text: X - 8 DORELLI ET AL.: HALL MHD GANYMEDE SIMULATIONS Figure 7. Comparisons of our simulations with the Galileo magnetometer observations for the G8 ﬂyby show good overall agreement. The left panels show the comparison using the published G8 ﬂyby coordinates. Green lines show the simulation results, and blue lines show the data. The right panels show the results of adding to the simulated probe an oﬀset of +0.05 R in the X and Z position components. G The top panels show the resistive MHD case, and the two bottom panels show the Hall MHD case. between simulated and observed B components, and the not attempted to identify it in Figure 8 (though it is obvious X oﬀset trajectory will be used in all further discussion below. in Figure 9). For reference, the GphiO magnetic ﬁeld com- Because of the G8 trajectory crosses the magnetopause ponents are reproduced in the top two panels, with resistive at a ﬁnite Y locatio−n, the reconnection plane is not aligned MHD on the left and Hall MHD on the right. As expected, with the GphiO X Z plane. Thus, the Hall out-of-plane transforming the data into inbound magnetopause BNC co- ﬁeld would be expected to produce large perturbations in ordinates eliminates the negative B excursion, leaving only both the X and Y components. However, rotating into a X an enhanced B excursion. This is consistent with what we “Boundary Normal Coordinate” (BNC) system (where the Y observe in the Hall simulation (right panels of Figure 8): new X unit vector points along the magnetopause outward The small negative B perturbation in the top right panel normal, the Z unit vector points northward tangent to the X at about 15.8hours has been transformed away in the bot- magnetopause, and the Y unit vector completes the right- handed system) would be expected to eliminate the pertur- tom right panel, and the small positive BY excursion in the bation in the new X component, leaving only the new Y top right panel has been enhanced in the lower right panel. (out of the local reconnection plane) component. This is Similarly, when viewed in the inbound BNC coordinate sys- exactly what we observe, as we show now below. tem, a large positive BX perturbation appears on the out- Figure 8 shows the data transformed into BNC coor- bound crossing. We have labelled these perturbations “Hall dinates corresponding to the inbound outer current sheet Fields” in Figure 8. crossing, and Figure 9 shows a three-dimensional view of the While these features are somewhat diﬃcult to see in the oﬀset G8 ﬂyby trajectory (green line) through our resistive line plots of Figure 8, they appear very clearly in the 3D (top panel) and Hall (bottom panel) simulations. We deter- view of Figure 9. The resistive MHD case in the top panel mined the BNC normal direction by inspection of this plot of Figure 9 shows that there is no ﬁeld line bending out of for the inbound magnetopause crossing. The corresponding the reconnection plane as the simulated probe crosses the coordinate axes are shown in white. Note that the axis near magnetopause boundary layer. In contrast, Hall case in the the outbound (upper) crossing is simply translated from the bottom panel shows clearly that there is signiﬁcant bend- inbound (lower) crossing, to simplify the interpretation of ing out of the reconnection planes of both the inbound and Figure 8, which plots the components in the single inbound outbound crossings. For the inbound crossing, it is obvi- current sheet BNC coordinate system. The yellow rectangles in Figure 8 show where the simu- ous that the perturbed ﬁeld is predominantly in the BNC lated probe enters and exits the outer magnetopause current Y direciton, while the perturbed ﬁeld is in the X direction sheet, which appears as an obvious reversal in the B com- for the outbound crossing when viewed in the local BNC Z ponent. The second current sheet crossing is less obvious in coordinate system for the inbound crossing. This is exactly both the simulations and the data and, therefore, we have what is observed in the data. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: X

text: DORELLI

text: ET

text: AL.:

text: HALL

text: MHD

text: GANYMEDE

text: SIMULATIONS

text: Figure

text: Comparisons

text: of

text: our

text: simulations

text: with

text: the

text: Galileo

text: magnetometer

text: observations

text: for

text: the

text: ﬂyby

text: show

text: good

text: overall

text: agreement.

text: The

text: left

text: panels

text: show

text: the

text: comparison

text: using

text: the

text: published

text: ﬂyby

text: coordinates.

text: Green

text: lines

text: show

text: the

text: simulation

text: results,

text: and

text: blue

text: lines

text: show

text: the

text: data.

text: The

text: right

text: panels

text: show

text: the

text: results

text: of

text: adding

text: to

text: the

text: simulated

text: probe

text: an

text: oﬀset

text: of

text: R

text: in

text: the

text: X

text: and

text: Z

text: position

text: components.

text: G

text: The

text: top

text: panels

text: show

text: the

text: resistive

text: MHD

text: case,

text: and

text: the

text: two

text: bottom

text: panels

text: show

text: the

text: Hall

text: MHD

text: case.

text: between

text: simulated

text: and

text: observed

text: B

text: components,

text: and

text: the

text: not

text: attempted

text: to

text: identify

text: it

text: in

text: Figure

text: (though

text: it

text: is

text: obvious

text: X

text: oﬀset

text: trajectory

text: will

text: be

text: used

text: in

text: all

text: further

text: discussion

text: below.

text: in

text: Figure

text: For

text: reference,

text: the

text: GphiO

text: magnetic

text: ﬁeld

text: com-

text: Because

text: of

text: the

text: trajectory

text: crosses

text: the

text: magnetopause

text: ponents

text: are

text: reproduced

text: in

text: the

text: top

text: two

text: panels,

text: with

text: resistive

text: at

text: a

text: ﬁnite

text: Y

text: locatio−n,

text: the

text: reconnection

text: plane

text: is

text: not

text: aligned

text: MHD

text: on

text: the

text: left

text: and

text: Hall

text: MHD

text: on

text: the

text: right.

text: As

text: expected,

text: with

text: the

text: GphiO

text: X

text: Z

text: plane.

text: Thus,

text: the

text: Hall

text: out-of-plane

text: transforming

text: the

text: data

text: into

text: inbound

text: magnetopause

text: BNC

text: co-

text: ﬁeld

text: would

text: be

text: expected

text: to

text: produce

text: large

text: perturbations

text: in

text: ordinates

text: eliminates

text: the

text: negative

text: B

text: excursion,

text: leaving

text: only

text: both

text: the

text: X

text: and

text: Y

text: components.

text: However,

text: rotating

text: into

text: a

text: X

text: an

text: enhanced

text: B

text: excursion.

text: This

text: is

text: consistent

text: with

text: what

text: we

text: “Boundary

text: Normal

text: Coordinate”

text: (BNC)

text: system

text: (where

text: the

text: Y

text: observe

text: in

text: the

text: Hall

text: simulation

text: (right

text: panels

text: of

text: Figure

text: new

text: X

text: unit

text: vector

text: points

text: along

text: the

text: magnetopause

text: outward

text: The

text: small

text: negative

text: B

text: perturbation

text: in

text: the

text: top

text: right

text: panel

text: normal,

text: the

text: Z

text: unit

text: vector

text: points

text: northward

text: tangent

text: to

text: the

text: X

text: at

text: about

text: has

text: been

text: transformed

text: away

text: in

text: the

text: bot-

text: magnetopause,

text: and

text: the

text: Y

text: unit

text: vector

text: completes

text: the

text: right-

text: handed

text: system)

text: would

text: be

text: expected

text: to

text: eliminate

text: the

text: pertur-

text: tom

text: right

text: panel,

text: and

text: the

text: small

text: positive

text: BY

text: excursion

text: in

text: the

text: bation

text: in

text: the

text: new

text: X

text: component,

text: leaving

text: only

text: the

text: new

text: Y

text: top

text: right

text: panel

text: has

text: been

text: enhanced

text: in

text: the

text: lower

text: right

text: panel.

