

text: high

text: beta

text: implies

text: in

text: more

text: consumption

text: of

text: the

text: family

text: and

text: less

text: money

text: saved

text: for

text: the

text: real

text: estate

text: market.

text: (d)

text: FPM

text: DISTRIBUTION:

text: FPM

text: is

text: a

text: municipal

text: transfer

text: tax

text: in

text: Brazil.

text: FPM

text: as

text: ‘true’,

text: the

text: default,

text: means

text: that

text: taxes

text: follow

text: FPM

text: distribution

text: rules.

text: (e)

text: HOUSE

text: VACANCY:

text: is

text: the

text: parameter

text: used

text: when

text: generating

text: houses

text: that

text: de-

text: termines

text: the

text: amount

text: of

text: houses

text: that

text: will

text: exceed

text: the

text: number

text: of

text: families

text: in

text: a

text: given

text: simulation

text: conﬁguration.

text: (f)

text: LABOR

text: MARKET:

text: is

text: the

text: frequency

text: the

text: ﬁrm

text: makes

text: decisions

text: on

text: the

text: labor

text: market.

text: (g)

text: MARKUP:

text: is

text: the

text: percentage

text: the

text: ﬁrms

text: use

text: when

text: they

text: decide

text: to

text: raise

text: prices.

text: (h)

text: MEMBERS

text: PER

text: FAMILY:

text: is

text: also

text: a

text: parameter

text: used

text: to

text: construct

text: the

text: agents.

text: (i)

text: PCT

text: DISTANCE

text: HIRING:

text: is

text: the

text: percentage

text: of

text: the

text: sample

text: of

text: candidates

text: in

text: a

text: given

text: month

text: that

text: will

text: be

text: selected

text: via

text: the

text: distance

text: to

text: the

text: ﬁrm

text: criteria.

text: The

text: remaining

text: of

text: the

text: sample

text: is

text: chosen

text: by

text: qualiﬁcation.

text: (j)

text: PERCENTAGE

text: ACTUAL

text: POP:

text: is

text: the

text: size

text: of

text: actual

text: population

text: to

text: be

text: included

text: in

text: the

text: model.

text: (k)

text: PERCENTAGE

text: CHECK

text: NEW

text: LOCATION:

text: is

text: the

text: sample

text: of

text: families

text: at

text: each

text: given

text: month

text: that

text: decide

text: to

text: enter

text: the

text: real

text: estate

text: market.

text: (l)

text: PROCESSING

text: ACPS:

text: is

text: the

text: choice

text: of

text: ACP

text: to

text: be

text: simulated.

text: (m)

text: SIZE

text: MARKET:

text: is

text: the

text: number

text: of

text: ﬁrms

text: that

text: families

text: check

text: when

text: deciding

text: on

text: where

text: to

text: make

text: purchases.

text: (n)

text: STICKY

text: PRICES:

text: is

text: the

text: factor

text: that

text: controls

text: the

text: frequency

text: in

text: which

text: the

text: ﬁrms

text: evaluates

text: whether

text: to

text: raise

text: prices.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 Concluding considerations and further development This paper has brieﬂy introduced PolicySpace, an ex-ante Public Policy anal- ysis tool, demonstrated its main parameters and usage and illustrated the model with an application of alternative taxes distribution testing. However, part of the appeal of PolicySpace is that other modelers and researchers may use the model and change, adapt or evolve it for their own interests and research questions. Tentatively, we can think of at least some suggestions: – Further detailing the ﬁrms, providing them with diﬀerentiated sectors and products, testing decision-making strategies. – Making full use of the transversal and economically integrated commuting demand embedded in the model, considering that families are mobile and geocoded and that workers move employment among ﬁrms and that com- muting is already computed within the model. – Make full use of all the data generated, especially real estate housing data, agents, families, ﬁrms, population dynamics and municipalities. – And, deeper testing and empirically detailing of taxes analysis and trickle eﬀects within municipalities. References 1. Abar, S., Theodoropoulos, G. K., Lemarinier, P., and O’Hare, G. M. P. Agent based modelling and simulation tools: A review of the state-of-art software. 13–33. 2. Ahrend, R., Farchy, E., Kaplanis, I., and Lembcke, A. C. What makes cities more productive? evidence on the role of urban governance from ﬁve OECD countries. 33. 3. Baptista, R., Farmer, J. D., Hinterschweiger, M., Low, K., Tang, D., and Uluc, A. Macroprudential policy in an agent-based model of the UK housing market. 4. Batty, M. A generic framework for computational spatial modelling. In Agent- based models of geographical systems. Springer, pp. 19–50. 5. Boero, R., Morini, M., Sonnesa, M., and Terna, P. Agent-based models of the economy: from theories to applications. Palgrave Macmillan. 6. Colander, D., and Kupers, R. Complexity and the Art of Public Policy: Solving Society’s Problems from the Bottom Up. Princeton University Press. 7. Dawid, H., Gemkow, S., Harting, P., Van der Hoog, S., and Neugart, M. Agent-based macroeconomic modeling and policy analysis: the eurace@ unibi model. 8. Dosi, G., Fagiolo, G., and Roventini, A. Schumpeter meeting keynes: A policy- friendly model of endogenous growth and business cycles. 1748–1767. 9. Edmonds, B., and Meyer, R. Simulating Social Complexity: A Handbook, 2013 edition ed. Springer. 10. Epstein, J. M., and Axtell, R. Growing artiﬁcial societies: social science from the bottom up. Brookings/MIT Press.

label: section

text: Concluding

text: considerations

text: and

text: further

text: development

text: This

label: paragraph

text: paper

text: has

text: brieﬂy

text: introduced

text: PolicySpace,

text: an

text: ex-ante

text: Public

text: Policy

text: anal-

text: ysis

text: tool,

text: demonstrated

text: its

text: main

text: parameters

text: and

text: usage

text: and

text: illustrated

text: the

text: model

text: with

text: an

text: application

text: of

text: alternative

text: taxes

text: distribution

text: testing.

text: However,

text: part

text: of

text: the

text: appeal

text: of

text: PolicySpace

text: is

text: that

text: other

text: modelers

text: and

text: researchers

text: may

text: use

text: the

text: model

text: and

text: change,

text: adapt

text: or

text: evolve

text: it

text: for

text: their

text: own

text: interests

text: and

text: research

text: questions.

text: Tentatively,

text: we

text: can

text: think

text: of

text: at

text: least

text: some

text: suggestions:

label: list

text: Further

text: detailing

text: the

text: ﬁrms,

text: providing

text: them

text: with

text: diﬀerentiated

text: sectors

text: and

text: products,

text: testing

text: decision-making

text: strategies.

text: Making

text: full

text: use

text: of

text: the

text: transversal

text: and

text: economically

text: integrated

text: commuting

text: demand

text: embedded

text: in

text: the

text: model,

text: considering

text: that

text: families

text: are

text: mobile

text: and

text: geocoded

text: and

text: that

text: workers

text: move

text: employment

text: among

text: ﬁrms

text: and

text: that

text: com-

text: muting

text: is

text: already

text: computed

text: within

text: the

text: model.

text: Make

text: full

text: use

text: of

text: all

text: the

text: data

text: generated,

text: especially

text: real

text: estate

text: housing

text: data,

text: agents,

text: families,

text: ﬁrms,

text: population

text: dynamics

text: and

text: municipalities.

text: And,

text: deeper

text: testing

text: and

text: empirically

text: detailing

text: of

text: taxes

text: analysis

text: and

text: trickle

text: eﬀects

text: within

text: municipalities.

text: References

text: Abar,

text: S.,

text: Theodoropoulos,

text: G.

text: K.,

text: Lemarinier,

text: P.,

text: and

text: O’Hare,

text: G.

text: M.

text: P.

text: Agent

text: based

text: modelling

text: and

text: simulation

text: tools:

text: A

text: review

text: of

text: the

text: state-of-art

text: software.

text: Ahrend,

text: R.,

text: Farchy,

text: E.,

text: Kaplanis,

text: I.,

text: and

text: Lembcke,

text: A.

text: C.

text: What

text: makes

text: cities

text: more

text: productive?

text: evidence

text: on

text: the

text: role

text: of

text: urban

text: governance

text: from

text: ﬁve

text: OECD

text: countries.

text: Baptista,

text: R.,

text: Farmer,

text: J.

text: D.,

text: Hinterschweiger,

text: M.,

text: Low,

text: K.,

text: Tang,

text: D.,

text: and

text: Uluc,

text: A.

text: Macroprudential

text: policy

text: in

text: an

text: agent-based

text: model

text: of

text: the

text: UK

text: housing

text: market.

text: Batty,

text: M.

text: A

text: generic

text: framework

text: for

text: computational

text: spatial

text: modelling.

text: In

text: Agent-

text: based

text: models

text: of

text: geographical

text: systems.

