text: 6 IV. CONSEQUENCES, OR LACK THEREOF, deﬁned in terms of operators and commutation2 relations FOR LIGHT-FRONT WAVE FUNCTIONS speciﬁed on a quantization surface (e.g., ﬁxed time or ﬁxed x+), and then evolution in t or x+ is applied. In the A conspicuous contrast between relativistic quantum Heisenberg picture, it is the operators that are evolved ﬁeld theory and non-relativistic quantum mechanics con- rather than the states. cerns the status and use of wave functions. Their use We now work in a ﬁeld theory framework. Suppose is routine in atomic, molecular, and condensed matter that we are able† to deﬁne annihilation and creation op- physics, and gives a lot of useful information about the erators ak and ak from the Fourier transform of the ﬁelds states. For example, crucial insights into new structures on the quantization surface, that they obey the standard emerging for interacting electrons in condensed matter commutation relations for annihilation and creation op- physics has arisen from thinking about wave functions, erators3, and that the annihilation operators give zero two prominent examples being superconductivity [17] when acting on the vacuum: and the fractional quantum Hall eﬀect [18]. In contrast, in work with relativistic quantum ﬁeld a |0(cid:105) = 0. (9) theory there is little systematic use of wave functions, k and correspondingly a paucity of information about the detailed microscopic nature of the quantum mechanical For the purposes of this discussion we assume there is states involved. This is a particular important issue for only one kind of each operator; the extension to multiple non-perturbative bound states in QCD, i.e., for hadrons ﬁelds is elementary. and nuclei. The most general state constructed by applying cre- The one known exception is with light-front quantiza- ation operators to the vacuum, has the form (cid:90) tion. Interesting progress has been made in applications, (cid:88) (cid:89) e.g., [3, 19–22] and references therein. However, the stan- dard formulation, as in [1], appears to rely on the vacuum 1 N † being trivial, so that a given state can be expressed as a |ψ(cid:105) = N! akj|0(cid:105)ψk1,...,kN , (10) non-pathological sum and integral over basis states that N j=1 are in the same Fock space as free particles, e.g., for a proton in QCD in terms of states of quarks and gluons. with an integration measure appropriate to the mo- In general, a wave function gives an expression for mentum variables and the normalization of the oper- a quantum-mechanical state in terms of a set of basis ators. The numerical coeﬃcient functions we call the states, which are ﬁxed independently of the presence and momentum-space wave functions of the state. Coordinate- nature of the interactions in a theory. Thus the basis is space wave functions are deﬁned by Fourier transforma- deﬁned before one has found a solution of the theory. In tion of the momentum-space wave functions. a non-relativistic system the basis is commonly of eigen- Then the wave functions of a state can be obtained states of the position operators, but a momentum-space from a matrix element of annihilation operators between basis could also be used. It gives a representation of the the vacuum and the state: (cid:89) canonical relations for the fundamental operators, which are coordinates and canonical momenta in a standard N non-relativistic system. ψ = (cid:104)0| a |ψ(cid:105). (11) But in a relativistic theory, these natural ideas appear k1,...,kN kj j=1 to conﬂict with Haag’s theorem. Inequivalent representa- tions of the canonical commutation relations are needed This is proved using the commutation relations for the in the free and interacting theories. In a sense, the states annihilation and creation operators, together with the for the interacting theory are in a diﬀerent space than vacuum-annihilation condition. those of a corresponding free theory. One appears to In the case of non-relativistic systems, wave func- need to solve the theory before knowing what the basis tions with the above deﬁnition are the same as ordinary states are; equivalently, ﬁnding a solution of the theory Schr¨odinger wave functions. This is shown by the equiv- includes a construction of its state space. alence, e.g., [23], between a theory of non-relativistic In this section, I show how light-front quantization Schr¨odinger ﬁeld(s) and a collection of Schr¨odinger wave evades this issue suﬃciently to allow a deﬁnition of wave function theories for arbitrarily many particles. functions even given non-triviality of the vacuum. In a Given the expression for the annihilation operators in sense the diﬃculties are all conﬁned to the nature of the terms of the ﬁelds, the right-hand side of Eq. (11) can vacuum. Haag’s theorem itself cannot be prevented from applying, but its consequences can be limited. A. Overall framework 2 Anticommutation relations in the case of fermionic ﬁelds. To be able to easily relate diﬀerent viewpoints, let us 3 Derived from the canonical commutation relations of the ﬁelds use the Heisenberg picture. We conceive of a theory being on the quantization surface. ##LTLine## ##LTLine##

text: IV.

text: CONSEQUENCES,

text: OR

text: LACK

text: THEREOF,

text: deﬁned

text: in

text: terms

text: of

text: operators

text: and

text: relations

text: FOR

text: LIGHT-FRONT

text: WAVE

text: FUNCTIONS

text: speciﬁed

text: on

text: a

text: quantization

text: surface

text: (e.g.,

text: ﬁxed

text: time

text: or

text: ﬁxed

text: x+),

text: and

text: then

text: evolution

text: in

text: t

text: or

text: x+

text: is

text: applied.

text: In

text: the

text: A

text: conspicuous

text: contrast

text: between

text: relativistic

text: quantum

text: Heisenberg

text: picture,

text: it

text: is

text: the

text: operators

text: that

text: are

text: evolved

text: ﬁeld

text: theory

text: and

text: non-relativistic

text: quantum

text: mechanics

text: con-

text: rather

text: than

text: the

text: states.

text: cerns

text: the

text: status

text: and

text: use

text: of

text: wave

text: functions.

text: Their

text: use

text: We

text: now

text: work

text: in

text: a

text: ﬁeld

text: theory

text: framework.

text: Suppose

text: is

text: routine

text: in

text: atomic,

text: molecular,

text: and

text: condensed

text: matter

text: that

text: we

text: are

text: able†

text: to

text: deﬁne

text: annihilation

text: and

text: creation

text: op-

text: physics,

text: and

text: gives

text: a

text: lot

text: of

text: useful

text: information

text: about

text: the

text: erators

text: ak

text: and

text: ak

text: from

text: the

text: Fourier

text: transform

text: of

text: the

text: ﬁelds

text: states.