text: (out

text: of

text: the

text: local

text: reconnection

text: plane)

text: component.

text: This

text: is

text: Similarly,

text: when

text: viewed

text: in

text: the

text: inbound

text: BNC

text: coordinate

text: sys-

text: exactly

text: what

text: we

text: observe,

text: as

text: we

text: show

text: now

text: below.

text: tem,

text: a

text: large

text: positive

text: BX

text: perturbation

text: appears

text: on

text: the

text: out-

text: Figure

text: shows

text: the

text: data

text: transformed

text: into

text: BNC

text: coor-

text: bound

text: crossing.

text: We

text: have

text: labelled

text: these

text: perturbations

text: “Hall

text: dinates

text: corresponding

text: to

text: the

text: inbound

text: outer

text: current

text: sheet

text: Fields”

text: in

text: Figure

text: crossing,

text: and

text: Figure

text: shows

text: a

text: three-dimensional

text: view

text: of

text: the

text: While

text: these

text: features

text: are

text: somewhat

text: diﬃcult

text: to

text: see

text: in

text: the

text: oﬀset

text: ﬂyby

text: trajectory

text: (green

text: line)

text: through

text: our

text: resistive

text: line

text: plots

text: of

text: Figure

text: they

text: appear

text: very

text: clearly

text: in

text: the

text: (top

text: panel)

text: and

text: Hall

text: (bottom

text: panel)

text: simulations.

text: We

text: deter-

text: view

text: of

text: Figure

text: The

text: resistive

text: MHD

text: case

text: in

text: the

text: top

text: panel

text: mined

text: the

text: BNC

text: normal

text: direction

text: by

text: inspection

text: of

text: this

text: plot

text: of

text: Figure

text: shows

text: that

text: there

text: is

text: no

text: ﬁeld

text: line

text: bending

text: out

text: of

text: for

text: the

text: inbound

text: magnetopause

text: crossing.

text: The

text: corresponding

text: the

text: reconnection

text: plane

text: as

text: the

text: simulated

text: probe

text: crosses

text: the

text: coordinate

text: axes

text: are

text: shown

text: in

text: white.

text: Note

text: that

text: the

text: axis

text: near

text: magnetopause

text: boundary

text: layer.

text: In

text: contrast,

text: Hall

text: case

text: in

text: the

text: the

text: outbound

text: (upper)

text: crossing

text: is

text: simply

text: translated

text: from

text: the

text: bottom

text: panel

text: shows

text: clearly

text: that

text: there

text: is

text: signiﬁcant

text: bend-

text: inbound

text: (lower)

text: crossing,

text: to

text: simplify

text: the

text: interpretation

text: of

text: ing

text: out

text: of

text: the

text: reconnection

text: planes

text: of

text: both

text: the

text: inbound

text: and

text: Figure

text: which

text: plots

text: the

text: components

text: in

text: the

text: single

text: inbound

text: outbound

text: crossings.

text: For

text: the

text: inbound

text: crossing,

text: it

text: is

text: obvi-

text: current

text: sheet

text: BNC

text: coordinate

text: system.

text: The

text: yellow

text: rectangles

text: in

text: Figure

text: show

text: where

text: the

text: simu-

text: ous

text: that

text: the

text: perturbed

text: ﬁeld

text: is

text: predominantly

text: in

text: the

text: BNC

text: lated

text: probe

text: enters

text: and

text: exits

text: the

text: outer

text: magnetopause

text: current

text: Y

text: direciton,

text: while

text: the

text: perturbed

text: ﬁeld

text: is

text: in

text: the

text: X

text: direction

text: sheet,

text: which

text: appears

text: as

text: an

text: obvious

text: reversal

text: in

text: the

text: B

text: com-

text: for

text: the

text: outbound

text: crossing

text: when

text: viewed

text: in

text: the

text: local

text: BNC

text: Z

text: ponent.

text: The

text: second

text: current

text: sheet

text: crossing

text: is

text: less

text: obvious

text: in

text: coordinate

text: system

text: for

text: the

text: inbound

text: crossing.

text: This

text: is

text: exactly

text: both

text: the

text: simulations

text: and

text: the

text: data

text: and,

text: therefore,

text: we

text: have

text: what

text: is

text: observed

text: in

text: the

text: data.

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text: Figure 4: Square error in online parameter estimation eferences R Theory and Practice . Marcel Dekker, Inc. [Corke, 2011] Peter Corke (2011). Robotics, [Chizhek, 2013] Howard J.Chizhek (2004). Vision and Control - Fundamental al- ME 585B Lectures. University of Wash- gorithms in matlab. Springer ington [Lewis, 2011] Frank L.Lewis, Darren [Astrom, 2008] Karl J. Astrom, Dr. Bjorn M.Dawson, Chaouki T.Abdallah Wittenmark (2008). Adaptive Control, (2004). Robot Manipulator Control - Second Edition Dover Publications, Inc 7 ##LTFigure##

text: Figure

label: paragraph

font: KGOMQI+URWPalladioL-Bold

text: Square

label: caption

font: OXOJOY+URWPalladioL-Ital

text: error

text: in

text: online

text: parameter

text: estimation

text: eferences

font: VKHWYA+TeXPalladioL-SC

text: R

font: NVMRQP+URWPalladioL-Roma

text: Theory

label: reference

text: and

text: Practice

text: Marcel

text: Dekker,

text: Inc.

text: [Corke,

text: Peter

text: Corke

text: Robotics,

text: [Chizhek,

text: Howard

text: J.Chizhek

text: Vision

text: and

text: Control

text: Fundamental

text: al-

text: ME

text: Lectures.

text: University

text: of

text: Wash-

text: gorithms

text: in

text: matlab.

text: Springer

text: ington

text: [Lewis,

text: Frank

text: L.Lewis,

text: Darren

text: [Astrom,

text: Karl

text: J.

text: Astrom,

text: Dr.

text: Bjorn

text: M.Dawson,

text: Chaouki

text: T.Abdallah

text: Wittenmark

text: Adaptive

text: Control,

text: Robot

text: Manipulator

text: Control

text: Second

text: Edition

text: Dover

text: Publications,

text: Inc

text: ##LTFigure##


text: Deﬁning and composing big state machines Victor Yodaiken May 27, 2020 Abstract A sequence function alternative representation of state machines. 1 Introduction State machines are usually presented in terms of a set of events E, a set of states × → S, and a map δ : S A S. Alternatively we can deﬁne a state variable as a function the set of ﬁnite sequences over E so that for any sequence s: x = F(s) The function F is an alternative presentation of a state machine in a way made precise in section 3. We can deﬁne a number of state variables (maybe a large number) all depending on the same sequence of events to describe a complex system. For example, given some sequence s that deﬁnes the current state, we might require that the number of processes that are executing code in a critical region never be more than one - where Processes and ProgramCounter(p) and Critical might all be state dependent. |{ ∈ ∈ }| ≤ p Process : ProgramCounter(p) Critical 1 Section 2 shows techniques for deﬁning and composing sequence functions. Section 3 shows the correspondence between sequence functions and standard state machine presentations. Examples of applications are in other papers. This work comes from a long process of attempting to reﬁne initial intuitions about the utility of recursive sequence presentations of state machines and the use of general state machine products to model composition and parallel/concurrent computation[Yod09] and [Yod08] 2 Sequence functions Many useful sequence functions can be deﬁned by primitive recursion on ∈ sequences. Let null be the empty sequence and if s is a sequence and a E, let sa be the sequence obtained by appending a to s. G(null) = x , G(sa) = g(a, G(s)) (1) 0 1

text: Deﬁning

label: title

text: and

text: composing

text: big

text: state

text: machines

text: Victor

label: author

text: Yodaiken

text: May

text: Abstract

text: A

label: abstract

text: sequence

text: function

text: alternative

text: representation

text: of

text: state

text: machines.