text: Springer,

text: pp.

text: Boero,

text: R.,

text: Morini,

text: M.,

text: Sonnesa,

text: M.,

text: and

text: Terna,

text: P.

text: Agent-based

text: models

text: of

text: the

text: economy:

text: from

text: theories

text: to

text: applications.

text: Palgrave

text: Macmillan.

text: Colander,

text: D.,

text: and

text: Kupers,

text: R.

text: Complexity

text: and

text: the

text: Art

text: of

text: Public

text: Policy:

text: Solving

text: Society’s

text: Problems

text: from

text: the

text: Bottom

text: Up.

text: Princeton

text: University

text: Press.

text: Dawid,

text: H.,

text: Gemkow,

text: S.,

text: Harting,

text: P.,

text: Van

text: der

text: Hoog,

text: S.,

text: and

text: Neugart,

text: M.

text: Agent-based

text: macroeconomic

text: modeling

text: and

text: policy

text: analysis:

text: the

text: eurace@

text: unibi

text: model.

text: Dosi,

text: G.,

text: Fagiolo,

text: G.,

text: and

text: Roventini,

text: A.

text: Schumpeter

text: meeting

text: keynes:

text: A

text: policy-

text: friendly

text: model

text: of

text: endogenous

text: growth

text: and

text: business

text: cycles.

text: Edmonds,

text: B.,

text: and

text: Meyer,

text: R.

text: Simulating

text: Social

text: Complexity:

text: A

text: Handbook,

text: edition

text: ed.

text: Springer.

text: Epstein,

text: J.

text: M.,

text: and

text: Axtell,

text: R.

text: Growing

text: artiﬁcial

text: societies:

text: social

text: science

text: from

text: the

text: bottom

text: up.

text: Brookings/MIT

text: Press.


text: Electronic States and Dissipations of Vortex Core in Quantum Limit Investigated by Microwave Complex Resistivity Measurements on Pure FeSe Single Crystals ∗ † ‡ § Yoshinori Imai, Takahiro Urata, Yoichi Tanabe, and Atsutaka Maeda ¶ ∗∗ †† Tatsunori Okada and Katsumi Tanigaki ¶ and ∗∗ 1 (Dated: May 30, 2020) Abstract In order to clarify the electronic state and the mechanism of energy dissipations by the motion of the vortex core in the quantum limit, we measured the microwave complex conductivity of pure FeSe single crystals in the zero-ﬁeld limit and under ﬁnite magnetic ﬁelds. The temperature dependence of the superﬂuid density changed as 1 − (T/T )1.2, indicating the presence of nodal lines in the c superconducting gap. From the magnetic-ﬁeld dependence of the ﬂux-ﬂow resistivity, we found that a barometer of the electronic state inside the vortex core turned out to be ω τ = 1 ± 0.5. 0 core This suggests that the vortex core of pure FeSe is still in the moderately clean regime, which is inconsistent with the expectation that the super-clean core is realized in this material. We also found that the mean-free path inside the vortex core is suppressed at the distance of the order of the core radius. Based on the observed results and previous reports, we discussed possible origins of rather small ω τ in terms of the multiple-bands nature and additional mechanisms giving 0 core extra energy dissipations speciﬁc to the vortex core in motion. PACS numbers: 1

text: Electronic

label: title

text: States

text: and

text: Dissipations

text: of

text: Vortex

text: Core

text: in

text: Quantum

text: Limit

text: Investigated

text: by

text: Microwave

text: Complex

text: Resistivity

text: Measurements

text: on

text: Pure

text: FeSe

text: Single

text: Crystals

text: Yoshinori

label: author

text: Imai,

text: Takahiro

text: Urata,

text: Yoichi

text: Tanabe,

text: and

text: Atsutaka

text: Maeda

text: Tatsunori

text: Okada

text: and

text: Katsumi

text: Tanigaki

text: and

text: (Dated:

text: May

label: date

text: Abstract

text: In

label: abstract

text: order

text: to

text: clarify

text: the

text: electronic

text: state

text: and

text: the

text: mechanism

text: of

text: energy

text: dissipations

text: by

text: the

text: motion

text: of

text: the

text: vortex

text: core

text: in

text: the

text: quantum

text: limit,

text: we

text: measured

text: the

text: microwave

text: complex

text: conductivity

text: of

text: pure

text: FeSe

text: single

text: crystals

text: in

text: the

text: zero-ﬁeld

text: limit

text: and

text: under

text: ﬁnite

text: magnetic

text: ﬁelds.

text: The

text: temperature

text: dependence

text: of

text: the

text: superﬂuid

text: density

text: changed

text: as

text: (T/T

text: indicating

text: the

text: presence

text: of

text: nodal

text: lines

text: in

text: the

text: c

text: superconducting

text: gap.

text: From

text: the

text: magnetic-ﬁeld

text: dependence

text: of

text: the

text: ﬂux-ﬂow

text: resistivity,

text: we

text: found

text: that

text: a

text: barometer

text: of

text: the

text: electronic

text: state

text: inside

text: the

text: vortex

text: core

text: turned

text: out

text: to

text: be

text: ω

text: τ

text: core

text: This

text: suggests

text: that

text: the

text: vortex

text: core

text: of

text: pure

text: FeSe

text: is

text: still

text: in

text: the

text: moderately

text: clean

text: regime,

text: which

text: is

text: inconsistent

text: with

text: the

text: expectation

text: that

text: the

text: super-clean

text: core

text: is

text: realized

text: in

text: this

text: material.

text: We

text: also

text: found

text: that

text: the

text: mean-free

text: path

text: inside

text: the

text: vortex

text: core

text: is

text: suppressed

text: at

text: the

text: distance

text: of

text: the

text: order

text: of

text: the

text: core

text: radius.

text: Based

text: on

text: the

text: observed

text: results

text: and

text: previous

text: reports,

text: we

text: discussed

text: possible

text: origins

text: of

text: rather

text: small

text: ω

text: τ

text: in

text: terms

text: of

text: the

text: multiple-bands

text: nature

text: and

text: additional

text: mechanisms

text: giving

text: core

text: extra

text: energy

text: dissipations

text: speciﬁc

text: to

text: the

text: vortex

text: core

text: in

text: motion.

text: PACS

text: numbers:


text: FIG. 4. (Color online) The magnetic-ﬁeld dependence of crossover frequency, ω /2π, FeSe mea- cr sured at T = 2 K (red circles). For comparison, ω (1.8 K)/2π of FeSe Te single crystal cr 0.4 0.6 synthesized by the melt-growth method is also shown as black diamonds13. By ﬁtting ρ (B) with linear function, we obtain a coeﬃcient of linear increment of ρ (B → f f − 0)/B = (3.2 ± 1) × 10 8 µΩcm/T. The Shubnikov-de Haas Oscillation33 and the Hall resistivity25 measurements for pure FeSe single crystals report the carrier density of n = − − 3.1 × 1020 cm 3 and 1.9 × 1020 cm 3, respectively. From these numbers, we evaluate a barometer of electronic state of ω τ = Φ B/nπh¯ρ = 1 ± 0.5. This number is the largest 0 core 0 f among other FeSCs we have investigated; LiFeAs (ω τ = 0.4 ± 0.1)10, NaFe Co As 0 core 0.97 0.03 (ω τ = 0.14 ± 0.06)12, BaFe (As P ) (ω τ = 0.11 ± 0.06, calculated by using 0 core 2 0.55 0.45 2 0 core data in Ref.32), and FeSe Te (ω τ = 0.14 ± 0.03, calculated by using data in Ref.13). 0.4 0.6 0 core Thus, we believe that ω τ (cid:39) 1 is the manifestation of features of high purity and large 0 core ∆ /E ratio of pure FeSe single crystal. However, it is surprising that ω τ is still in the 0 F 0 core moderately clean regime. This is inconsistent with the expectation that the super-clean core is realized in pure FeSe. Here we consider the reason why ω τ of pure FeSe resulted in 0 core 8 ##LTFigure##

text: FIG.

text: (Color

label: caption

text: online)

text: The

text: magnetic-ﬁeld

text: dependence

text: of

text: crossover

text: frequency,

text: ω

text: FeSe

text: mea-

text: cr

text: sured

text: at

text: T

text: K

text: (red

text: circles).

text: For

text: comparison,

text: ω

text: of

text: FeSe

text: Te

text: single

text: crystal

text: cr

text: synthesized

text: by

text: the

text: melt-growth

text: method

text: is

text: also

text: shown

text: as

text: black

text: By

text: ﬁtting

text: ρ

text: (B)

text: with

text: linear

text: function,

text: we

text: obtain

text: a

text: coeﬃcient

text: of

text: linear

text: increment

text: of

text: ρ

text: (B

text: f

text: f

text: µΩcm/T.