text: For

text: example,

text: crucial

text: insights

text: into

text: new

text: structures

text: on

text: the

text: quantization

text: surface,

text: that

text: they

text: obey

text: the

text: standard

text: emerging

text: for

text: interacting

text: electrons

text: in

text: condensed

text: matter

text: commutation

text: relations

text: for

text: annihilation

text: and

text: creation

text: op-

text: physics

text: has

text: arisen

text: from

text: thinking

text: about

text: wave

text: functions,

text: and

text: that

text: the

text: annihilation

text: operators

text: give

text: zero

text: two

text: prominent

text: examples

text: being

text: superconductivity

text: when

text: acting

text: on

text: the

text: vacuum:

text: and

text: the

text: fractional

text: quantum

text: Hall

text: eﬀect

text: In

text: contrast,

text: in

text: work

text: with

text: relativistic

text: quantum

text: ﬁeld

text: a

text: theory

text: there

text: is

text: little

text: systematic

text: use

text: of

text: wave

text: functions,

text: k

text: and

text: correspondingly

text: a

text: paucity

text: of

text: information

text: about

text: the

text: detailed

text: microscopic

text: nature

text: of

text: the

text: quantum

text: mechanical

text: For

text: the

text: purposes

text: of

text: this

text: discussion

text: we

text: assume

text: there

text: is

text: states

text: involved.

text: This

text: is

text: a

text: particular

text: important

text: issue

text: for

text: only

text: one

text: kind

text: of

text: each

text: operator;

text: the

text: extension

text: to

text: multiple

text: non-perturbative

text: bound

text: states

text: in

text: QCD,

text: i.e.,

text: for

text: hadrons

text: ﬁelds

text: is

text: elementary.

text: and

text: nuclei.

text: The

text: most

text: general

text: state

text: constructed

text: by

text: applying

text: cre-

text: The

text: one

text: known

text: exception

text: is

text: with

text: light-front

text: quantiza-

text: ation

text: operators

text: to

text: the

text: vacuum,

text: has

text: the

text: form

text: tion.

text: Interesting

text: progress

text: has

text: been

text: made

text: in

text: applications,

text: e.g.,

text: and

text: references

text: therein.

text: However,

text: the

text: stan-

text: dard

text: formulation,

text: as

text: in

text: appears

text: to

text: rely

text: on

text: the

text: vacuum

text: N

text: being

text: trivial,

text: so

text: that

text: a

text: given

text: state

text: can

text: be

text: expressed

text: as

text: a

text: N!

text: non-pathological

text: sum

text: and

text: integral

text: over

text: basis

text: states

text: that

text: N

text: are

text: in

text: the

text: same

text: Fock

text: space

text: as

text: free

text: particles,

text: e.g.,

text: for

text: a

text: proton

text: in

text: QCD

text: in

text: terms

text: of

text: states

text: of

text: quarks

text: and

text: gluons.

text: with

text: an

text: integration

text: measure

text: appropriate

text: to

text: the

text: mo-

text: In

text: general,

text: a

text: wave

text: function

text: gives

text: an

text: expression

text: for

text: mentum

text: variables

text: and

text: the

text: normalization

text: of

text: the

text: oper-

text: a

text: quantum-mechanical

text: state

text: in

text: terms

text: of

text: a

text: set

text: of

text: basis

text: ators.

text: The

text: numerical

text: coeﬃcient

text: functions

text: we

text: call

text: the

text: states,

text: which

text: are

text: ﬁxed

text: independently

text: of

text: the

text: presence

text: and

text: momentum-space

text: wave

text: functions

text: of

text: the

text: state.

text: Coordinate-

text: nature

text: of

text: the

text: interactions

text: in

text: a

text: theory.

text: Thus

text: the

text: basis

text: is

text: space

text: wave

text: functions

text: are

text: deﬁned

text: by

text: Fourier

text: transforma-

text: deﬁned

text: before

text: one

text: has

text: found

text: a

text: solution

text: of

text: the

text: theory.

text: In

text: tion

text: of

text: the

text: momentum-space

text: wave

text: functions.

text: a

text: non-relativistic

text: system

text: the

text: basis

text: is

text: commonly

text: of

text: eigen-

text: Then

text: the

text: wave

text: functions

text: of

text: a

text: state

text: can

text: be

text: obtained

text: states

text: of

text: the

text: position

text: operators,

text: but

text: a

text: momentum-space

text: from

text: a

text: matrix

text: element

text: of

text: annihilation

text: operators

text: between

text: basis

text: could

text: also

text: be

text: used.

text: It

text: gives

text: a

text: representation

text: of

text: the

text: the

text: vacuum

text: and

text: the

text: state:

text: canonical

text: relations

text: for

text: the

text: fundamental

text: operators,

text: which

text: are

text: coordinates

text: and

text: canonical

text: momenta

text: in

text: a

text: standard

text: N

text: non-relativistic

text: system.

text: ψ

text: a

text: But

text: in

text: a

text: relativistic

text: theory,

text: these

text: natural

text: ideas

text: appear

text: kj

text: to

text: conﬂict

text: with

text: Haag’s

text: theorem.

text: Inequivalent

text: representa-

text: tions

text: of

text: the

text: canonical

text: commutation

text: relations

text: are

text: needed

text: This

text: is

text: proved

text: using

text: the

text: commutation

text: relations

text: for

text: the

text: in

text: the

text: free

text: and

text: interacting

text: theories.

text: In

text: a

text: sense,

text: the

text: states

text: annihilation

text: and

text: creation

text: operators,

text: together

text: with

text: the

text: for

text: the

text: interacting

text: theory

text: are

text: in

text: a

text: diﬀerent

text: space

text: than

text: vacuum-annihilation

text: condition.

text: those

text: of

text: a

text: corresponding

text: free

text: theory.

text: One

text: appears

text: to

text: In

text: the

text: case

text: of

text: non-relativistic

text: systems,

text: wave

text: func-

text: need

text: to

text: solve

text: the

text: theory

text: before

text: knowing

text: what

text: the

text: basis

text: tions

text: with

text: the

text: above

text: deﬁnition

text: are

text: the

text: same

text: as

text: ordinary

text: states

text: are;

text: equivalently,

text: ﬁnding

text: a

text: solution

text: of

text: the

text: theory

text: Schr¨odinger

text: wave

text: functions.

text: This

text: is

text: shown

text: by

text: the

text: equiv-

text: includes

text: a

text: construction

text: of

text: its

text: state

text: space.