label: section

text: Introduction

text: State

text: machines

text: are

text: usually

text: presented

text: in

text: terms

text: of

text: a

text: set

text: of

text: events

text: E,

text: a

text: set

text: of

text: states

text: S,

text: and

text: a

text: map

text: δ

text: S

text: A

text: S.

text: Alternatively

text: we

text: can

text: deﬁne

text: a

text: state

text: variable

text: as

text: a

text: function

text: the

text: set

text: of

text: ﬁnite

text: sequences

text: over

text: E

text: so

text: that

text: for

text: any

text: sequence

text: s:

text: x

text: F(s)

text: The

text: function

text: F

text: is

text: an

text: alternative

text: presentation

text: of

text: a

text: state

text: machine

text: in

text: a

text: way

text: made

text: precise

text: in

text: section

text: We

text: can

text: deﬁne

text: a

text: number

text: of

text: state

text: variables

text: (maybe

text: a

text: large

text: number)

text: all

text: depending

text: on

text: the

text: same

text: sequence

text: of

text: events

text: to

text: describe

text: a

text: complex

text: system.

text: For

text: example,

text: given

text: some

text: sequence

text: s

text: that

text: deﬁnes

text: the

text: current

text: state,

text: we

text: might

text: require

text: that

text: the

text: number

text: of

text: processes

text: that

text: are

text: executing

text: code

text: in

text: a

text: critical

text: region

text: never

text: be

text: more

text: than

text: one

text: where

text: Processes

text: and

text: ProgramCounter(p)

text: and

text: Critical

text: might

text: all

text: be

text: state

text: dependent.

text: p

text: Process

text: ProgramCounter(p)

text: Critical

text: Section

text: shows

text: techniques

text: for

text: deﬁning

text: and

text: composing

text: sequence

text: functions.

text: Section

text: shows

text: the

text: correspondence

text: between

text: sequence

text: functions

text: and

text: standard

text: state

text: machine

text: presentations.

text: Examples

text: of

text: applications

text: are

text: in

text: other

text: papers.

text: This

text: work

text: comes

text: from

text: a

text: long

text: process

text: of

text: attempting

text: to

text: reﬁne

text: initial

text: intuitions

text: about

text: the

text: utility

text: of

text: recursive

text: sequence

text: presentations

text: of

text: state

text: machines

text: and

text: the

text: use

text: of

text: general

text: state

text: machine

text: products

text: to

text: model

text: composition

text: and

text: parallel/concurrent

text: and

text: Sequence

text: functions

text: Many

text: useful

text: sequence

text: functions

text: can

text: be

text: deﬁned

text: by

text: primitive

text: recursion

text: on

text: sequences.

text: Let

text: be

text: the

text: empty

text: sequence

text: and

text: if

text: s

text: is

text: a

text: sequence

text: and

text: a

text: E,

text: let

text: sa

text: be

text: the

text: sequence

text: obtained

text: by

text: appending

text: a

text: to

text: s.

text: G(null)

text: x

text: G(sa)

text: g(a,

text: G(s))


text: g(i, a, x . . . x ) = r gives the sequence of events component i sees when a single 1 n i event a is appended to the global sequence and the component current state ∗ values are given by x . . . x . The function g determines a map g (i, s) = s by 1 n i recursion. Let s concat r be the sequence obtained by concatenating sequences ∗ ∗ s and r. We set g (i, null) = null and then if g(i, s) = s we deﬁne g (i, sa) = i s concat r where r = g(i, a, G (s ) . . . G (s )). i i i 1 1 n n ∗ ∗ G(s) = (G (g (1, s)) . . . G (g (n, s))) ∗ 1 n g (i, null) = null and (3) and ∗ ∗ ∗ ∗ g (i, sa) = g (i, s) concat g(a, G (g (1, s)), . . . G(g (n, s))) 1 3 State machines Obviously, ﬁnite state machines are an important class but I do not here assume state machines are ﬁnite. ∈ A Moore type state machine is M = (X, E, S, σ , δ, λ) where σ S is the × → 0 0→ “start state” and δ : S E S is the transition function λ : S X is the output map and λ(σ) is the output of the state machine in state σ. A standard state machine can be considered to be a more machine where λ(σ) = σ. ∗ Given Moore machine M = (X, E, S, s , δ, λ) let E be the set of ﬁnite 0 sequences over E including null and let ∗ δ (σ, null) = σ and ∗ ∗ δ (σ, sa) = δ(δ (σ, s), a). ∗ Let M be deﬁned by ∗ ∗ M (s) = δ (σ , s) 0 ∗ Then λ (M (s)) is the output of M in the state reached by following s from M the initial state. ∗ → ∗ Say M is an implementation of G : E X if and only if λ (M (s)) = G(s) ∈ ∗ M for all s E . Say G is ﬁnite state if and only if there is a M that implements G where the state set of M is ﬁnite. × → If E and X are ﬁnite sets and g : E X X then G deﬁned by G(null) = ∈ x X and G(sa) = g(a, G(s)) is ﬁnite state. 0 Suppose that M , . . . M implement G , . . . G and M = (X , E , S , σ , δ , λ ). 1 n 1 n i i i i 0,i i i Deﬁne G using deﬁnition form 3. Then we can construct a product of the M i that implements G as follows: × The state set S = S . . . S 1 n The output map λ((σ . . . σ ) = (λ (σ ), . . . λ (σ )) 1 n 1 1 n n ∗ The transition map δ(σ, a) = (. . . δ (σ , g(i, a, λ(s)) . . . ) where σ = (σ , . . . σ ) i i 1 n 3

text: g(i,

text: a,

text: x

text: x

text: r

text: gives

text: the

text: sequence

text: of

text: events

text: component

text: i

text: sees

text: when

text: a

text: single

text: n

text: i

text: event

text: a

text: is

text: appended

text: to

text: the

text: global

text: sequence

text: and

text: the

text: component

text: current

text: state

text: values

text: are

text: given

text: by

text: x

text: x

text: The

text: function

text: g

text: determines

text: a

text: map

text: g

text: (i,

text: s)

text: s

text: by

text: n

text: i

text: recursion.

text: Let

text: s

text: concat

text: r

text: be

text: the

text: sequence

text: obtained

text: by

text: concatenating

text: sequences

text: s

text: and

text: r.

text: We

text: set

text: g

text: (i,

text: null)

text: and

text: then

text: if

text: g(i,

text: s)

text: s

text: we

text: deﬁne

text: g

text: (i,

text: sa)

text: i

text: s

text: concat

text: r

text: where

text: r

text: g(i,

text: a,

text: G

text: (s

text: G

text: (s

text: i

text: i

text: i

text: n

text: n

text: G(s)

text: (G

text: (g

text: s))

text: G

text: (g

text: (n,

text: s)))

text: n

text: g

text: (i,

text: null)

text: and

text: and

text: g

text: (i,

text: sa)

text: g

text: (i,

text: s)

text: concat

text: g(a,

text: G

text: (g

text: s)),

text: G(g

text: (n,

text: s)))

text: State

text: machines

text: Obviously,

text: ﬁnite

text: state

text: machines

text: are

text: an

text: important

text: class

text: but

text: I

text: do

text: not

text: here

text: assume

text: state

text: machines

text: are

text: ﬁnite.