text: The

text: Shubnikov-de

text: Haas

text: and

text: the

text: Hall

text: measurements

text: for

text: pure

text: FeSe

text: single

text: crystals

text: report

text: the

text: carrier

text: density

text: of

text: n

text: cm

text: and

text: cm

text: respectively.

text: From

text: these

text: numbers,

text: we

text: evaluate

text: a

text: barometer

text: of

text: electronic

text: state

text: of

text: ω

text: τ

text: Φ

text: B/nπh¯ρ

text: This

text: number

text: is

text: the

text: largest

text: core

text: f

text: among

text: other

text: FeSCs

text: we

text: have

text: investigated;

text: LiFeAs

text: (ω

text: τ

text: NaFe

text: Co

text: As

text: core

text: (ω

text: τ

text: BaFe

text: (As

text: P

text: (ω

text: τ

text: calculated

text: by

text: using

text: core

text: core

text: data

text: in

text: and

text: FeSe

text: Te

text: (ω

text: τ

text: calculated

text: by

text: using

text: data

text: in

text: core

text: Thus,

text: we

text: believe

text: that

text: ω

text: τ

text: is

text: the

text: manifestation

text: of

text: features

text: of

text: high

text: purity

text: and

text: large

text: core

text: /E

text: ratio

text: of

text: pure

text: FeSe

text: single

text: crystal.

text: However,

text: it

text: is

text: surprising

text: that

text: ω

text: τ

text: is

text: still

text: in

text: the

text: F

text: core

text: moderately

text: clean

text: regime.

text: This

text: is

text: inconsistent

text: with

text: the

text: expectation

text: that

text: the

text: super-clean

text: core

text: is

text: realized

text: in

text: pure

text: FeSe.

text: Here

text: we

text: consider

text: the

text: reason

text: why

text: ω

text: τ

text: of

text: pure

text: FeSe

text: resulted

text: in

text: core

text: ##LTFigure##


text: ( ) ≠ ∅ ∶ → ( ) ∩ ∏ ( ) ′ ′ ′ ∈ ( ) ∈ Theore⊕m 4.9 ([10] 4.41 (with 4.32)). Suppose I ∆,∆ . Then there is a( u)nique bijection ζ I I such that th′e set I ∆,∆ i I HomD V i,V ζ i is not empty. The latter non-emptiness condition is equivalent(i)n saying that for all i I the D-dimensions of V i and V ζ i coincide and the direct sum ∆ ∆ is equivalent to a sim(ple str)atum. i ζ i ′ ( ) ≠ ∅ ′ The map ζ is called the matching of ∆,∆ ,′ and we denote it also as ζ . ∆.∆ Theore(m 4.1)0= ([1(0] 4).3(6)(. Sup)po∩s∏e I ∆,∆ ( with m))atc(hin)g ζ. Then the following holds. ′ ′ ′ ( ) ∈ = (i) I ∆,∆ M ∆ I ∆,∆ Hom V i,V ζ i− M ′ ∆ . i I D ( ) = ( ) ( ) ( ) (ii) Suppose there is an element g˜ of G˜ suc′h that g˜β′g˜ 1 β , t×hen we have I ∆,∆ M ∆ g˜C β M ∆ . A (∗) Proof. The second assertion follows from [10, 4.36] and the ﬁrst assertion follows from the second by diagonalization, Proposition 4.2, and the fact that M only depends on the equivalence class of the semisimple stratum. We now can come to the self-dual case: ′ Theorem 4.11 (see [11] 6.22 for the case over F). Suppose ∆ and ∆ are intertwined by an element of G wit(h matc)h=in(g ζ∩. Th(en t)h)(e fo(llowing) h∩o∏lds. ( ))( ∩ ( )) ′ ′ ′ ( ) ∈ = (i) I ∆,∆ G M ∆ I ∆,∆ Hom − V i,V′ ζ i G M ∆ . G G i I D ( ) = ( ∩ ( ))( ∩ ( ) )( ∩ ( )) (ii) Suppose there is an element g˜ ′of G˜ such that′ g˜βg˜ 1 β , the×n we have I ∆,∆ G M ∆ G g˜C β G M ∆ . G A = { } Proof. This Theorem follows from Theorem 4.10, Proposition 4.8 and Proposition 3.3 for the group Γ σ,1 . The last Theorem has now consequence(s for th)e∩∏matching. ( ′ ) ′ ( ) ∈ Corollary 4.12 (see [6] 9.5 for the case over F). Suppose ∆ and ∆ are self-dual and intertwined by an ( ) element of G with matching ζ. Then I ∆,∆ Hom V i,V ζ i is non-empty and h is isometric G i I D J to h for all σ-invariant subse(ts J of)I∩. ∏In particula(r ζ and σ) commute. ζ J ′ ( ) ∈ ( ) ( ) Proof. The non-emptiness of I ∆,∆ Hom V i,V ζ i follows from 4.11(i). Every element of G i I D this intersection restricts to an isometry from h to h . If g is an element of the above intersection J ζ J then σ g too, and the uniqueness of the matching im(plies t)hat ζ and σ commute. ′ Later we are going to see that the non-emptiness of I ∆,∆ is not needed in Corollary 4.12 for σ and ζ G to commute. 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Theore⊕m

text: (with

text: Suppose

text: I

text: Then

text: there

text: is

text: a(

text: u)nique

text: bijection

text: ζ

text: I

text: I

text: such

text: that

text: th′e

text: set

text: I

text: i

text: I

text: HomD

text: V

text: i,V

text: ζ

text: i

text: is

text: not

text: empty.

text: The

text: latter

text: non-emptiness

text: condition

text: is

text: equivalent(i)n

text: saying

text: that

text: for

text: all

text: i

text: I

text: the

text: D-dimensions

text: of

text: V

text: i

text: and

text: V

text: ζ

text: i

text: coincide

text: and

text: the

text: direct

text: sum

text: is

text: equivalent

text: to

text: a

text: sim(ple

text: str)atum.

text: i

text: ζ

text: i

text: The

text: map

text: ζ

text: is

text: called

text: the

text: matching

text: of

text: and

text: we

text: denote

text: it

text: also

text: as

text: ζ

text: Theore(m

text: Sup)po∩s∏e

text: I

text: with

text: m))atc(hin)g

text: ζ.

text: Then

text: the

text: following

text: holds.

text: (i)

text: I

text: M

text: I

text: Hom

text: V

text: i,V

text: ζ

text: i−

text: M

text: i

text: I

text: D

text: (ii)

text: Suppose

text: there

text: is

text: an

text: element

text: g˜

text: of

text: G˜

text: suc′h

text: that

text: g˜β′g˜

text: β

text: t×hen

text: we

text: have

text: I

text: M

text: g˜C

text: β

text: M

text: A

text: Proof.

text: The

text: second

text: assertion

text: follows

text: from

text: and

text: the

text: ﬁrst

text: assertion

text: follows

text: from

text: the

text: second

text: by

text: diagonalization,

text: Proposition

text: and

text: the

text: fact

text: that

text: M

text: only

text: depends

text: on

text: the

text: equivalence

text: class

text: of

text: the

text: semisimple

text: stratum.

text: We

text: now

text: can

text: come

text: to

text: the

text: self-dual

text: case:

text: Theorem

text: (see

text: for

text: the

text: case

text: over

text: F).

text: Suppose

text: and

text: are

text: intertwined

text: by

text: an

text: element

text: of

text: G

text: wit(h

text: matc)h=in(g

text: ζ∩.

text: Th(en

text: t)h)(e

text: fo(llowing)

text: h∩o∏lds.

text: (i)

text: I

text: G

text: M

text: I

text: Hom

text: V

text: i,V′

text: ζ

text: i

text: G

text: M

text: G

text: G

text: i

text: I

text: D

text: (ii)

text: Suppose

text: there

text: is

text: an

text: element

text: g˜

text: ′of

text: G˜

text: such

text: that′

text: g˜βg˜

text: β

text: the×n

text: we

text: have

text: I

text: G

text: M

text: G

text: g˜C

text: β

text: G

text: M

text: G

text: A

text: Proof.

text: This

text: Theorem

text: follows

text: from

text: Theorem

text: Proposition

text: and

text: Proposition

text: for

text: the

text: group

text: Γ

text: The

text: last

text: Theorem

text: has

text: now

text: consequence(s

text: for

text: th)e∩∏matching.

text: Corollary

text: (see

text: for

text: the

text: case

text: over

text: F).

text: Suppose

text: and

text: are

text: self-dual

text: and

text: intertwined

text: by

text: an

text: element

text: of

text: G

text: with

text: matching

text: ζ.

text: Then

text: I

text: Hom

text: V

text: i,V

text: ζ

text: i

text: is

text: non-empty

text: and

text: h

text: is

text: isometric

text: G

text: i

text: I

text: D

text: J

text: to

text: h

text: for

text: all

text: σ-invariant

text: subse(ts

text: J

text: of)I∩.