text: alence,

text: e.g.,

text: between

text: a

text: theory

text: of

text: non-relativistic

text: In

text: this

text: section,

text: I

text: show

text: how

text: light-front

text: quantization

text: Schr¨odinger

text: ﬁeld(s)

text: and

text: a

text: collection

text: of

text: Schr¨odinger

text: wave

text: evades

text: this

text: issue

text: suﬃciently

text: to

text: allow

text: a

text: deﬁnition

text: of

text: wave

text: function

text: theories

text: for

text: arbitrarily

text: many

text: particles.

text: functions

text: even

text: given

text: non-triviality

text: of

text: the

text: vacuum.

text: In

text: a

text: Given

text: the

text: expression

text: for

text: the

text: annihilation

text: operators

text: in

text: sense

text: the

text: diﬃculties

text: are

text: all

text: conﬁned

text: to

text: the

text: nature

text: of

text: the

text: terms

text: of

text: the

text: ﬁelds,

text: the

text: right-hand

text: side

text: of

text: Eq.

text: can

text: vacuum.

text: Haag’s

text: theorem

text: itself

text: cannot

text: be

text: prevented

text: from

text: applying,

text: but

text: its

text: consequences

text: can

text: be

text: limited.

text: A.

text: Overall

text: framework

text: Anticommutation

text: relations

text: in

text: the

text: case

text: of

text: fermionic

text: ﬁelds.

text: To

text: be

text: able

text: to

text: easily

text: relate

text: diﬀerent

text: viewpoints,

text: let

text: us

text: Derived

text: from

text: the

text: canonical

text: commutation

text: relations

text: of

text: the

text: ﬁelds

text: use

text: the

text: Heisenberg

text: picture.

text: We

text: conceive

text: of

text: a

text: theory

text: being

text: on

text: the

text: quantization

text: surface.

text: ##LTLine##

text: ##LTLine##


text: CNS MDC PC 0 PC 1 PC 0 PC 1 Activity space size, C -0.58 0.02 0.55 0.12 New ties/week, n -0.48 -0.19 0.51 -0.09 tie Activity space entropy, H -0.50 -0.08 0.43 0.20 AS Activity space stability, J -0.02 0.94 -0.19 0.95 AS Activity space rank turnover, R 0.43 -0.25 -0.47 -0.16 AS Table 11: Principal Components (only spatial data). The weight of each metric in the ﬁrst two principal components, for both datasets. PC 0 PC 1 R2 = 0.10, F = 12.83, p = 0.0 R2 = 0.03, F = 3.50, p = 0.0 F F coeﬀ p val LMG coeﬀ p val LMG − ± − − ± extraversion 0.50 0.07 < 10 10 0.77 0.11 0.05 0.02 0.3 ± − ± openness 0.19 0.07 0.004 0.08 0.11 0.04 0.009 0.45 − ± − ± neuroticism 0.07 0.07 0.4 0.03 0.10 0.05 0.03 0.21 − ± ± agreeableness 0.10 0.07 0.2 0.07 0.01 0.05 0.8 0.01 − ± ± conscientiousness 0.05 0.07 0.5 0.05 0.03 0.04 0.4 0.03 Table 12: Extraversion, openness, and neuroticism explain spatial behaviour. The result of a multiple linear regression explaining principal components of spatial data (see Table 7). The value of each coeﬃcient (coeﬀ) is reported together with the probability (p val) that the coeﬃcient is not relevant for the model. The relative importance of each coeﬃcient (LMG) is computed using the LMG method [79]. and persistent manner. We have found that, to a signiﬁcant extent, individuals adopt a similar strategy in the social and spatial sphere. These strategies are heterogeneous across the two samples considered, and there is no evidence suggesting that there exist distinct classes of individuals. Finally, we have shown that the big ﬁve personality traits explain related aspects of both social and spatial behaviour. In particular, we have found that extraverted individuals are more explorative and have diverse routines in both the social and the spatial sphere while neuroticism and openness associate with high level of routine instability in the social and spatial domain. Our ﬁndings conﬁrm the usefulness of mobile phone data to study the connections between behaviour and personality [29,40,44,84–86]. The results are in line with previous ﬁndings on the relation between personality and social behaviour: extraversion correlates with social network size [18, 41, 43], openness to experience to social network turnover [29] and neuroticism does not correlate with social network size [11]. Finally, our ﬁndings establish a relation between personality and spatial behaviour, validating the theories suggesting that spatial choices are partially dictated by personality dispositions [15] and that a single set of personality traits underlies all aspect of a person’s behaviour [2, 3]. The individual characterisation of spatial behaviour is also fundamental to develop conceptual [55] and predictive [87] models of travel behaviour accounting for individual-level diﬀerences. 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CNS

label: paragraph

text: MDC

text: PC

text: PC

text: PC

text: PC

text: Activity

text: space

text: size,

text: C

text: New

text: ties/week,

text: n

text: tie

text: Activity

text: space

text: entropy,

text: H

text: AS

text: Activity

text: space

text: stability,

text: J

text: AS

text: Activity

text: space

text: rank

text: turnover,

text: R

text: AS

text: Table

text: Principal

label: caption

text: Components

text: (only

text: spatial

text: data).

text: The

text: weight

text: of

text: each

text: metric

text: in

text: the

text: ﬁrst

text: two

text: principal

text: components,

text: for

text: both

text: datasets.

text: PC

text: PC

text: F

text: p

text: F

text: p

text: F

text: F

text: coeﬀ

text: p

text: val

text: LMG

text: coeﬀ

text: p

text: val

text: LMG

text: extraversion

text: openness

text: neuroticism

text: agreeableness

text: conscientiousness

text: Table

text: Extraversion,

text: openness,

text: and

text: neuroticism

text: explain

text: spatial

text: behaviour.

text: The

text: result

text: of

text: a

text: multiple

text: linear

text: regression

text: explaining

text: principal

text: components

text: of

text: spatial

text: data

text: (see

text: Table

text: The

text: value

text: of

text: each

text: coeﬃcient

text: (coeﬀ)

text: is

text: reported

text: together

text: with

text: the

text: probability

text: (p

text: val)

text: that

text: the

text: coeﬃcient

text: is

text: not

text: relevant

text: for

text: the

text: model.

text: The

text: relative

text: importance

text: of

text: each

text: coeﬃcient

text: (LMG)

text: is

text: computed

text: using

text: the

text: LMG

text: method

text: and

text: persistent

text: manner.

text: We

text: have

text: found

text: that,

text: to

text: a

text: signiﬁcant

text: extent,

text: individuals

text: adopt

text: a

text: similar

text: strategy

text: in

text: the

text: social

text: and

text: spatial

text: sphere.