text: A

text: Moore

text: type

text: state

text: machine

text: is

text: M

text: (X,

text: E,

text: S,

text: σ

text: δ,

text: λ)

text: where

text: σ

text: S

text: is

text: the

text: 0→

text: “start

text: state”

text: and

text: δ

text: S

text: E

text: S

text: is

text: the

text: transition

text: function

text: λ

text: S

text: X

text: is

text: the

text: output

text: map

text: and

text: λ(σ)

text: is

text: the

text: output

text: of

text: the

text: state

text: machine

text: in

text: state

text: σ.

text: A

text: standard

text: state

text: machine

text: can

text: be

text: considered

text: to

text: be

text: a

text: more

text: machine

text: where

text: λ(σ)

text: σ.

text: Given

text: Moore

text: machine

text: M

text: (X,

text: E,

text: S,

text: s

text: δ,

text: λ)

text: let

text: E

text: be

text: the

text: set

text: of

text: ﬁnite

text: sequences

text: over

text: E

text: including

text: and

text: let

text: δ

text: (σ,

text: null)

text: σ

text: and

text: δ

text: (σ,

text: sa)

text: δ(δ

text: (σ,

text: s),

text: a).

text: Let

text: M

text: be

text: deﬁned

text: by

text: M

text: (s)

text: δ

text: (σ

text: s)

text: Then

text: λ

text: (M

text: (s))

text: is

text: the

text: output

text: of

text: M

text: in

text: the

text: state

text: reached

text: by

text: following

text: s

text: from

text: M

text: the

text: initial

text: state.

text: Say

text: M

text: is

text: an

text: implementation

text: of

text: G

text: E

text: X

text: if

text: and

text: only

text: if

text: λ

text: (M

text: (s))

text: G(s)

text: M

text: for

text: all

text: s

text: E

text: Say

text: G

text: is

text: ﬁnite

text: state

text: if

text: and

text: only

text: if

text: there

text: is

text: a

text: M

text: that

text: implements

text: G

text: where

text: the

text: state

text: set

text: of

text: M

text: is

text: ﬁnite.

text: If

text: E

text: and

text: X

text: are

text: ﬁnite

text: sets

text: and

text: g

text: E

text: X

text: X

text: then

text: G

text: deﬁned

text: by

text: G(null)

text: x

text: X

text: and

text: G(sa)

text: g(a,

text: G(s))

text: is

text: ﬁnite

text: state.

text: Suppose

text: that

text: M

text: M

text: implement

text: G

text: G

text: and

text: M

text: (X

text: E

text: S

text: σ

text: δ

text: λ

text: n

text: n

text: i

text: i

text: i

text: i

text: i

text: i

text: Deﬁne

text: G

text: using

text: deﬁnition

text: form

text: Then

text: we

text: can

text: construct

text: a

text: product

text: of

text: the

text: M

text: i

text: that

text: implements

text: G

text: as

text: follows:

text: The

text: state

text: set

text: S

text: S

text: S

text: n

text: The

text: output

text: map

text: λ((σ

text: σ

text: (λ

text: (σ

text: λ

text: (σ

text: n

text: n

text: n

text: The

text: transition

text: map

text: δ(σ,

text: a)

text: δ

text: (σ

text: g(i,

text: a,

text: λ(s))

text: where

text: σ

text: (σ

text: σ

text: i

text: i

text: n


text: 12 ANDREW SALCH C generated by the coﬁbrant objects. Then the Waldhausen K-theory of C is trivial, that is, | | K∗(C) = π∗ (Ω wS•C ) (cid:27) 0. Proof. First suppose that C is a Bousﬁeld localization of the discrete model structure. Then all objects are coﬁbrant, and we have the exact (in the sense of Waldhausen) functor 1 : → C C sending every object of C to the terminal object, and sending every morphism to the identity morphism on the terminal object. Furthermore, since all maps in C are → coﬁbrations, we have a coﬁber sequence of exact functors C C: → → idC 1 1. Now by Waldhausen’s additivity theorem (Proposition 1.3.2 and Theorem 1.4.2 of [11]), the functors ∨ → (wS• idC) (wS•1), wS•1 : wS•C wS•C are homotopic. Hence the identity map on wS•C is nulhomotopic, hence wS•C is con- | | tractible, hence Kn(C) = πnΩ wS•C (cid:27) 0 for all n. Now suppose that C is a Bousﬁeld colocalization of the discrete model structure. By the dual of Lemma 4.6, all maps between coﬁbrant objects in C are coﬁbrations. So the same argument as above, using Waldhausen’s additivity theorem, implies contractibility of wS•W, hence triviality of all K-groups. (cid:3) References [1] Francis Borceux. Handbook of categorical algebra. 2, volume 51 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. Categories and structures. [2] A. K. Bousﬁeld. On the telescopic homotopy theory of spaces. Trans. Amer. Math. Soc., 353(6):2391–2426 (electronic), 2001. [3] A. K. Bousﬁeld and E. M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80–130. Springer, Berlin, 1978. [4] C. Cassidy, M. He´bert, and G. M. Kelly. Reﬂective subcategories, localizations and factorization systems. J. Austral. Math. Soc. Ser. A, 38(3):287–329, 1985. [5] W. G. Dwyer and J. Spalin´ski. Homotopy theories and model categories. In Handbook of algebraic topology, pages 73–126. North-Holland, Amsterdam, 1995. [6] P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag New York, Inc., New York, 1967. [7] Philip S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. [8] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathemat- ics. Springer-Verlag, New York, second edition, 1998. [9] Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin- New York, 1967. [10] Jiˇr´ı Rosicky´ and Walter Tholen. Factorization, ﬁbration and torsion. J. Homotopy Relat. Struct., 2(2):295– 314, 2007. [11] Friedhelm Waldhausen. Algebraic K-theory of spaces. In Algebraic and geometric topology (New Brunswick, N.J., 1983), volume 1126 of Lecture Notes in Math., pages 318–419. Springer, Berlin, 1985.

text: ANDREW

text: SALCH

text: C

font: YVNAWL+URWChanceryL-MediItal

text: generated

text: by

text: the

text: coﬁbrant

text: objects.

text: Then

text: the

text: Waldhausen

text: K-theory

text: of

text: C

text: is

text: trivial,

text: that

text: is,

font: BJNAZM+txsy

text: K∗(C)

font: ZEDRXW+rtxr

text: π∗

text: (Ω

text: wS•C

font: VYFONL+txsyc

text: Proof.

text: First

text: suppose

text: that

text: C

text: is

text: a

text: Bousﬁeld

text: localization

text: of

text: the

text: discrete

text: model

text: structure.

text: Then

text: all

text: objects

text: are

text: coﬁbrant,

text: and

text: we

text: have

text: the

text: exact

text: (in

text: the

text: sense

text: of

text: Waldhausen)

text: functor

text: C

text: C

text: sending

text: every

text: object

text: of

text: C

text: to

text: the

text: terminal

text: object,

text: and

text: sending

text: every

text: morphism

text: to

text: the

text: identity

text: morphism

text: on

text: the

text: terminal

text: object.

text: Furthermore,

text: since

text: all

text: maps

text: in

text: C

text: are

text: coﬁbrations,

text: we

text: have

text: a

text: coﬁber

text: sequence

text: of

text: exact

text: functors

text: C

text: C:

text: idC

text: Now

text: by

text: Waldhausen’s

text: additivity

text: theorem

text: (Proposition

text: and

text: Theorem

text: of

text: the

text: functors

text: (wS•

text: idC)

text: wS•C

text: wS•C

text: are

text: homotopic.