text: ∏In

text: particula(r

text: ζ

text: and

text: σ)

text: commute.

text: ζ

text: J

text: Proof.

text: The

text: non-emptiness

text: of

text: I

text: Hom

text: V

text: i,V

text: ζ

text: i

text: follows

text: from

text: Every

text: element

text: of

text: G

text: i

text: I

text: D

text: this

text: intersection

text: restricts

text: to

text: an

text: isometry

text: from

text: h

text: to

text: h

text: If

text: g

text: is

text: an

text: element

text: of

text: the

text: above

text: intersection

text: J

text: ζ

text: J

text: then

text: σ

text: g

text: too,

text: and

text: the

text: uniqueness

text: of

text: the

text: matching

text: im(plies

text: t)hat

text: ζ

text: and

text: σ

text: commute.

text: Later

text: we

text: are

text: going

text: to

text: see

text: that

text: the

text: non-emptiness

text: of

text: I

text: is

text: not

text: needed

text: in

text: Corollary

text: for

text: σ

text: and

text: ζ

text: G

text: to

text: commute.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∣ ( ) ( ) ( ) ( ) ∶= ( −) ( ) ∶= ( ) ′ − ′ − ′ − of θ to H ∆ . The elements of C ∆ are called self-dual sem∈isim(ple) character∣s. For a stratum ∆ H ∆ equiva(len)t to ∆ we deﬁne C ∆ C ∆ and C ∆ C ∆ . ( ) − − Proposition 6.2. Let ∆ be a self-dual semisimple stratum and θ C ∆ . Then θ is an element H ∆ of C ∆ . ( ) ( ) ( ) ( )− ( ) ( ) − Proof. If θ ,θ and θ are elements of C ∆ then θ θ 1θ is an element of C ∆ by the deﬁnition of C ∆ , 1 2 3 ∈ ( ) 1 2 3 ( ) see [10, 5.6]. Thus the number of elements of C ∆ which have the same restriction as θ on H ∆ is independent θ. The number of elements of C ∆ is a power of p and thus there is a non-negative integer s such that for every element of θ C ∆ there are exactly ps elements in C ∆ with the same restriction as θ. Thus the action of σ on the set of those characters must have a ﬁxed point. This ﬁnishes the proof. ( ) ↦ ∈ ∶ → ∶ → ∩ Tdehneotgerobuyp G′g˜,χac(ts′ bgy.)χCc.onAjungaelteiomnenotngtheG˜seits osfa′′iadlltcoo(imntpelre′tx)wcinhearaaccthearsraχcteo−rnχsubgK′roupsCo(wf iGt˜h′w)ahsicehcownde chara(cter χ) K ′ if the restrictions of χ and g.χ coincide on′ gKg 1 K . We denote the set of all elem′ ents of G˜ which intertwine χ with χ by I χ,χ . We adapt the notation IH χ,χ for the intersection of I χ,χ with a subgroup H of G˜. We say that χ and χ intertwine by some element of H if I χ,χ is non-empty. H 6.2 Transfers We now recall the no′tion of transfer. For more detail consult section [10, section 6.2]. Transfers are deﬁned between strata over diﬀerent vectors spaces over diﬀerent skew-ﬁelds central and of ﬁnite degree over F. Let ∆ and ∆ be two endo-equivalent ∶sem(isim⊕ple)st→rata(. L)et us at ﬁrst recall the transfer over the same skew-ﬁeld. The canonically deﬁned restriction m′aps ⊕ ′ ∶=res∆ ∆ ,∆ ○C ∆ ∆ ∶ C( ∆) → ( ) − ′ ⊕ ⊕ deﬁne a bijection ′ ′ ′ ′ τ r′es res 1 C ∆ C ∆ ∆,∆ ∆ ∆ ,∆ ∆ ∆ ,∆ called the transfer map from ∆ to ∆ . ( ) ∶= ( )∣ In the case of diﬀerent skew-ﬁelds the transfer map is deﬁned as follows: ⊗ ⊗ ( ) ( ⊗ ′) ′ ( ) τ∆,∆ θ τ∆ L,∆ L θL H ∆ , = ( ) = ( )′⊆ ( ) ′ ′ ′ w′here θL is any extensio′n of θ to H′ ∆ L . Here we use that L splits the skew-ﬁelds. We call τ∆,∆ θ′ the transfer of θ from ∆ to ∆ . Let us recall that if ∆ and ∆ are endo-equivalent and V V then τ θ ∆,∆ θ if and( only) if I ∆(,∆ )I θ,θ , see [10, 6.10, 5.9]. ′ ′ ′ Propos○ition 6=.3. Transf○er commutes with the adjoint involutions, i.e. ′ given two sig′ned hermitian spaces ′ h,V and h ,V of the ﬁrst kind and the same type with adjoint involutions σ and σ , respectively, a′nd given′ t′wo endo-equivalent semisimple strata ∆ and ∆ in V and V , respectively, then σ τ τ σ. ∆,∆ #∆,# ∆ ○ ( ⊕ ) = Pro○of. Without loss of generality we can assume that we work over the same s⊕kew-ﬁeld, because ′the tr′ansfer is deﬁned by reducing to that case, i.e. from diﬀerent skew-ﬁelds to L. Now′ ′the′ r′esult follows, ⊕ because the′ re′striction maps commute with the adjoint involutions, i.e. res σ σ #∆ # ∆ ,# ∆ σ res . ∆ ∆ ,∆ ( ) ( ) ( )∣ ′ ′ − − − ( ) The transfer map induces for self-dual semisimple strata ∆′ and−∆ ′ via the Glauberman correspondenc′e a bijection from C ∆ to C ∆ which maps θ to τ θ . We denote this map still by τ . ∆,∆ H ∆ ∆,∆ Let us recall: 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: θ

text: to

text: H

text: The

text: elements

text: of

text: C

text: are

text: called

text: self-dual

text: sem∈isim(ple)

text: character∣s.

text: For

text: a

text: stratum

text: H

text: equiva(len)t

text: to

text: we

text: deﬁne

text: C

text: C

text: and

text: C

text: C

text: Proposition

text: Let

text: be

text: a

text: self-dual

text: semisimple

text: stratum

text: and

text: θ

text: C

text: Then

text: θ

text: is

text: an

text: element

text: H

text: of

text: C

text: Proof.

text: If

text: θ

text: ,θ

text: and

text: θ

text: are

text: elements

text: of

text: C

text: then

text: θ

text: θ

text: 1θ

text: is

text: an

text: element

text: of

text: C

text: by

text: the

text: deﬁnition

text: of

text: C

text: see

text: Thus

text: the

text: number

text: of

text: elements

text: of

text: C

text: which

text: have

text: the

text: same

text: restriction

text: as

text: θ

text: on

text: H

text: is

text: independent

text: θ.

text: The

text: number

text: of

text: elements

text: of

text: C

text: is

text: a

text: power

text: of

text: p

text: and

text: thus

text: there

text: is

text: a

text: non-negative

text: integer

text: s

text: such

text: that

text: for

text: every

text: element

text: of

text: θ

text: C

text: there

text: are

text: exactly

text: ps

text: elements

text: in

text: C

text: with

text: the

text: same

text: restriction

text: as

text: θ.

text: Thus

text: the

text: action

text: of

text: σ

text: on

text: the

text: set

text: of

text: those

text: characters

text: must

text: have

text: a

text: ﬁxed

text: point.

text: This

text: ﬁnishes

text: the

text: proof.

text: Tdehneotgerobuyp

text: G′g˜,χac(ts′

text: iGt˜h′w)ahsicehcownde

text: chara(cter

text: χ)

text: K

text: if

text: the

text: restrictions

text: of

text: χ

text: and

text: g.χ

text: coincide

text: on′

text: gKg

text: K

text: We

text: denote

text: the

text: set

text: of

text: all

text: elem′

text: ents

text: of

text: G˜

text: which

text: intertwine

text: χ

text: with

text: χ

text: by

text: I

text: χ,χ

text: We

text: adapt

text: the

text: notation

text: IH

text: χ,χ

text: for

text: the

text: intersection

text: of

text: I

text: χ,χ

text: with

text: a

text: subgroup

text: H

text: of

text: G˜.

text: We

text: say

text: that

text: χ

text: and

text: χ

text: intertwine

text: by

text: some

text: element

text: of

text: H

text: if

text: I

text: χ,χ

text: is

text: non-empty.

text: H

text: Transfers

text: We

text: now

text: recall

text: the

text: no′tion

text: of

text: transfer.

text: For

text: more

text: detail

text: consult

text: section

text: section

text: Transfers

text: are

text: deﬁned

text: between

text: strata

text: over

text: diﬀerent

text: vectors

text: spaces

text: over

text: diﬀerent

text: skew-ﬁelds

text: central

text: and

text: of

text: ﬁnite

text: degree

text: over

text: F.