text: These

text: strategies

text: are

text: heterogeneous

text: across

text: the

text: two

text: samples

text: considered,

text: and

text: there

text: is

text: no

text: evidence

text: suggesting

text: that

text: there

text: exist

text: distinct

text: classes

text: of

text: individuals.

text: Finally,

text: we

text: have

text: shown

text: that

text: the

text: big

text: ﬁve

text: personality

text: traits

text: explain

text: related

text: aspects

text: of

text: both

text: social

text: and

text: spatial

text: behaviour.

text: In

text: particular,

text: we

text: have

text: found

text: that

text: extraverted

text: individuals

text: are

text: more

text: explorative

text: and

text: have

text: diverse

text: routines

text: in

text: both

text: the

text: social

text: and

text: the

text: spatial

text: sphere

text: while

text: neuroticism

text: and

text: openness

text: associate

text: with

text: high

text: level

text: of

text: routine

text: instability

text: in

text: the

text: social

text: and

text: spatial

text: domain.

text: Our

text: ﬁndings

text: conﬁrm

text: the

text: usefulness

text: of

text: mobile

text: phone

text: data

text: to

text: study

text: the

text: connections

text: between

text: behaviour

text: and

text: personality

text: The

text: results

text: are

text: in

text: line

text: with

text: previous

text: ﬁndings

text: on

text: the

text: relation

text: between

text: personality

text: and

text: social

text: behaviour:

text: extraversion

text: correlates

text: with

text: social

text: network

text: size

text: openness

text: to

text: experience

text: to

text: social

text: network

text: turnover

text: and

text: neuroticism

text: does

text: not

text: correlate

text: with

text: social

text: network

text: size

text: Finally,

text: our

text: ﬁndings

text: establish

text: a

text: relation

text: between

text: personality

text: and

text: spatial

text: behaviour,

text: validating

text: the

text: theories

text: suggesting

text: that

text: spatial

text: choices

text: are

text: partially

text: dictated

text: by

text: personality

text: dispositions

text: and

text: that

text: a

text: single

text: set

text: of

text: personality

text: traits

text: underlies

text: all

text: aspect

text: of

text: a

text: person’s

text: behaviour

text: The

text: individual

text: characterisation

text: of

text: spatial

text: behaviour

text: is

text: also

text: fundamental

text: to

text: develop

text: conceptual

text: and

text: predictive

text: models

text: of

text: travel

text: behaviour

text: accounting

text: for

text: individual-level

text: diﬀerences.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Figure S5: T=30, Relative importance of personality traits for socio-spatial behaviour LMG of each regressor computed using the Lindeman, Merenda and Gold method for the multiple regression model of the principal components (table S7). Figure S6: T=30, Relative importance of personalitry traits for spatial behaviour LMG of each regressor computed using the Lindeman, Merenda and Gold method for the multiple regression model of the principal components (table S10). 26 ##LTFigure## ##LTFigure##

text: Figure

text: Relative

text: importance

text: of

text: personality

text: traits

text: for

text: socio-spatial

text: behaviour

text: LMG

text: of

text: each

text: regressor

text: computed

text: using

text: the

text: Lindeman,

text: Merenda

text: and

text: Gold

text: method

text: for

text: the

text: multiple

text: regression

text: model

text: of

text: the

text: principal

text: components

text: (table

text: Figure

text: Relative

text: importance

text: of

text: personalitry

text: traits

text: for

text: spatial

text: behaviour

text: LMG

text: of

text: each

text: regressor

text: computed

text: using

text: the

text: Lindeman,

text: Merenda

text: and

text: Gold

text: method

text: for

text: the

text: multiple

text: regression

text: model

text: of

text: the

text: principal

text: components

text: (table

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##


text: d d d (i) < d (i, j) self ref self ref ± ± Social circle size, k 0.04 0.13 15 6 100% ± ± Activity space size, C 0.04 0.07 8 3 99% ± ± New locations/week, n 0.06 0.12 0.9 0.5 96% loc ± ± New ties/week, n 0.1 0.2 1 1 95% tie ± ± Social circle entropy, H 0.002 0.005 0.7 0.3 99% SC ± ± Activity space entropy, H 0.002 0.006 0.4 0.1 99% AS ± · − ± Social circle stability, J (6 15) 10 4 0.14 0.05 100% SC ± · − ± Activity space stability, J (6 11) 10 4 0.10 0.04 100% AS ± ± Social circle rank turnover, R 0.04 0.11 2 1 99% SC ± ± Activity space rank turnover, R 0.04 0.20 2 1 99% AS Table S3: T=30, CNS dataset: Persistence of social and spatial behaviour.For each of the social and spatial metrics, d is the average self-distance and d is the reference distance between self ref an individual and all others, averaged across individuals. The third column reports the fraction of cases where d (i) < d (i, j), averaged across the population. self ref 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: d

text: d

text: d

text: (i)

text: d

text: (i,

text: j)

text: self

text: ref

text: self

text: ref

text: Social

text: circle

text: size,

text: k

text: Activity

text: space

text: size,

text: C

text: New

text: locations/week,

text: n

text: loc

text: New

text: ties/week,

text: n

text: tie

text: Social

text: circle

text: entropy,

text: H

text: SC

text: Activity

text: space

text: entropy,

text: H

text: AS

text: Social

text: circle

text: stability,

text: J

text: SC

text: Activity

text: space

text: stability,

text: J

text: AS

text: Social

text: circle

text: rank

text: turnover,

text: R

text: SC

text: Activity

text: space

text: rank

text: turnover,

text: R

text: AS

text: Table

text: CNS

text: dataset:

text: Persistence

text: of

text: social

text: and

text: spatial

text: behaviour.For

text: each

text: of

text: the

text: social

text: and

text: spatial

text: metrics,

text: d

text: is

text: the

text: average

text: self-distance

text: and

text: d

text: is

text: the

text: reference

text: distance

text: between

text: self

text: ref

text: an

text: individual

text: and

text: all

text: others,

text: averaged

text: across

text: individuals.

text: The

text: third

text: column

text: reports

text: the

text: fraction

text: of

text: cases

text: where

text: d

text: (i)

text: d

text: (i,

text: j),

text: averaged

text: across

text: the

text: population.