text: Hence

text: the

text: identity

text: map

text: on

text: wS•C

text: is

text: nulhomotopic,

text: hence

text: wS•C

text: is

text: con-

text: tractible,

text: hence

text: Kn(C)

text: πnΩ

text: wS•C

text: for

text: all

text: n.

text: Now

text: suppose

text: that

text: C

text: is

text: a

text: Bousﬁeld

text: colocalization

text: of

text: the

text: discrete

text: model

text: structure.

text: By

text: the

text: dual

text: of

text: Lemma

text: all

text: maps

text: between

text: coﬁbrant

text: objects

text: in

text: C

text: are

text: coﬁbrations.

text: So

text: the

text: same

text: argument

text: as

text: above,

text: using

text: Waldhausen’s

text: additivity

text: theorem,

text: implies

text: contractibility

text: of

text: wS•W,

text: hence

text: triviality

text: of

text: all

text: K-groups.

text: References

text: Francis

text: Borceux.

text: Handbook

text: of

text: categorical

text: algebra.

text: volume

text: of

text: Encyclopedia

text: of

text: Mathematics

text: and

text: its

text: Applications.

text: Cambridge

text: University

text: Press,

text: Cambridge,

text: Categories

text: and

text: structures.

text: A.

text: K.

text: Bousﬁeld.

text: On

text: the

text: telescopic

text: homotopy

text: theory

text: of

text: spaces.

text: Trans.

text: Amer.

text: Math.

text: Soc.,

text: (electronic),

text: A.

text: K.

text: Bousﬁeld

text: and

text: E.

text: M.

text: Friedlander.

text: Homotopy

text: theory

text: of

text: Γ-spaces,

text: spectra,

text: and

text: bisimplicial

text: sets.

text: In

text: Geometric

text: applications

text: of

text: homotopy

text: theory

text: (Proc.

text: Conf.,

text: Evanston,

text: Ill.,

text: II,

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Math.,

text: pages

text: Springer,

text: Berlin,

text: C.

text: Cassidy,

text: M.

text: He´bert,

text: and

text: G.

text: M.

text: Kelly.

text: Reﬂective

text: subcategories,

text: localizations

text: and

text: factorization

text: systems.

text: J.

text: Austral.

text: Math.

text: Soc.

text: Ser.

text: A,

text: W.

text: G.

text: Dwyer

text: and

text: J.

text: Spalin´ski.

text: Homotopy

text: theories

text: and

text: model

text: categories.

text: In

text: Handbook

text: of

text: algebraic

text: topology,

text: pages

text: North-Holland,

text: Amsterdam,

text: P.

text: Gabriel

text: and

text: M.

text: Zisman.

text: Calculus

text: of

text: fractions

text: and

text: homotopy

text: theory.

text: Ergebnisse

text: der

text: Mathematik

text: und

text: ihrer

text: Grenzgebiete,

text: Band

text: Springer-Verlag

text: New

text: York,

text: Inc.,

text: New

text: York,

text: Philip

text: S.

text: Hirschhorn.

text: Model

text: categories

text: and

text: their

text: localizations,

text: volume

text: of

text: Mathematical

text: Surveys

text: and

text: Monographs.

text: American

text: Mathematical

text: Society,

text: Providence,

text: RI,

text: Saunders

text: Mac

text: Lane.

text: Categories

text: for

text: the

text: working

text: mathematician,

text: volume

text: of

text: Graduate

text: Texts

text: in

text: Mathemat-

text: ics.

text: Springer-Verlag,

text: New

text: York,

text: second

text: edition,

text: Daniel

text: G.

text: Quillen.

text: Homotopical

text: algebra.

text: Lecture

text: Notes

text: in

text: Mathematics,

text: No.

text: Springer-Verlag,

text: Berlin-

text: New

text: York,

text: Jiˇr´ı

text: Rosicky´

text: and

text: Walter

text: Tholen.

text: Factorization,

text: ﬁbration

text: and

text: torsion.

text: J.

text: Homotopy

text: Relat.

text: Struct.,

text: Friedhelm

text: Waldhausen.

text: Algebraic

text: K-theory

text: of

text: spaces.

text: In

text: Algebraic

text: and

text: geometric

text: topology

text: (New

text: Brunswick,

text: N.J.,

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Math.,

text: pages

text: Springer,

text: Berlin,


text: THE BOUSFIELD LOCALIZATIONS AND COLOCALIZATIONS OF THE DISCRETE MODEL STRUCTURE. 3 (6) If C is pointed, then any Bousﬁeld localization of the discrete model structure makes C into a Waldhausen category, and the Waldhausen K-groups K∗(C) = | | π∗ (Ω wS•C ) are all trivial. The arguments used in this paper dualize without diﬃculties; in particular, we do not make any use of the small object argument or any coﬁbrant generation hypotheses. Hence the dual statements of Answers 1.2 also hold, if we assume that C is ﬁnitely co-well- complete. → As an amusing application, we have Corollary 4.4: let R S be a morphism of com- mutative rings, and equip Mod(R) with the discrete model structure. Then there exists a − ⊗ Bousﬁeld localization of Mod(R) for which the base change functor S is a ﬁbrant ⊗ → R replacement monad if and only if the natural multiplication map S S S is an isomor- R phism. Remark 1.3. A lot of the work of proving the results in this paper is done by the construc- tion of a factorization system from a reﬂective subcategory from the paper [4] (see also the more recent paper [10] of Rosicky´ and Tholen, in which the ideas of [4] were revisited and expanded upon); we recall that result in Theorem 3.4. With the aid of that theorem, none of the statements in Answers 1.2 are diﬃcult to prove. We wrote an early version of this paper in December 2011, but we were not sure if the results were already known. (The paper has been completely rewritten and simpliﬁed since then.) After encouragement from several other homotopy theorists who work with model categories and found the results of this paper novel, we decided to submit this paper for publication, since neither [4] nor [10] give clean results in familiar homotopy-theoretic language (instead of model categories and Bousﬁeld localization, [4] and [10] work with torsion theories and factorization sys- tems), so one does not ﬁnd the classiﬁcation of the Bousﬁeld localizations of the discrete model structure on a category, or the computation of the homotopy category and algebraic K-theory of all such localizations, in the existing literature. So we feel that the present paper adds something worthwhile to the literature on the subject. We are grateful to C. Weibel for ﬁnding errors in two lemmas in an earlier version of this paper. Conventions 1.4. It is worth being clear on some deﬁnitions which are not totally consis- tent in the literature: • In this paper, a “model category” is a closed model category in the sense of Quillen’s monograph [9], that is (brieﬂy): a ﬁnitely complete, ﬁnitely co-complete category equipped with three closed-under-retracts classes of morphisms called “coﬁbrations,” “weak equivalences,” and “ﬁbrations,” such that each morphism factors into an acyclic coﬁbration followed by a ﬁbration, each morphism factors as a coﬁbration followed by an acyclic ﬁbration, weak equivalences have the two- out-of-three property, the coﬁbrations are precisely the maps which have the left lifting property with respect to acyclic ﬁbrations, and the acyclic coﬁbrations are precisely the maps which have the left lifting property with respect ot the ﬁbra- tions. We do not assume that a model category has all limits and colimits, we do not include a speciﬁc choice of factorization systems as part of the structure of a model category, and we do not assume that the factorization systems are functo- rial. (We will show, in fact, that in the model categories considered in this paper, factorization systems can be found which are functorial.)