text: Let

text: and

text: be

text: two

text: endo-equivalent

text: ∶sem(isim⊕ple)st→rata(.

text: L)et

text: us

text: at

text: ﬁrst

text: recall

text: the

text: transfer

text: over

text: the

text: same

text: skew-ﬁeld.

text: The

text: canonically

text: deﬁned

text: restriction

text: m′aps

text: ∶=res∆

text: ○C

text: C(

text: deﬁne

text: a

text: bijection

text: τ

text: r′es

text: res

text: C

text: C

text: called

text: the

text: transfer

text: map

text: from

text: to

text: In

text: the

text: case

text: of

text: diﬀerent

text: skew-ﬁelds

text: the

text: transfer

text: map

text: is

text: deﬁned

text: as

text: follows:

text: τ∆,∆

text: θ

text: τ∆

text: L,∆

text: L

text: θL

text: H

text: w′here

text: θL

text: is

text: any

text: extensio′n

text: of

text: θ

text: to

text: H′

text: L

text: Here

text: we

text: use

text: that

text: L

text: splits

text: the

text: skew-ﬁelds.

text: We

text: call

text: τ∆,∆

text: θ′

text: the

text: transfer

text: of

text: θ

text: from

text: to

text: Let

text: us

text: recall

text: that

text: if

text: and

text: are

text: endo-equivalent

text: and

text: V

text: V

text: then

text: τ

text: θ

text: θ

text: if

text: and(

text: only)

text: if

text: I

text: )I

text: θ,θ

text: see

text: Propos○ition

text: Transf○er

text: commutes

text: with

text: the

text: adjoint

text: involutions,

text: i.e.

text: given

text: two

text: sig′ned

text: hermitian

text: spaces

text: h,V

text: and

text: h

text: ,V

text: of

text: the

text: ﬁrst

text: kind

text: and

text: the

text: same

text: type

text: with

text: adjoint

text: involutions

text: σ

text: and

text: σ

text: respectively,

text: a′nd

text: given′

text: t′wo

text: endo-equivalent

text: semisimple

text: strata

text: and

text: in

text: V

text: and

text: V

text: respectively,

text: then

text: σ

text: τ

text: τ

text: σ.

text: Pro○of.

text: Without

text: loss

text: of

text: generality

text: we

text: can

text: assume

text: that

text: we

text: work

text: over

text: the

text: same

text: s⊕kew-ﬁeld,

text: because

text: ′the

text: tr′ansfer

text: is

text: deﬁned

text: by

text: reducing

text: to

text: that

text: case,

text: i.e.

text: from

text: diﬀerent

text: skew-ﬁelds

text: to

text: L.

text: Now′

text: ′the′

text: r′esult

text: follows,

text: because

text: the′

text: re′striction

text: maps

text: commute

text: with

text: the

text: adjoint

text: involutions,

text: i.e.

text: res

text: σ

text: σ

text: σ

text: res

text: The

text: transfer

text: map

text: induces

text: for

text: self-dual

text: semisimple

text: strata

text: and−∆

text: via

text: the

text: Glauberman

text: correspondenc′e

text: a

text: bijection

text: from

text: C

text: to

text: C

text: which

text: maps

text: θ

text: to

text: τ

text: θ

text: We

text: denote

text: this

text: map

text: still

text: by

text: τ

text: H

text: Let

text: us

text: recall:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ⟨ ⟩ = ⟨ ( )⟩ = ⟨ ⟩ = ⟨ ⟩ So, its residue class is a square and hence the cla−im. Thus −we ge−t − ≡ ≡ ≡ ≡ ⟨ ⟩ −d gdρ g −gdy2g 1 gyg 1gdg 1gyg −1 . ′ ≡ The latter term is gdg 1 , because gyg 1 is ρ-symmetric. Now, gdg 1 and d are ρ-symmetric, congruent and commute, and hence we get (ii) by Case 1. ′ × Remark 2.8. The proof of Lemma 2.7 shows that if two ρ-symmetric elements d and d are conjugate by an element of D then they deﬁne the same Witt class. ( ) Proof of Proposition 2.3. Using [4, 1.14] every signed hermitian space h,V over D has a Witt basis, i.e. a basis which has a Gram matrix with two diagonal blocks: ( ) ( ) (i) An anti-diagonal matrix having 1 and (cid:15) in the anti-diagonal (block 1,1 ), ( ) ( ) (ii) A diagonal matrix which is the Gram matrix of an anisotropic subspace of V (block 2,2 ). (iii) The blocks 2,1 and 1,2 have only zero entries. ( ) Thus to classify all equivalence classes of signed hermitian forms, one only needs to classify the possible + diagonal blocks 2,2 , and for them only the diagonal entries. The F-vector space of skew-symmetric ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ elements of D is Fπ l . Thus, from Lemma 2.5 follows that there is only one non-trivial Witt class in D s ≡ ≡ ≡ ( ) ≡ ( ) the symplectic case. The F-vector space of symmetric elements of D is L π F. By Lemma 2.7 and D Lemma 2.5, the class d is π or 1 or α . They are pairwise diﬀerent because of valuation D reasons and because α cannot be of the form ρ x x because the residue class of a unit of the form ρ x x is a square. ⊆ ( ) 2.4 G SL V ( ) = ∣ = ( ) − D ∣ An element g of G satisﬁes gσ g 1, and ρ id leaves for Nrd g only 1 or 1. It is remarkable h F A F that in fact all elements of G have reduced norm 1. We are going to prove this fact in this section. Proposition 2.9. Every element of G has reduced n=orm 1. ( ) Proof. We prove the assertion by induction on m dim V . We only need to consider orthogonal G D groups, because i(n th=e sy)mplectic case the group L is a split symplectic group where every element has determinant 1. T[ h]us let us assume that G is an inner form of an orthogonal grou(p.) = = ∈ ∈ { − } ( ) ≠ ∣ Induction start m 1 : Let g be an element of G. We take the canonical isomorphism from D [ ]∣ ( ) = ( ) = ( ) = [ ] to End D. Then F g is a ﬁeld in D, which is invariant under the action of σ . If σ g g then g2 1, D h h [ ]∣ because g G a(nd>thu)s g 1, 1 , i.e[. t]he reduced norm of g would be 1. If σh g g then σh F g is the Galois generator of F g F. Thus Nrd g N g σ g g 1. F g F h = Induction step m 1 : We consider F g generated by an element g of G . The minimal polynomial µ g,F of g over F has a prime factorization [ ] ≅µ [ ]/Pν1 ...Pν[l, ]/ g,F 1 l which gives a decomposition: F g F X Pν1 ...F X Pνl 1 l The factors are permuted by σ . For every orbit of this action we get a σ -ﬁxed idempotent. In the case h h of at least two orbits we conclude that g is an element of the product of at least two quaternionic inner forms of classical groups and the induction hypothesis implies that the reduced norm of the restrictions of g to the factors is 1. Thus we are left with the case of one orbit. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: So,

text: its

text: residue

text: class

text: is

text: a

text: square

text: and

text: hence

text: the

text: cla−im.

text: Thus

text: −we

text: ge−t

text: −d

text: gdρ

text: g

text: gyg

text: −1

text: The

text: latter

text: term

text: is

text: gdg

text: because

text: gyg

text: is

text: ρ-symmetric.

text: Now,

text: gdg

text: and

text: d

text: are

text: ρ-symmetric,

text: congruent

text: and

text: commute,

text: and

text: hence

text: we

text: get

text: (ii)

text: by

text: Case

text: Remark

text: The

text: proof

text: of

text: Lemma

text: shows

text: that

text: if

text: two

text: ρ-symmetric

text: elements

text: d

text: and

text: d

text: are

text: conjugate

text: by

text: an

text: element

text: of

text: D

text: then

text: they

text: deﬁne

text: the

text: same

text: Witt

text: class.

text: Proof

text: of

text: Proposition

text: Using

text: every

text: signed

text: hermitian

text: space

text: h,V

text: over

text: D

text: has

text: a

text: Witt

text: basis,

text: i.e.

text: a

text: basis

text: which

text: has

text: a

text: Gram

text: matrix

text: with

text: two

text: diagonal

text: blocks:

text: (i)

text: An

text: anti-diagonal

text: matrix

text: having

text: and

text: in

text: the

text: anti-diagonal

text: (block

text: (ii)

text: A

text: diagonal

text: matrix

text: which

text: is

text: the

text: Gram

text: matrix

text: of

text: an

text: anisotropic

text: subspace

text: of

text: V

text: (block

text: (iii)

text: The

text: blocks

text: and

text: have

text: only

text: zero

text: entries.