text: self

text: ref

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PC 0 PC 1 PC 2 PC 3 PC 4 CNS 0.57 0.21 0.10 0.08 0.04 MDC 0.55 0.24 0.12 0.06 0.03 Table S8: T=30, Variance explained by principal components (only spatial data). The fraction of variance explained by each principal component for the CNS and MDC dataset. CNS MDC PC 0 PC 1 PC 0 PC 1 Activity space size, C -0.55 0.01 -0.55 -0.10 New ties/week, n -0.48 -0.16 -0.48 -0.35 tie Activity space entropy, H -0.48 -0.14 -0.44 0.20 AS Activity space stability, J -0.06 0.96 -0.02 0.88 AS Activity space rank turnover, R 0.48 -0.16 0.51 -0.22 AS Table S9: T=30, Principal Components (only spatial data). The weight of each metric in the ﬁrst two principal components, for both datasets. PC 0 PC 1 R2 = 0.11, F = 11.55, p = 0.0 R2 = 0.02, F = 2.02, p = 0.1 F F coeﬀ p val LMG coeﬀ p val LMG − ± − − ± extraversion 0.56 0.09 < 10 9 0.8 0.12 0.05 0.03 0.38 ± − ± openness 0.20 0.08 0.01 0.07 0.04 0.05 0.5 0.08 ± − ± neuroticism 0.03 0.08 0.7 0.07 0.14 0.05 0.01 0.48 − ± − ± agreeableness 0.05 0.08 0.5 0.03 0.002 0.052 1.0 0.01 − ± − ± conscientiousness 0.005 0.081 1.0 0.03 0.03 0.05 0.6 0.04 Table S10: T=30, Extraversion, openness, and neuroticism explain spatial behaviour. The result of a multiple linear regression explaining principal components of spatial data (Table S6). The value of each coeﬃcient (coeﬀ) is reported together with the probability (p val) that the coeﬃcient is not relevant for the model. The relative importance of each coeﬃcient (LMG) is computed using the LMG method. 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PC

text: PC

text: PC

text: PC

text: PC

text: CNS

text: MDC

text: Table

text: Variance

text: explained

text: by

text: principal

text: components

text: (only

text: spatial

text: data).

text: The

text: fraction

text: of

text: variance

text: explained

text: by

text: each

text: principal

text: component

text: for

text: the

text: CNS

text: and

text: MDC

text: dataset.

text: CNS

text: MDC

text: PC

text: PC

text: PC

text: PC

text: Activity

text: space

text: size,

text: C

text: New

text: ties/week,

text: n

text: tie

text: Activity

text: space

text: entropy,

text: H

text: AS

text: Activity

text: space

text: stability,

text: J

text: AS

text: Activity

text: space

text: rank

text: turnover,

text: R

text: AS

text: Table

text: Principal

text: Components

text: (only

text: spatial

text: data).

text: The

text: weight

text: of

text: each

text: metric

text: in

text: the

text: ﬁrst

text: two

text: principal

text: components,

text: for

text: both

text: datasets.

text: PC

text: PC

text: F

text: p

text: F

text: p

text: F

text: F

text: coeﬀ

text: p

text: val

text: LMG

text: coeﬀ

text: p

text: val

text: LMG

text: extraversion

text: openness

text: neuroticism

text: agreeableness

text: conscientiousness

text: Table

text: Extraversion,

text: openness,

text: and

text: neuroticism

text: explain

text: spatial

text: behaviour.

text: The

text: result

text: of

text: a

text: multiple

text: linear

text: regression

text: explaining

text: principal

text: components

text: of

text: spatial

text: data

text: (Table

text: The

text: value

text: of

text: each

text: coeﬃcient

text: (coeﬀ)

text: is

text: reported

text: together

text: with

text: the

text: probability

text: (p

text: val)

text: that

text: the

text: coeﬃcient

text: is

text: not

text: relevant

text: for

text: the

text: model.

text: The

text: relative

text: importance

text: of

text: each

text: coeﬃcient

text: (LMG)

text: is

text: computed

text: using

text: the

text: LMG

text: method.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where c is the speed of light, ∆ν is the channel width, and ν is the central channel observing ∼ ◦ c frequency. At 4.5 GHz this corresponds to 9 , and the depolarization fraction is − sin(2∆θ) f = 1 = 1.6%. (4) depol 2∆θ We supplemented our above analysis with a combination of RM Synthesis and RMCLEAN (e.g. Extended Data Fig. 4). Ensuring the presence of minimal Faraday complexity is possible by integrating across the full bandwidth and taking advantage of a Fourier transform relation between the observed L(cid:126) (λ2) values and the Faraday spectrum (the polarized brightness as a function of RM). This approach is commonly known as RM Synthesis37, and can be coupled with a deconvo- lution procedure (RMCLEAN) to estimate the intrinsic Faraday spectrum38. While RM Synthesis and RMCLEAN can have difﬁculty in properly reconstructing the intrinsic Faraday spectrum under certain circumstances, the spread of clean components is a reliable indicator of spectra that contain more than a single Faraday-unresolved source39. At each observed frequency, we integrated Stokes Q and U values across the pulse width and normalized using Stokes I. Due to the normalization we only used frequency bins that had a Stokes I signal-to-noise ratio of at least 5. We computed a deconvolved Faraday spectrum for each burst ≈ − separately, on a highly oversampled RM axis (δRM 10 4 of the nominal FWHM of the RM resolution element). We used a relatively small gain parameter (0.02) and terminated the decon- volution when the peak of the residual decreased to 2σ above the mean. The algorithm typically − required 50 80 iterations to converge. This combination of settings permits us to carefully con- sider the cumulative distribution of RMCLEAN components along the RM axis, and thus constrain (cid:46) the intrinsic width of the polarized emission to 0.1% of the typical RM uncertainty. We found that this value scales with δRM because the peak of the Faraday spectrum rarely lands precisely on an individual pixel. To a high degree of conﬁdence, there is no evidence for emission at more than one RM value, nor for a broadened (“Faraday thick”) emitting region; we therefore forego more complicated QU-ﬁtting19. Results of this analysis are shown in Extended Data Table 2, and are consistent with the simpliﬁed QU-ﬁtting results described above. Calculation of burst properties As in previous studies5,15, a search for periodicity in the burst arrival times remains incon- clusive. Determining the exact DMs of the bursts is complicated by their changing morphology with radio frequency5,15. Measuring DM based on maximizing the peak S/N of the burst often leads to the blurring of burst structure and, in the case of FRB 121102, an overestimation of DM. We have thus chosen to display all bursts dedispersed to the same nominal DM from Burst #6 (Fig. 1 and Extended Data Fig. 1). Taking advantage of the narrowness of Burst #6, we estimated its optimal DM by minimizing its width at different DM trials. We measured burst widths at half the maximum by ﬁtting von Mises functions using the PSRCHIVE routine paas (Table 1). These widths correspond to the burst envelope in the case of multi-component bursts. 16 ##LTLine##

text: where

text: c

text: is

text: the

text: speed

text: of

text: light,

text: ∆ν

text: is

text: the

text: channel

text: width,

text: and

text: ν

text: is

text: the

text: central

text: channel

text: observing

text: c

text: frequency.