text: THE

text: BOUSFIELD

text: LOCALIZATIONS

text: AND

text: COLOCALIZATIONS

text: OF

text: THE

text: DISCRETE

text: MODEL

text: STRUCTURE.

label: list

text: If

text: C

text: is

text: pointed,

text: then

text: any

text: Bousﬁeld

text: localization

text: of

text: the

text: discrete

text: model

text: structure

text: makes

text: C

text: into

text: a

text: Waldhausen

text: category,

text: and

text: the

text: Waldhausen

text: K-groups

text: K∗(C)

text: π∗

text: (Ω

text: wS•C

text: are

text: all

text: trivial.

text: The

text: arguments

text: used

text: in

text: this

text: paper

text: dualize

text: without

text: diﬃculties;

text: in

text: particular,

text: we

text: do

text: not

text: make

text: any

text: use

text: of

text: the

text: small

text: object

text: argument

text: or

text: any

text: coﬁbrant

text: generation

text: hypotheses.

text: Hence

text: the

text: dual

text: statements

text: of

text: Answers

text: also

text: hold,

text: if

text: we

text: assume

text: that

text: C

text: is

text: ﬁnitely

text: co-well-

text: complete.

text: As

text: an

text: amusing

text: application,

text: we

text: have

text: Corollary

text: let

text: R

text: S

text: be

text: a

text: morphism

text: of

text: com-

text: mutative

text: rings,

text: and

text: equip

text: Mod(R)

text: with

text: the

text: discrete

text: model

text: structure.

text: Then

text: there

text: exists

text: a

text: Bousﬁeld

text: localization

text: of

text: Mod(R)

text: for

text: which

text: the

text: base

text: change

text: functor

text: S

text: is

text: a

text: ﬁbrant

text: R

text: replacement

text: monad

text: if

text: and

text: only

text: if

text: the

text: natural

text: multiplication

text: map

text: S

text: S

text: S

text: is

text: an

text: isomor-

text: R

text: phism.

text: Remark

text: A

text: lot

text: of

text: the

text: work

text: of

text: proving

text: the

text: results

text: in

text: this

text: paper

text: is

text: done

text: by

text: the

text: construc-

text: tion

text: of

text: a

text: factorization

text: system

text: from

text: a

text: reﬂective

text: subcategory

text: from

text: the

text: paper

text: (see

text: also

text: the

text: more

text: recent

text: paper

text: of

text: Rosicky´

text: and

text: Tholen,

text: in

text: which

text: the

text: ideas

text: of

text: were

text: revisited

text: and

text: expanded

text: upon);

text: we

text: recall

text: that

text: result

text: in

text: Theorem

text: With

text: the

text: aid

text: of

text: that

text: theorem,

text: of

text: the

text: statements

text: in

text: Answers

text: are

text: diﬃcult

text: to

text: prove.

text: We

text: wrote

text: an

text: early

text: version

text: of

text: this

text: paper

text: in

text: December

text: but

text: we

text: were

text: not

text: sure

text: if

text: the

text: results

text: were

text: already

text: known.

text: (The

text: paper

text: has

text: been

text: completely

text: rewritten

text: and

text: simpliﬁed

text: since

text: then.)

text: After

text: encouragement

text: from

text: several

text: other

text: homotopy

text: theorists

text: who

text: work

text: with

text: model

text: categories

text: and

text: found

text: the

text: results

text: of

text: this

text: paper

text: novel,

text: we

text: decided

text: to

text: submit

text: this

text: paper

text: for

text: publication,

text: since

text: neither

text: nor

text: give

text: clean

text: results

text: in

text: familiar

text: homotopy-theoretic

text: language

text: (instead

text: of

text: model

text: categories

text: and

text: Bousﬁeld

text: localization,

text: and

text: work

text: with

text: torsion

text: theories

text: and

text: factorization

text: sys-

text: tems),

text: so

text: one

text: does

text: not

text: ﬁnd

text: the

text: classiﬁcation

text: of

text: the

text: Bousﬁeld

text: localizations

text: of

text: the

text: discrete

text: model

text: structure

text: on

text: a

text: category,

text: or

text: the

text: computation

text: of

text: the

text: homotopy

text: category

text: and

text: algebraic

text: K-theory

text: of

text: all

text: such

text: localizations,

text: in

text: the

text: existing

text: literature.

text: So

text: we

text: feel

text: that

text: the

text: present

text: paper

text: adds

text: something

text: worthwhile

text: to

text: the

text: literature

text: on

text: the

text: subject.

text: We

text: are

text: grateful

text: to

text: C.

text: Weibel

text: for

text: ﬁnding

text: errors

text: in

text: two

text: lemmas

text: in

text: an

text: earlier

text: version

text: of

text: this

text: paper.

text: Conventions

text: It

text: is

text: worth

text: being

text: clear

text: on

text: some

text: deﬁnitions

text: which

text: are

text: not

text: totally

text: consis-

text: tent

text: in

text: the

text: literature:

text: In

text: this

text: paper,

text: a

text: “model

text: category”

text: is

text: a

text: closed

text: model

text: category

text: in

text: the

text: sense

text: of

text: Quillen’s

text: monograph

text: that

text: is

text: (brieﬂy):

text: a

text: ﬁnitely

text: complete,

text: ﬁnitely

text: co-complete

text: category

text: equipped

text: with

text: three

text: closed-under-retracts

text: classes

text: of

text: morphisms

text: called

text: “coﬁbrations,”

text: “weak

text: equivalences,”

text: and

text: “ﬁbrations,”

text: such

text: that

text: each

text: morphism

text: factors

text: into

text: an

text: acyclic

text: coﬁbration

text: followed

text: by

text: a

text: ﬁbration,

text: each

text: morphism

text: factors

text: as

text: a

text: coﬁbration

text: followed

text: by

text: an

text: acyclic

text: ﬁbration,

text: weak

text: equivalences

text: have

text: the

text: two-

text: out-of-three

text: property,

text: the

text: coﬁbrations

text: are

text: precisely

text: the

text: maps

text: which

text: have

text: the

text: left

text: lifting

text: property

text: with

text: respect

text: to

text: acyclic

text: ﬁbrations,

text: and

text: the

text: acyclic

text: coﬁbrations

text: are

text: precisely

text: the

text: maps

text: which

text: have

text: the

text: left

text: lifting

text: property

text: with

text: respect

text: ot

text: the

text: ﬁbra-

text: tions.

text: We

text: do

text: not

text: assume

text: that

text: a

text: model

text: category

text: has

text: all

text: limits

text: and

text: colimits,

text: we

text: do

text: not

text: include

text: a

text: speciﬁc

text: choice

text: of

text: factorization

text: systems

text: as

text: part

text: of

text: the

text: structure

text: of

text: a

text: model

text: category,

text: and

text: we

text: do

text: not

text: assume

text: that

text: the

text: factorization

text: systems

text: are

text: functo-

text: rial.

text: (We

text: will

text: show,

text: in

text: fact,

text: that

text: in

text: the

text: model

text: categories

text: considered

text: in

text: this

text: paper,

text: factorization

text: systems

text: can

text: be

text: found

text: which

text: are

text: functorial.)