text: Thus

text: to

text: classify

text: all

text: equivalence

text: classes

text: of

text: signed

text: hermitian

text: forms,

text: one

text: only

text: needs

text: to

text: classify

text: the

text: possible

text: diagonal

text: blocks

text: and

text: for

text: them

text: only

text: the

text: diagonal

text: entries.

text: The

text: F-vector

text: space

text: of

text: skew-symmetric

text: elements

text: of

text: D

text: is

text: Fπ

text: l

text: Thus,

text: from

text: Lemma

text: follows

text: that

text: there

text: is

text: only

text: one

text: non-trivial

text: Witt

text: class

text: in

text: D

text: s

text: the

text: symplectic

text: case.

text: The

text: F-vector

text: space

text: of

text: symmetric

text: elements

text: of

text: D

text: is

text: L

text: π

text: F.

text: By

text: Lemma

text: and

text: D

text: Lemma

text: the

text: class

text: d

text: is

text: π

text: or

text: or

text: α

text: They

text: are

text: pairwise

text: diﬀerent

text: because

text: of

text: valuation

text: D

text: reasons

text: and

text: because

text: α

text: cannot

text: be

text: of

text: the

text: form

text: ρ

text: x

text: x

text: because

text: the

text: residue

text: class

text: of

text: a

text: unit

text: of

text: the

text: form

text: ρ

text: x

text: x

text: is

text: a

text: square.

text: G

text: SL

text: V

text: D

text: An

text: element

text: g

text: of

text: G

text: satisﬁes

text: gσ

text: g

text: and

text: ρ

text: id

text: leaves

text: for

text: Nrd

text: g

text: only

text: or

text: It

text: is

text: remarkable

text: h

text: F

text: A

text: F

text: that

text: in

text: fact

text: all

text: elements

text: of

text: G

text: have

text: reduced

text: norm

text: We

text: are

text: going

text: to

text: prove

text: this

text: fact

text: in

text: this

text: section.

text: Proposition

text: Every

text: element

text: of

text: G

text: has

text: reduced

text: n=orm

text: Proof.

text: We

text: prove

text: the

text: assertion

text: by

text: induction

text: on

text: m

text: dim

text: V

text: We

text: only

text: need

text: to

text: consider

text: orthogonal

text: G

text: D

text: groups,

text: because

text: i(n

text: th=e

text: sy)mplectic

text: case

text: the

text: group

text: L

text: is

text: a

text: split

text: symplectic

text: group

text: where

text: every

text: element

text: has

text: determinant

text: T[

text: h]us

text: let

text: us

text: assume

text: that

text: G

text: is

text: an

text: inner

text: form

text: of

text: an

text: orthogonal

text: grou(p.)

text: Induction

text: start

text: m

text: Let

text: g

text: be

text: an

text: element

text: of

text: G.

text: We

text: take

text: the

text: canonical

text: isomorphism

text: from

text: D

text: to

text: End

text: D.

text: Then

text: F

text: g

text: is

text: a

text: ﬁeld

text: in

text: D,

text: which

text: is

text: invariant

text: under

text: the

text: action

text: of

text: σ

text: If

text: σ

text: g

text: g

text: then

text: D

text: h

text: h

text: because

text: g

text: G

text: a(nd>thu)s

text: g

text: i.e[.

text: t]he

text: reduced

text: norm

text: of

text: g

text: would

text: be

text: If

text: σh

text: g

text: g

text: then

text: σh

text: F

text: g

text: is

text: the

text: Galois

text: generator

text: of

text: F

text: g

text: F.

text: Thus

text: Nrd

text: g

text: N

text: g

text: σ

text: g

text: g

text: F

text: g

text: F

text: h

text: Induction

text: step

text: m

text: We

text: consider

text: F

text: g

text: generated

text: by

text: an

text: element

text: g

text: of

text: G

text: The

text: minimal

text: polynomial

text: µ

text: g,F

text: of

text: g

text: over

text: F

text: has

text: a

text: prime

text: factorization

text: ≅µ

text: ...Pν[l,

text: g,F

text: l

text: which

text: gives

text: a

text: decomposition:

text: F

text: g

text: F

text: X

text: ...F

text: X

text: Pνl

text: l

text: The

text: factors

text: are

text: permuted

text: by

text: σ

text: For

text: every

text: orbit

text: of

text: this

text: action

text: we

text: get

text: a

text: σ

text: -ﬁxed

text: idempotent.

text: In

text: the

text: case

text: h

text: h

text: of

text: at

text: least

text: two

text: orbits

text: we

text: conclude

text: that

text: g

text: is

text: an

text: element

text: of

text: the

text: product

text: of

text: at

text: least

text: two

text: quaternionic

text: inner

text: forms

text: of

text: classical

text: groups

text: and

text: the

text: induction

text: hypothesis

text: implies

text: that

text: the

text: reduced

text: norm

text: of

text: the

text: restrictions

text: of

text: g

text: to

text: the

text: factors

text: is

text: Thus

text: we

text: are

text: left

text: with

text: the

text: case

text: of

text: one

text: orbit.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 M. De Donno – Z. Palmowski — J. Tumilewicz Figure 4. − The American put option value for diﬀerent discounting factor: q = 0.01 with double continuation region and q = 0.01 with single continuation region. Parameters: σ = 0.2,λ = 0.2, ρ = 7.5, µ = 0.06. The main vehicle for deriving all results was based on ﬂuctuation theory of L´evy processes and general theory of stopping problems. This kind of study is still at its ﬁrst step and several further extension can be done. First of all, the characterization of the price and critical prices of the perpetual American options is the starting point for the analysis of ﬁnite-maturity American options and Canadized options. Moreover, one can think of more general set-up by adding Markov regime switching to the asset price modeling. These type of results could be achieved using the work of Ivanovs and Palmowski [35] where some exit identities related with the ﬁrst passage times (2.5) are given. It would be also interesting to consider multi-dimensional exponential L´evy process (see Klimsiak and Rozkosz [36]). Finally, the analysis of a market in presenct of a negative discounting rate should address also other typs of options, like e.g. the so-called π-options described by Guo and Zervos [33]. All of these problems are quite complex and left for future research. Acknowledgments This paper was conceived while the ﬁrst author was visiting the Hugo Steinhaus Center at Wroc(cid:32)law University of Science and Technology. A special thank is due to prof. A. Weron and his group for their kind hospitality. References [1] http://www.gold.org/investment/statistics/. [2] Knut K. Aase. The perpetual american put option for jump-diﬀusions. In Endre Bjørndal, Mette Bjørndal, Panos M. Pardalos, and Mikael R¨onnqvist, editors, Energy, Natural Resources and Environmental Economics, pages 493–507. Springer Berlin Heidelberg, 2010. [3] L. Alili and A.E. Kyprianou. Some remarks on ﬁrst passage of L´evy process, the American put and pasting principles. Annals of Applied Probability, 15:2062–2080, 2005. [4] S. Asmussen, F. Avram, and M.R. Pistorius. Russian and American put options under exponential phase-type L´evy models. Stochastic Processes and their applications, 109:79–111, 2004. [5] F. Avram, A.E. Kyprianou, and M.R. Pistorius. Exit problems for spectrally negative L´evy processes and applications to russian, American and canadized options. (unabridged version of AvramKyprianouPistorius. Utrecht University Preprint, 2002. [6] O. Bandorﬀ-Nielssen. The McKean stochastic game driven by a spectrally negative L´evy process. Finance Stoch., 1:41–68, 1998. [7] P. Barrieu and N. Bellamy. Impact of a market crisis on real options. In Wilmott Kyprianou, Schoutens, editor, Exotic Option Pricing and Advanced L´evy Models. Wiley Finance, 2005. [8] A. Battauz, M. De Donno, and A. Sbuelz. Real options with a double continuation region. Quantitative Finance, 12(3):465–475, 2012. ##LTFigure##

text: M.

text: De

text: Donno

text: Z.

text: Palmowski

text: J.

text: Tumilewicz

text: Figure

text: The

text: American

text: put

text: option

text: value

text: for

text: diﬀerent

text: discounting

text: factor:

text: q

text: with

text: double

text: continuation

text: region

text: and

text: q

text: with

text: single

text: continuation

text: region.

text: Parameters:

text: σ

text: ρ

text: µ

text: The

text: main

text: vehicle

text: for

text: deriving

text: all

text: results

text: was

text: based

text: on

text: ﬂuctuation

text: theory

text: of

text: L´evy

text: processes

text: and

text: general

text: theory

text: of

text: stopping

text: problems.

text: This

text: kind

text: of

text: study

text: is

text: still

text: at

text: its

text: ﬁrst

text: step

text: and

text: several

text: further

text: extension

text: can

text: be

text: done.