text: At

text: GHz

text: this

text: corresponds

text: to

text: and

text: the

text: depolarization

text: fraction

text: is

text: f

text: depol

text: 2∆θ

text: We

text: supplemented

text: our

text: above

text: analysis

text: with

text: a

text: combination

text: of

text: RM

text: Synthesis

text: and

text: RMCLEAN

font: LHMILZ+NimbusMonL-Regu

text: (e.g.

text: Extended

text: Data

text: Fig.

text: Ensuring

text: the

text: presence

text: of

text: minimal

text: Faraday

text: complexity

text: is

text: possible

text: by

text: integrating

text: across

text: the

text: full

text: bandwidth

text: and

text: taking

text: advantage

text: of

text: a

text: Fourier

text: transform

text: relation

text: between

text: the

text: observed

text: values

text: and

text: the

text: Faraday

text: spectrum

text: (the

text: polarized

text: brightness

text: as

text: a

text: function

text: of

text: RM).

text: This

text: approach

text: is

text: commonly

text: known

text: as

text: RM

text: and

text: can

text: be

text: coupled

text: with

text: a

text: deconvo-

text: lution

text: procedure

text: (RMCLEAN)

text: to

text: estimate

text: the

text: intrinsic

text: Faraday

text: While

text: RM

text: Synthesis

text: and

text: RMCLEAN

text: can

text: have

text: difﬁculty

text: in

text: properly

text: reconstructing

text: the

text: intrinsic

text: Faraday

text: spectrum

text: under

text: certain

text: circumstances,

text: the

text: spread

text: of

text: clean

text: components

text: is

text: a

text: reliable

text: indicator

text: of

text: spectra

text: that

text: contain

text: more

text: than

text: a

text: single

text: Faraday-unresolved

text: At

text: each

text: observed

text: frequency,

text: we

text: integrated

text: Stokes

text: Q

text: and

text: U

text: values

text: across

text: the

text: pulse

text: width

text: and

text: normalized

text: using

text: Stokes

text: I.

text: Due

text: to

text: the

text: normalization

text: we

text: only

text: used

text: frequency

text: bins

text: that

text: had

text: a

text: Stokes

text: I

text: signal-to-noise

text: ratio

text: of

text: at

text: least

text: We

text: computed

text: a

text: deconvolved

text: Faraday

text: spectrum

text: for

text: each

text: burst

text: separately,

text: on

text: a

text: highly

text: oversampled

text: RM

text: axis

text: (δRM

text: of

text: the

text: nominal

text: FWHM

text: of

text: the

text: RM

text: resolution

text: element).

text: We

text: used

text: a

text: relatively

text: small

text: gain

text: parameter

text: and

text: terminated

text: the

text: decon-

text: volution

text: when

text: the

text: peak

text: of

text: the

text: residual

text: decreased

text: to

text: 2σ

text: above

text: the

text: mean.

text: The

text: algorithm

text: typically

text: required

text: iterations

text: to

text: converge.

text: This

text: combination

text: of

text: settings

text: permits

text: us

text: to

text: carefully

text: con-

text: sider

text: the

text: cumulative

text: distribution

text: of

text: RMCLEAN

text: components

text: along

text: the

text: RM

text: axis,

text: and

text: thus

text: constrain

text: the

text: intrinsic

text: width

text: of

text: the

text: polarized

text: emission

text: to

text: of

text: the

text: typical

text: RM

text: uncertainty.

text: We

text: found

text: that

text: this

text: value

text: scales

text: with

text: δRM

text: because

text: the

text: peak

text: of

text: the

text: Faraday

text: spectrum

text: rarely

text: lands

text: precisely

text: on

text: an

text: individual

text: pixel.

text: To

text: a

text: high

text: degree

text: of

text: conﬁdence,

text: there

text: is

text: no

text: evidence

text: for

text: emission

text: at

text: more

text: than

text: one

text: RM

text: value,

text: nor

text: for

text: a

text: broadened

text: (“Faraday

text: thick”)

text: emitting

text: region;

text: we

text: therefore

text: forego

text: more

text: complicated

text: Results

text: of

text: this

text: analysis

text: are

text: shown

text: in

text: Extended

text: Data

text: Table

text: and

text: are

text: consistent

text: with

text: the

text: simpliﬁed

text: QU-ﬁtting

text: results

text: described

text: above.

text: Calculation

text: of

text: burst

text: properties

text: As

text: in

text: previous

text: a

text: search

text: for

text: periodicity

text: in

text: the

text: burst

text: arrival

text: times

text: remains

text: incon-

text: clusive.

text: Determining

text: the

text: exact

text: DMs

text: of

text: the

text: bursts

text: is

text: complicated

text: by

text: their

text: changing

text: morphology

text: with

text: radio

text: Measuring

text: DM

text: based

text: on

text: maximizing

text: the

text: peak

text: S/N

text: of

text: the

text: burst

text: often

text: leads

text: to

text: the

text: blurring

text: of

text: burst

text: structure

text: and,

text: in

text: the

text: case

text: of

text: FRB

text: an

text: overestimation

text: of

text: DM.

text: We

text: have

text: thus

text: chosen

text: to

text: display

text: all

text: bursts

text: dedispersed

text: to

text: the

text: same

text: nominal

text: DM

text: from

text: Burst

text: (Fig.

text: and

text: Extended

text: Data

text: Fig.

text: Taking

text: advantage

text: of

text: the

text: narrowness

text: of

text: Burst

text: we

text: estimated

text: its

text: optimal

text: DM

text: by

text: minimizing

text: its

text: width

text: at

text: different

text: DM

text: trials.