text: 4 ANDREW SALCH • In this paper, a subcategory A of a category C is said to be reﬂective if the inclusion → functor A (cid:44) C is full and admits a left adjoint. (Not all sources include fullness in the deﬁnition of a reﬂective subcategory; see for example [8].) 2. From Bousfield localizations of discrete model structures to replete reflective subcategories. Lemma 2.1. Let C be a model category which is a Bousﬁeld localization of a discrete model category. (1) A map in C is an acyclic ﬁbration if and only if it is an isomorphism. (2) Two maps in C with ﬁbrant codomain are left-homotopic if and only if they are equal. (3) Two maps in C with ﬁbrant codomain are right-homotopic if and only if they are equal. Proof. (1) Bousﬁeld localization does not change the coﬁbrations of the underlying model category, hence it does not change the acyclic ﬁbrations (since they are determined by the coﬁbrations). In a discrete model category, the acyclic ﬁbrations are the isomorphisms; hence in any localization of a discrete model category, the acyclic ﬁbrations are still the isomorphisms. −→ (2) Let f, g : A B be two maps in C, and suppose that B is ﬁbrant. Then any two maps that are left-homotopic are left-homotopic by a “very good” cylinder object (see [5], particularly Lemma 4.6), so if we assume that f, g are left-homotopic, (cid:97) −→h then we have a map Cyl(A) B and a factorization (cid:96) −→i −→j (cid:96) (cid:96) A A Cyl(A) A → (cid:96) of the codiagonal map A A A, with i a coﬁbration and j an acyclic ﬁbration, ◦ (cid:96) → and h i = f g : A A B. But every acyclic ﬁbration in C is an isomorphism; so the map j is an isomorphism, so we can factor f g all the way through the → codiagonal map A A A. So f = g. (3) Maps with ﬁbrant codomain and coﬁbrant domain are left-homotopic if and only if they are right-homotopic; see Lemma 4.21 in [5] a proof of this. The statement then follows immediately from the previous parts of this lemma. (cid:3) −→f Lemma 2.2. Let C a model category and let A B be a map in C. Let g1, g2 be two maps making the diagram A cof,we(cid:47)(cid:47) PA fib (cid:47)(cid:47) 1 f gi (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) B cof,we(cid:47)(cid:47) PB fib (cid:47)(cid:47) 1 commute for i = 1, 2, where maps labelled cof, we, fib are coﬁbrations, weak equiva- lences, and ﬁbrations in C, respectively. Then g and g are right-homotopic. 1 2 Proof. We have the factorization of the diagonal map PB c−o→f,we Path(PB) −f→ib PB × PB ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ANDREW

text: SALCH

text: In

text: this

text: paper,

text: a

text: subcategory

text: A

text: of

text: a

text: category

text: C

text: is

text: said

text: to

text: be

text: reﬂective

text: if

text: the

text: inclusion

text: functor

text: A

font: PWXRCB+rtxmi

text: C

text: is

text: full

text: and

text: admits

text: a

text: left

text: adjoint.

text: (Not

text: all

text: sources

text: include

text: fullness

text: in

text: the

text: deﬁnition

text: of

text: a

text: reﬂective

text: subcategory;

text: see

text: for

text: example

text: From

font: UHCMTT+rtxsc

text: Bousfield

text: localizations

text: of

text: discrete

text: model

text: structures

text: to

text: replete

text: reflective

text: subcategories.

text: Lemma

text: Let

text: C

text: be

text: a

text: model

text: category

text: which

text: is

text: a

text: Bousﬁeld

text: localization

text: of

text: a

text: discrete

text: model

text: category.

text: A

text: map

text: in

text: C

text: is

text: an

text: acyclic

text: ﬁbration

text: if

text: and

text: only

text: if

text: it

text: is

text: an

text: isomorphism.

text: Two

text: maps

text: in

text: C

text: with

text: ﬁbrant

text: codomain

text: are

text: left-homotopic

text: if

text: and

text: only

text: if

text: they

text: are

text: equal.

text: Two

text: maps

text: in

text: C

text: with

text: ﬁbrant

text: codomain

text: are

text: right-homotopic

text: if

text: and

text: only

text: if

text: they

text: are

text: equal.

text: Proof.

text: Bousﬁeld

text: localization

text: does

text: not

text: change

text: the

text: coﬁbrations

text: of

text: the

text: underlying

text: model

text: category,

text: hence

text: it

text: does

text: not

text: change

text: the

text: acyclic

text: ﬁbrations

text: (since

text: they

text: are

text: determined

text: by

text: the

text: coﬁbrations).

text: In

text: a

text: discrete

text: model

text: category,

text: the

text: acyclic

text: ﬁbrations

text: are

text: the

text: isomorphisms;

text: hence

text: in

text: any

text: localization

text: of

text: a

text: discrete

text: model

text: category,

text: the

text: acyclic

text: ﬁbrations

text: are

text: still

text: the

text: isomorphisms.

text: Let

text: f,

text: g

text: A

text: B

text: be

text: two

text: maps

text: in

text: C,

text: and

text: suppose

text: that

text: B

text: is

text: ﬁbrant.

text: Then

text: any

text: two

text: maps

text: that

text: are

text: left-homotopic

text: are

text: left-homotopic

text: by

text: a

text: “very

text: good”

text: cylinder

text: object

text: (see

text: particularly

text: Lemma

text: so

text: if

text: we

text: assume

text: that

text: f,

text: g

text: are

text: left-homotopic,

text: −→h

text: then

text: we

text: have

text: a

text: map

text: Cyl(A)

text: B

text: and

text: a

text: factorization

text: −→i

text: −→j

text: A

text: A

text: Cyl(A)

text: A

text: of

text: the

text: codiagonal

text: map

text: A

text: A

text: A,

text: with

text: i

text: a

text: coﬁbration

text: and

text: j

text: an

text: acyclic

text: ﬁbration,

text: and

text: h

text: i

text: f

text: g

text: A

text: A

text: B.

text: But

text: every

text: acyclic

text: ﬁbration

text: in

text: C

text: is

text: an

text: isomorphism;

text: so

text: the

text: map

text: j

text: is

text: an

text: isomorphism,

text: so

text: we

text: can

text: factor

text: f

text: g

text: all

text: the

text: way

text: through

text: the

text: codiagonal

text: map

text: A

text: A

text: A.

text: So

text: f

text: g.

text: Maps

text: with

text: ﬁbrant

text: codomain

text: and

text: coﬁbrant

text: domain

text: are

text: left-homotopic

text: if

text: and

text: only

text: if

text: they

text: are

text: right-homotopic;

text: see

text: Lemma

text: in

text: a

text: proof

text: of

text: this.

text: The

text: statement

text: then

text: follows

text: immediately

text: from

text: the

text: previous

text: parts

text: of

text: this

text: lemma.

text: −→f

text: Lemma

text: Let

text: C

text: a

text: model

text: category

text: and

text: let

text: A

text: B

text: be

text: a

text: map

text: in

text: C.

text: Let

text: be

text: two

text: maps

text: making

text: the

text: diagram

text: A

text: PA

text: fib

text: f

text: gi

text: B

text: PB

text: fib

text: commute

text: for

text: i

text: where

text: maps

text: labelled

text: cof,

text: we,

text: fib

text: are

text: coﬁbrations,

text: weak

text: equiva-

text: lences,

text: and

text: ﬁbrations

text: in

text: C,

text: respectively.

text: Then

text: g

text: and

text: g

text: are

text: right-homotopic.

text: Proof.