text: First

text: of

text: all,

text: the

text: characterization

text: of

text: the

text: price

text: and

text: critical

text: prices

text: of

text: the

text: perpetual

text: American

text: options

text: is

text: the

text: starting

text: point

text: for

text: the

text: analysis

text: of

text: ﬁnite-maturity

text: American

text: options

text: and

text: Canadized

text: options.

text: Moreover,

text: one

text: can

text: think

text: of

text: more

text: general

text: set-up

text: by

text: adding

text: Markov

text: regime

text: switching

text: to

text: the

text: asset

text: price

text: modeling.

text: These

text: type

text: of

text: results

text: could

text: be

text: achieved

text: using

text: the

text: work

text: of

text: Ivanovs

text: and

text: Palmowski

text: where

text: some

text: exit

text: identities

text: related

text: with

text: the

text: ﬁrst

text: passage

text: times

text: are

text: given.

text: It

text: would

text: be

text: also

text: interesting

text: to

text: consider

text: multi-dimensional

text: exponential

text: L´evy

text: process

text: (see

text: Klimsiak

text: and

text: Rozkosz

text: Finally,

text: the

text: analysis

text: of

text: a

text: market

text: in

text: presenct

text: of

text: a

text: negative

text: discounting

text: rate

text: should

text: address

text: also

text: other

text: typs

text: of

text: options,

text: like

text: e.g.

text: the

text: so-called

text: π-options

text: described

text: by

text: Guo

text: and

text: Zervos

text: All

text: of

text: these

text: problems

text: are

text: quite

text: complex

text: and

text: left

text: for

text: future

text: research.

text: Acknowledgments

text: This

text: paper

text: was

text: conceived

text: while

text: the

text: ﬁrst

text: author

text: was

text: visiting

text: the

text: Hugo

text: Steinhaus

text: Center

text: at

text: University

text: of

text: Science

text: and

text: Technology.

text: A

text: special

text: thank

text: is

text: due

text: to

text: prof.

text: A.

text: Weron

text: and

text: his

text: group

text: for

text: their

text: kind

text: hospitality.

text: References

text: Knut

text: K.

text: Aase.

text: The

text: perpetual

text: american

text: put

text: option

text: for

text: jump-diﬀusions.

text: In

text: Endre

text: Bjørndal,

text: Mette

text: Bjørndal,

text: Panos

text: M.

text: Pardalos,

text: and

text: Mikael

text: R¨onnqvist,

text: editors,

text: Energy,

text: Natural

text: Resources

text: and

text: Environmental

text: Economics,

text: pages

text: Springer

text: Berlin

text: Heidelberg,

text: L.

text: Alili

text: and

text: A.E.

text: Kyprianou.

text: Some

text: remarks

text: on

text: ﬁrst

text: passage

text: of

text: L´evy

text: process,

text: the

text: American

text: put

text: and

text: pasting

text: principles.

text: Annals

text: of

text: Applied

text: Probability,

text: S.

text: Asmussen,

text: F.

text: Avram,

text: and

text: M.R.

text: Pistorius.

text: Russian

text: and

text: American

text: put

text: options

text: under

text: exponential

text: phase-type

text: L´evy

text: models.

text: Stochastic

text: Processes

text: and

text: their

text: applications,

text: F.

text: Avram,

text: A.E.

text: Kyprianou,

text: and

text: M.R.

text: Pistorius.

text: Exit

text: problems

text: for

text: spectrally

text: negative

text: L´evy

text: processes

text: and

text: applications

text: to

text: russian,

text: American

text: and

text: canadized

text: options.

text: (unabridged

text: version

text: of

text: AvramKyprianouPistorius.

text: Utrecht

text: University

text: Preprint,

text: O.

text: Bandorﬀ-Nielssen.

text: The

text: McKean

text: stochastic

text: game

text: driven

text: by

text: a

text: spectrally

text: negative

text: L´evy

text: process.

text: Finance

text: Stoch.,

text: P.

text: Barrieu

text: and

text: N.

text: Bellamy.

text: Impact

text: of

text: a

text: market

text: crisis

text: on

text: real

text: options.

text: In

text: Wilmott

text: Kyprianou,

text: Schoutens,

text: editor,

text: Exotic

text: Option

text: Pricing

text: and

text: Advanced

text: L´evy

text: Models.

text: Wiley

text: Finance,

text: A.

text: Battauz,

text: M.

text: De

text: Donno,

text: and

text: A.

text: Sbuelz.

text: Real

text: options

text: with

text: a

text: double

text: continuation

text: region.

text: Quantitative

text: Finance,

text: ##LTFigure##


text: 7 by a macroscopic black hole. The temperature of a black diﬀerent radii, the central mass “seen” by these observers hole is, in this limit, inversely proportional to its mass25 will be the same. However, the evaporated energy moves outward so the mass should increase, if r is increased at ﬁxed time, because a mass shell that has already passed (cid:126) TBH(M) = c3 , (27) an inner observer will still be inside the sphere on which a 8πGk M more outward observer is sitting, who therefore will “see” B a bigger mass. In the Vaidya metric, the time coordinate where Planck’s constant (divided by 2π) and Boltz- is null, so moving outward at ﬁxed time means moving mann’s constant appear in standard notations. The ther- outward with the speed of light, staying on the surface mal radiation of a black body at this temperature is pro- of a volume containing a ﬁxed mass, if we assume the portional to T4 but also to the surface of the black hole, radiation to move at the speed of light. In this respect, which goes asBrH2 ∝ M2 ∝ T−2, so the total power out- the Vaidya metric is consistent, while models (25) and put of a black hoSle due to HaBwHking ratdiation behaves as (31) are not. But they are more easily interpreted, both TB2H ∝ M−2: having a time-like time coordinate (for r > rs). This is at least true for the model (31) proposed here, whereas the Aste-Trautmann model has a somewhat se- (cid:126) PBH(M) = 240 · 6c46πG2 M−2 (28) vhearsencoonmceepatnuinalgpfroorbrlem<. rT(hte )t,imbeeccaouosrediintadteoetsS noof t(2e5s)- s S tablish a simultaneity relationship between the outside leading to and the inside of the Schwarzschild radius. However, our causal picture of Hawking radiation is that it transports energy from inside the volume delimited by r to its out- (cid:126) s dM = − c4 (29) side. Now, given the temporal law rs(tS), how is a mass dt 15360πG2M2 element inside r to “know” what time it should “tunnel” s outside (to keep the law going, so to speak)? The sin- and M3(t) = k˜ (t0 − t), where gmualassr innastidueretoofittsS afpoprbeiadrsanrceelatoiuntgsitdhee. dTihsaisppmeeaarnansctehaotf besides the weak point already mentioned, consisting in t0 = 5120M03 π(cid:126)Gc42 (30) adoteesmnpootrgailvleawa mofeatnhienfgortmo thrse =lawrsi(ntSt)h,etrheegimonodoeflspevaecne where the radiation may be thought to originate! This is the life time of the black hole and k˜ = (cid:126)c4/(5120πG2). is diﬀerent in the case of the Vaidya metric and for the (cid:48) suggestion made here. In both of these cases at least k is then just 2Gk˜1/3/c2. In Ref. [26], it is argued that the phenomenological temporal course of the dynamics the dependency (26), based on a ﬁxed-background calcu- is clear, including the relationship between events inside lation, cannot hold down to mass zero, so the functional and outside the “horizon”. law must be modiﬁed near t = t . As we shall see, our 0 Next, we wish to describe the dynamics of a test par- calculations suggest a similar conclusion. ticle falling radially towards the black hole. To obtain Instead of (25), I propose the following toy model for equations(cid:18)of motion,(cid:19)we use th(cid:114)e eﬀective Lagrangian the metric of (cid:18)an evapora(cid:19)ting black h(cid:114)ole: L − r (t) − r (t) − − r (t) − r (t) = 1 s c2t˙2 2 s c t˙r˙ r˙2 . (32) ds2 = 1 s c2dt2 2 s c dtdr r r r r L L − − d ∂ − ∂ dr2 r2dΩ2 , (31) The Euler-Lagrange equations = 0 (q = dτ ∂q˙ ∂q t,(cid:16)r) then(cid:17)read ((cid:114)r ≡(cid:18)∂rs ) (cid:19) with r (t) again given by (26). s,t ∂t s Two remarks are in order: First, at some suﬃciently large distance from the black hole, the observed decay 1 − rs ct¨= (cid:114)rs r¨ − r˙2 − rs ct˙r˙ + rs,t ct˙2 , (33) law will be the same as in the Aste-Trautmann model, r r 2r r2 2r so this new model is as compatible with Hawking radia- − r − r − √r tion as the former. Second, the two models are no longer r¨ = s ct¨ s,t c2t˙2 s,t ct˙2 , (34) r 2r2 2 r r related to each other by a coordinate transformation, al- s L though they “almost” are as long as the time dependence and from = (ds/dτ)2 = c2, we obtain a constant of of r (t) is weak. Nevertheless, with r time dependent, motion (th(cid:18)e modulus(cid:19)of the fou(cid:114)r-velocity) s s they describe two diﬀerent physical situations. Both models are less realistic than the Vaidya model of Ref. [26] in the following respect: If we consider the met- c2 = 1 − rs(t) c2t˙2 − 2 rs(t) c t˙r˙ − r˙2 . (35) ric of either model at ﬁxed global time and observers at r r ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: by

text: a

text: macroscopic

text: black

text: hole.