text: We

text: measured

text: burst

text: widths

text: at

text: half

text: the

text: maximum

text: by

text: ﬁtting

text: von

text: Mises

text: functions

text: using

text: the

text: PSRCHIVE

text: routine

text: paas

text: (Table

text: These

text: widths

text: correspond

text: to

text: the

text: burst

text: envelope

text: in

text: the

text: case

text: of

text: multi-component

text: bursts.

text: ##LTLine##


text: down luminosity of the magnetar13,48. Extended Data Fig. 6 describes a range of sizes, densities, and temperatures for the Faraday-rotating medium that are consistent with Crab-like pulsar wind nebulae, known supernova remnants, and a simple model for swept-up supernova ejecta. Data availability The calibrated burst data are available, upon request, from the Corresponding Author. Code availability The code used to analyse the data is available at the following sites: PRESTO (https://github.com/scottransom/presto), PSRCHIVE (http://psrchive.sourceforge.net), DSPSR (http://dspsr.sourceforge.net). 20

text: down

text: luminosity

text: of

text: the

text: Extended

text: Data

text: Fig.

text: describes

text: a

text: range

text: of

text: sizes,

text: densities,

text: and

text: temperatures

text: for

text: the

text: Faraday-rotating

text: medium

text: that

text: are

text: consistent

text: with

text: Crab-like

text: pulsar

text: wind

text: nebulae,

text: known

text: supernova

text: remnants,

text: and

text: a

text: simple

text: model

text: for

text: swept-up

text: supernova

text: ejecta.

text: Data

label: section

text: availability

text: The

text: calibrated

text: burst

text: data

text: are

text: available,

text: upon

text: request,

text: from

text: the

text: Corresponding

text: Author.

text: Code

text: availability

text: The

text: code

text: used

text: to

text: analyse

text: the

text: data

text: is

text: available

text: at

text: the

text: following

text: sites:

text: PRESTO

text: PSRCHIVE

text: DSPSR


text: Extended Data Table 2: Results of RM Synthesis and RMCLEAN. RMs were determined by ﬁtting a quadratic function to the peak of the deconvolved Faraday spectrum. RM uncertainties were determined by dividing the nominal FWHM of the RM resolution element by twice the signal- to-noise ratio at the peak of the RM spectrum. RMdisp is the second moment (dispersion) of the RMCLEAN clean components discovered during the Faraday spectrum deconvolution. Upper limits indicate that the value scales with RM pixel size. A value of zero means that all clean components fell within the same pixel, and indicates a Faraday spectrum that is indistinguishable from being inﬁnitely thin. Burst RM RMdisp (radm 2) (radm 2) 1 +102805± 37 < 0.12 2 +102685± 70 < 0.05 3 +102667± 37 < 0.12 6 +102642± 73 < 0.11 7 +102643± 105 < 0.04 8 +102680± 43 < 0.12 12 +102585± 67 < 0.02 13 +102484± 53 0 14 +102440± 51 0 15 +102701± 211 < 0.05 16 +102986± 27 < 0.10 GBT-1 +93572± 2885 0 GBT-2 +93523± 237 0 26 (cid:1) (cid:1) ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Extended

text: Data

text: Table

text: Results

text: of

text: RM

text: Synthesis

text: and

text: RMCLEAN.

text: RMs

text: were

text: determined

text: by

text: ﬁtting

text: a

text: quadratic

text: function

text: to

text: the

text: peak

text: of

text: the

text: deconvolved

text: Faraday

text: spectrum.

text: RM

text: uncertainties

text: were

text: determined

text: by

text: dividing

text: the

text: nominal

text: FWHM

text: of

text: the

text: RM

text: resolution

text: element

text: by

text: twice

text: the

text: signal-

text: to-noise

text: ratio

text: at

text: the

text: peak

text: of

text: the

text: RM

text: spectrum.

text: RMdisp

text: is

text: the

text: second

text: moment

text: (dispersion)

text: of

text: the

text: RMCLEAN

text: clean

text: components

text: discovered

text: during

text: the

text: Faraday

text: spectrum

text: deconvolution.

text: Upper

text: limits

text: indicate

text: that

text: the

text: value

text: scales

text: with

text: RM

text: pixel

text: size.

text: A

text: value

text: of

text: zero

text: means

text: that

text: all

text: clean

text: components

text: fell

text: within

text: the

text: same

text: pixel,

text: and

text: indicates

text: a

text: Faraday

text: spectrum

text: that

text: is

text: indistinguishable

text: from

text: being

text: inﬁnitely

text: thin.

text: Burst

text: RM

text: RMdisp

text: (radm

text: (radm

text: ##LTFigure##

text: ##LTLine##

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text: Figure 6 Physical constraints from source parameters. Parameter space for electron den- sity (n ) and length scale (L ) of the Faraday region for three different temperature e RM regimes, T = [104, 106, 108] K. The shaded red region indicates parameter space ex- e cluded by optical depth considerations (τ > 5). The solid black line gives the maximum ff − DM permitted, while the shaded grey region shows the DM down to 1 pc cm 3. The Host − solid blue line gives RM . The shaded blue region gives the range 10 4 <= β <= 1. The src intersection of grey and blue regions outside of the red region are physically permitted. The arrows indicates the upper limits on the sizes of the persistent source (left) and the star-forming region (right), respectively8,10. The parallel dashed lines represent ﬁts to a range of galactic and extragalactic HII regions21. The parallel dotted lines represent the − evolution of 1 and 10 M(cid:12) of ejecta evolving up to 1000 years at a velocity of 104 km s 1 in the blast-wave phase following a supernova53. The ﬁlled downwards triangle and dia- mond are for the supernova remnants Cas A54 and SN 1987A55, respectively. The ﬁlled circle represents the mean density and diameter of the Crab Nebula, whereas the ﬁlled square represents the characteristic density and length scale of a dense ﬁlament in the Crab Nebula24. The star indicates the density at the Bondi radius of Sgr A*22. 33 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