text: The

text: temperature

text: of

text: a

text: black

text: diﬀerent

text: radii,

text: the

text: central

text: mass

text: “seen”

text: by

text: these

text: observers

text: hole

text: is,

text: in

text: this

text: limit,

text: inversely

text: proportional

text: to

text: its

text: will

text: be

text: the

text: same.

text: However,

text: the

text: evaporated

text: energy

text: moves

text: outward

text: so

text: the

text: mass

text: should

text: increase,

text: if

text: r

text: is

text: increased

text: at

text: ﬁxed

text: time,

text: because

text: a

text: mass

text: shell

text: that

text: has

text: already

text: passed

label: equation

text: TBH(M)

text: an

text: inner

text: observer

text: will

text: still

text: be

text: inside

text: the

text: sphere

text: on

text: which

text: a

text: M

text: more

text: outward

text: observer

text: is

text: sitting,

text: who

text: therefore

text: will

text: “see”

text: B

text: a

text: bigger

text: mass.

text: In

text: the

text: Vaidya

text: metric,

text: the

text: time

text: coordinate

text: where

text: Planck’s

text: constant

text: (divided

text: by

text: and

text: Boltz-

text: is

text: null,

text: so

text: moving

text: outward

text: at

text: ﬁxed

text: time

text: means

text: moving

text: mann’s

text: constant

text: appear

text: in

text: standard

text: notations.

text: The

text: ther-

text: outward

text: with

text: the

text: speed

text: of

text: light,

text: staying

text: on

text: the

text: surface

text: mal

text: radiation

text: of

text: a

text: black

text: body

text: at

text: this

text: temperature

text: is

text: pro-

text: of

text: a

text: volume

text: containing

text: a

text: ﬁxed

text: mass,

text: if

text: we

text: assume

text: the

text: portional

text: to

text: but

text: also

text: to

text: the

text: surface

text: of

text: the

text: black

text: hole,

text: radiation

text: to

text: move

text: at

text: the

text: speed

text: of

text: light.

text: In

text: this

text: respect,

text: which

text: goes

text: so

text: the

text: total

text: power

text: out-

text: the

text: Vaidya

text: metric

text: is

text: consistent,

text: while

text: models

text: and

text: put

text: of

text: a

text: black

text: hoSle

text: due

text: to

text: HaBwHking

text: ratdiation

text: behaves

text: as

text: are

text: not.

text: But

text: they

text: are

text: more

text: easily

text: interpreted,

text: both

text: having

text: a

text: time-like

text: time

text: coordinate

text: (for

text: r

text: rs).

text: This

text: is

text: at

text: least

text: true

text: for

text: the

text: model

text: proposed

text: here,

text: whereas

text: the

text: Aste-Trautmann

text: model

text: has

text: a

text: somewhat

text: se-

text: PBH(M)

text: vhearsencoonmceepatnuinalgpfroorbrlem<.

text: rT(hte

text: )t,imbeeccaouosrediintadteoetsS

text: noof

text: s

text: S

text: tablish

text: a

text: simultaneity

text: relationship

text: between

text: the

text: outside

text: leading

text: to

text: and

text: the

text: inside

text: of

text: the

text: Schwarzschild

text: radius.

text: However,

text: our

text: causal

text: picture

text: of

text: Hawking

text: radiation

text: is

text: that

text: it

text: transports

text: energy

text: from

text: inside

text: the

text: volume

text: delimited

text: by

text: r

text: to

text: its

text: out-

text: s

text: dM

text: side.

text: Now,

text: given

text: the

text: temporal

text: law

text: rs(tS),

text: how

text: is

text: a

text: mass

text: dt

text: element

text: inside

text: r

text: to

text: “know”

text: what

text: time

text: it

text: should

text: “tunnel”

text: s

text: outside

text: (to

text: keep

text: the

text: law

text: going,

text: so

text: to

text: speak)?

text: The

text: sin-

text: and

text: k˜

text: t),

text: where

text: gmualassr

text: innastidueretoofittsS

text: afpoprbeiadrsanrceelatoiuntgsitdhee.

text: dTihsaisppmeeaarnansctehaotf

text: besides

text: the

text: weak

text: point

text: already

text: mentioned,

text: consisting

text: in

text: adoteesmnpootrgailvleawa

text: mofeatnhienfgortmo

text: thrse

text: where

text: the

text: radiation

text: may

text: be

text: thought

text: to

text: originate!

text: This

text: is

text: the

text: life

text: time

text: of

text: the

text: black

text: hole

text: and

text: k˜

text: is

text: diﬀerent

text: in

text: the

text: case

text: of

text: the

text: Vaidya

text: metric

text: and

text: for

text: the

text: suggestion

text: made

text: here.

text: In

text: both

text: of

text: these

text: cases

text: at

text: least

text: k

text: is

text: then

text: just

text: In

text: Ref.

text: it

text: is

text: argued

text: that

text: the

text: phenomenological

text: temporal

text: course

text: of

text: the

text: dynamics

text: the

text: dependency

text: based

text: on

text: a

text: ﬁxed-background

text: calcu-

text: is

text: clear,

text: including

text: the

text: relationship

text: between

text: events

text: inside

text: lation,

text: cannot

text: hold

text: down

text: to

text: mass

text: zero,

text: so

text: the

text: functional

text: and

text: outside

text: the

text: “horizon”.

text: law

text: must

text: be

text: modiﬁed

text: near

text: t

text: t

text: As

text: we

text: shall

text: see,

text: our

text: Next,

text: we

text: wish

text: to

text: describe

text: the

text: dynamics

text: of

text: a

text: test

text: par-

text: calculations

text: suggest

text: a

text: similar

text: conclusion.

text: ticle

text: falling

text: radially

text: towards

text: the

text: black

text: hole.

text: To

text: obtain

text: Instead

text: of

text: I

text: propose

text: the

text: following

text: toy

text: model

text: for

text: use

text: eﬀective

text: Lagrangian

text: the

text: metric

text: of

text: black

text: L

text: r

text: (t)

text: r

text: (t)

text: r

text: (t)

text: r

text: (t)

text: s

text: s

text: c

text: t˙r˙

text: s

text: s

text: c

text: dtdr

text: r

text: r

text: r

text: r

text: L

text: L

text: d

text: The

text: Euler-Lagrange

text: equations

text: (q

text: dτ

text: ∂q˙

text: ∂q

text: with

text: r

text: (t)

text: again

text: given

text: by

text: s,t

text: ∂t

text: s

text: Two

text: remarks

text: are

text: in

text: order:

text: First,

text: at

text: some

text: suﬃciently

text: large

text: distance

text: from

text: the

text: black

text: hole,

text: the

text: observed

text: decay

text: rs

text: ct¨=

text: r¨

text: rs

text: ct˙r˙

text: rs,t

text: law

text: will

text: be

text: the

text: same

text: as

text: in

text: the

text: Aste-Trautmann

text: model,

text: r

text: r

text: so

text: this

text: new

text: model

text: is

text: as

text: compatible

text: with

text: Hawking

text: radia-

text: r

text: r

text: √r

text: tion

text: as

text: the

text: former.

text: Second,

text: the

text: two

text: models

text: are

text: no

text: longer

text: r¨

text: s

text: ct¨

text: s,t

text: s,t

text: r

text: r

text: r

text: related

text: to

text: each

text: other

text: by

text: a

text: coordinate

text: transformation,

text: al-

text: s

text: L

text: though

text: they

text: “almost”

text: are

text: as

text: long

text: as

text: the

text: time

text: dependence

text: and

text: from

text: we

text: obtain

text: a

text: constant

text: of

text: of

text: r

text: (t)

text: is

text: weak.

text: Nevertheless,

text: with

text: r

text: time

text: dependent,

text: motion

text: the

text: s

text: s

text: they

text: describe

text: two

text: diﬀerent

text: physical

text: situations.

text: Both

text: models

text: are

text: less

text: realistic

text: than

text: the

text: Vaidya

text: model

text: of

text: Ref.

text: in

text: the

text: following

text: respect:

text: If

text: we

text: consider

text: the

text: met-

text: rs(t)

text: rs(t)

text: c

text: t˙r˙

text: ric

text: of

text: either

text: model

text: at

text: ﬁxed

text: global

text: time

text: and

text: observers

text: at

text: r

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##