font: GQACHI+NimbusSanL-Bold

text: Physical

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text: Parameter

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text: 90 65 40 #6 #13 4.8 4.4 (a) (b) 90 65 ×107 ×107 40 #GBT-1 #GBT-2 8.0 7.6 7.2 6.8 6.4 6.0 5.6 5.2 (c) (d) 4.8 −1.0 −0.5 0.0 0.5 1.0−1.0 −0.5 0.0 0.5 1.0 Time (ms) Time (ms) 10 )ged( AP )zHG( ycneuqerF )ged( AP )zHG( ycneuqerF ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: I (cid:104) − − = p s p s r q + r q , 6 0 0 7 7 0 0 7 7 − − b p b p + a q a q , 7 7 0 0 0 0 7 7 − − b r b r + a s a s , 7 7 0 0 0 0 7 7 − − − b ( p s + r q ) + b (p s r q ) + a ( s q + s q ), 7 7 0 7 0 0 7 7 7 7 7 7 0 0 7 − − − b ( p r + p r ) + a ( p s + r q ) + a (p s r q ), 7 7 0 0 7 7 0 7 0 7 0 7 7 7 7 − − − b p ( p s + r q ) + b (p2s p r q + p (r q r q )) + 0 0 7 7 7 7 − 7 7−7 0 7 0 (cid:105)7 0 0 7 7 a (p s q + q ( p s r q + r q )) 7 0 7 0 7 7 7 0 0 7 7 I I | Note that we now reached the ﬁxed point as = ← . 6 4 v(5) v(7) Corollary 22 provides a bound on the number of iterations in Algorithm 1. Therefore, we know at which stage we have to reach the ﬁxed point of the I computation at the latest, viz. after computing · . This fact allows us to r m I construct a new algorithm which computes the ideal · directly instead of r m doing a ﬁxed point computation. The beneﬁt of Algorithm 2 is that we have to perform only one Gr¨obner basis computation in the end, although the new algorithm might performs more iterations than Algorithm 1. Algorithm 2 Invariant generation without ﬁxed point computation Input: Loop bodies B , . . . , B as described. 1 r ∗ ∗ ∗ ∗ Output: The ideal of all polynomial invariants of (B ; B ; . . . ; B ) . (cid:80) 1 2 r 1: Compute I := I(θ, ζ) as described above I (cid:104) − (cid:105) 2: = m v(1) v(0) + I new j=1 j i 3: FOR j = 1, . . . , m DO 4: FOR i = 1, . . . , r DO I ← I 5: (J · + ) new i j new I ∩ K · 6: RETURN [v(m r+1), v(0)] new The proof of termination of the invariant generation method of [10] assumes that the ideal of algebraic dependencies is prime. In general, this does not hold. Consider the following loop and its closed forms with exponential sequences 2n − and ( 2)n: while . . . do · x := 2x x(n) = 2n x(0) − − · y := 2y y(n) = ( 2)n y(0) end while The ideal of algebraic dependencies among the before-mentioned exponential (cid:104) − (cid:105) sequences is given by a2 b2 which is obviously not prime. As a consequence, the termination proof of [10] is incorrect. This paper closes this gap by providing a new algorithm and a corresponding termination proof. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I

text: p

text: s

text: p

text: s

text: r

text: q

text: r

text: q

text: b

text: p

text: b

text: p

text: a

text: q

text: a

text: q

text: b

text: r

text: b

text: r

text: a

text: s

text: a

text: s

text: b

text: p

text: s

text: r

text: q

text: b

text: (p

text: s

text: r

text: q

text: a

text: s

text: q

text: s

text: q

text: b

text: p

text: r

text: p

text: r

text: a

text: p

text: s

text: r

text: q

text: a

text: (p

text: s

text: r

text: q

text: b

text: p

text: p

text: s

text: r

text: q

text: b

text: p

text: r

text: q

text: p

text: (r

text: q

text: r

text: q

text: a

text: (p

text: s

text: q

text: q

text: p

text: s

text: r

text: q

text: r

text: q

text: I

text: I

text: Note

text: that

text: we

text: now

text: reached

text: the

text: ﬁxed

text: point

text: as

text: Corollary

text: provides

text: a

text: bound

text: on

text: the

text: number

text: of

text: iterations

text: in

text: Algorithm

text: Therefore,

text: we

text: know

text: at

text: which

text: stage

text: we

text: have

text: to

text: reach

text: the

text: ﬁxed

text: point

text: of

text: the

text: I

text: computation

text: at

text: the

text: latest,

text: viz.

text: after

text: computing

text: This

text: fact

text: allows

text: us

text: to

text: r

text: m

text: I

text: construct

text: a

text: new

text: algorithm

text: which

text: computes

text: the

text: ideal

text: directly

text: instead

text: of

text: r

text: m

text: doing

text: a

text: ﬁxed

text: point

text: computation.

text: The

text: beneﬁt

text: of

text: Algorithm

text: is

text: that

text: we

text: have

text: to

text: perform

text: only

text: one

text: Gr¨obner

text: basis

text: computation

text: in

text: the

text: end,

text: although

text: the

text: new

text: algorithm

text: might

text: performs

text: more

text: iterations

text: than

text: Algorithm

text: Algorithm

text: Invariant

text: generation

text: without

text: ﬁxed

text: point

text: computation

text: Input:

text: Loop

text: bodies

text: B

text: B

text: as

text: described.

text: r

text: Output:

text: The

text: ideal

text: of

text: all

text: polynomial

text: invariants

text: of

text: (B

text: B

text: B

text: r

text: Compute

text: I

text: I(θ,

text: ζ)

text: as

text: described

text: above

text: I

text: m

text: I

text: new

text: j

text: i

text: FOR

text: j

text: m

text: DO

text: FOR

text: i

text: r

text: DO

text: I

text: I

text: (J

text: new

text: i

text: j

text: new

text: I

text: K

text: RETURN

text: [v(m

text: new

text: The

text: proof

text: of

text: termination

text: of

text: the

text: invariant

text: generation

text: method

text: of

text: assumes

text: that

text: the

text: ideal

text: of

text: algebraic

text: dependencies

text: is

text: prime.

text: In

text: general,

text: this

text: does

text: not

text: hold.

text: Consider

text: the

text: following

text: loop

text: and

text: its

text: closed

text: forms

text: with

text: exponential

text: sequences

text: and

text: while

text: do

text: x

text: x(n)

text: y

text: y(n)

text: end

text: while

text: The

text: ideal

text: of

text: algebraic

text: dependencies

text: among

text: the

text: before-mentioned

text: exponential

text: sequences

text: is

text: given

text: by

text: which

text: is

text: obviously

text: not

text: prime.

text: As

text: a

text: consequence,

text: the

text: termination

text: proof

text: of

text: is

text: incorrect.

text: This

text: paper

text: closes

text: this

text: gap

text: by

text: providing

text: a

text: new

text: algorithm

text: and

text: a

text: corresponding

text: termination

text: proof.

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