text: We

text: have

text: the

text: factorization

text: of

text: the

text: diagonal

text: map

text: PB

text: c−o→f,we

text: Path(PB)

text: −f→ib

text: PB

text: PB

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE BOUSFIELD LOCALIZATIONS AND COLOCALIZATIONS OF THE DISCRETE MODEL STRUCTURE. 5 and we arrange this composable pair of maps as well as the given maps into the commuta- tive diagram: A f f (cid:122)(cid:122) (cid:15)(cid:15) (cid:36)(cid:36) B PA B g g 1 2 (cid:15)(cid:15) (cid:122)(cid:122) (cid:36)(cid:36) (cid:122)(cid:122) (cid:36)(cid:36) (cid:15)(cid:15) PB(cid:91)(cid:91) PB P(cid:67)(cid:67) B (cid:15)(cid:15) Path(PB) π1 π2 (cid:15)(cid:15) × PB PB and, redrawing a portion of the central axis of this commutative diagram, we have A cof,we (cid:47)(cid:47) PA (cid:15)(cid:15) (cid:15)(cid:15) fib (cid:47)(cid:47) × Path(PB) PB PB. → So there exists a lift PA Path(PB) of this acyclic coﬁbration over this ﬁbration. This lift is a right homotopy of g and g . (cid:3) 1 2 Proposition 2.3. Let C be a model category which is a Bousﬁeld localization of a discrete −→f model category. Then, given a map A B in C, there exists a unique map g making the following diagram commute: (2.0.1) A cof,we(cid:47)(cid:47) PA fib (cid:47)(cid:47) 1 f g (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) B cof,we(cid:47)(cid:47) PB fib (cid:47)(cid:47) 1, where maps labelled cof, we, fib are coﬁbrations, weak equivalences, and ﬁbrations in C, respectively. As a consequence, C admits functorial ﬁbrant replacement. More speciﬁ- cally, every choice of ﬁbrant replacements in C extends to a ﬁbrant replacement functor. Proof. Existence of the desired map is given trivially by the lifting property of an acyclic coﬁbration over a ﬁbration, and the commutative square A cof,we (cid:47)(cid:47) PA (cid:15)(cid:15) (cid:15)(cid:15) fib (cid:47)(cid:47) PB 1. → (cid:48) → The real question is why such a map PA PB is unique. Let g, g : PA PB be two (cid:48) maps making the above diagram commute. By Lemma 2.2, g and g are right-homotopic. (cid:48) By Lemma 2.1, g and g are hence left-homotopic (since they have ﬁbrant codomain), ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: BOUSFIELD

text: LOCALIZATIONS

text: AND

text: COLOCALIZATIONS

text: OF

text: THE

text: DISCRETE

text: MODEL

text: STRUCTURE.

text: and

text: we

text: arrange

text: this

text: composable

text: pair

text: of

text: maps

text: as

text: well

text: as

text: the

text: given

text: maps

text: into

text: the

text: commuta-

text: tive

text: diagram:

text: A

text: f

text: f

font: default

text: B

text: PA

text: B

text: g

text: g

text: PB

text: B

text: Path(PB)

text: π1

text: π2

text: PB

text: PB

text: and,

text: redrawing

text: a

text: portion

text: of

text: the

text: central

text: axis

text: of

text: this

text: commutative

text: diagram,

text: we

text: have

text: A

text: cof,we

text: PA

text: fib

text: Path(PB)

text: PB

text: PB.

text: So

text: there

text: exists

text: a

text: lift

text: PA

text: Path(PB)

text: of

text: this

text: acyclic

text: coﬁbration

text: over

text: this

text: ﬁbration.

text: This

text: lift

text: is

text: a

text: right

text: homotopy

text: of

text: g

text: and

text: g

text: Proposition

text: Let

text: C

text: be

text: a

text: model

text: category

text: which

text: is

text: a

text: Bousﬁeld

text: localization

text: of

text: a

text: discrete

text: −→f

text: model

text: category.

text: Then,

text: given

text: a

text: map

text: A

text: B

text: in

text: C,

text: there

text: exists

text: a

text: unique

text: map

text: g

text: making

text: the

text: following

text: diagram

text: commute:

text: A

text: PA

text: fib

text: f

text: g

text: B

text: PB

text: fib

text: where

text: maps

text: labelled

text: cof,

text: we,

text: fib

text: are

text: coﬁbrations,

text: weak

text: equivalences,

text: and

text: ﬁbrations

text: in

text: C,

text: respectively.

text: As

text: a

text: consequence,

text: C

text: admits

text: functorial

text: ﬁbrant

text: replacement.

text: More

text: speciﬁ-

text: cally,

text: every

text: choice

text: of

text: ﬁbrant

text: replacements

text: in

text: C

text: extends

text: to

text: a

text: ﬁbrant

text: replacement

text: functor.

text: Proof.

text: Existence

text: of

text: the

text: desired

text: map

text: is

text: given

text: trivially

text: by

text: the

text: lifting

text: property

text: of

text: an

text: acyclic

text: coﬁbration

text: over

text: a

text: ﬁbration,

text: and

text: the

text: commutative

text: square

text: A

text: cof,we

text: PA

text: fib

text: PB

text: The

text: real

text: question

text: is

text: why

text: such

text: a

text: map

text: PA

text: PB

text: is

text: unique.

text: Let

text: g,

text: g

text: PA

text: PB

text: be

text: two

text: maps

text: making

text: the

text: above

text: diagram

text: commute.

text: By

text: Lemma

text: g

text: and

text: g

text: are

text: right-homotopic.

text: By

text: Lemma

text: g

text: and

text: g

text: are

text: hence

text: left-homotopic

text: (since

text: they

text: have

text: ﬁbrant

text: codomain),

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: 12 S. RAMAWADH AND S. J. TATE (cid:18)(cid:90) (cid:19) We wish to show that the full weight (cid:89)of a tree also factorises as we require. The full weight of a tree is given by 1 ω˜(τ) = lim | | dx ω(τ). (6.8) Λ→Rd Λ ∈ Λ i i V (τ) Since we are assuming that the pair potentials are centred, the integrand de- pends only upon the distances between the particles. The thermodynamic limit is independent of boundary conditions and so we consider periodic boundary condi- tions. With periodic boundary conditions, we may make the change of variables from the spatial points to the diﬀerences. We may choose the change of variables · · · (cid:55)→ − · · · − to be (x , , x ) (x , x x , , x x ) and in this case all diﬀerences can 1 N 1 2 1 −N 1 be expressed in terms of the latter N 1 variables and so is independent of x 1 integral. The key point is the periodic boundary conditions, which allows for the change in coordinates to preserve the area over which we integrate. The factorisation of the integrals occurs analogously to the block factorisation M of connected graph weights as explained in [16]. The graph ((V ) ∈ ) acts in i i [k] the same way as the block cutpoint tree, where the vertices corresponding to trees bear the analogy with blocks and the vertices corresponding to the common label bear the analogy with cutpoints. First we give the deﬁnitions required to rigorously convey how to treat the factorisation of these integrals. Deﬁnition 7 (The Centre of a Tree). The eccentricity of a vertex v in a tree τ is: { | ∈ } ε(v) = max d(v, w) w V (τ) . { | ∈ } The radius of a tree is: radius(τ) = min ε(v) v V (τ) . { ∈ | } The centre of a tree is: v V (τ) ε(v) = radius(τ) . Remark 2. A priori the centre of a tree may be a set of cardinality strictly greater M than 1. We are working with a special class of trees for ((V ) ∈ ), which are i i [k] bipartite and importantly have all their leaves in a single set. This is suﬃcient to get a unique centre. The idea is that the vertex w for which the eccentricity of a vertex is attained is necessarily a leaf. If not then we always have a neighbour away from the original vertex which allows us to increase eccentricity by 1. So any intersection point has odd eccentricity and any tree vertex has even eccentricity and so the centre can contain only vertices from one of the two sets in the bipartite graph. The centre cannot contain two points of distance at least two from each other and hence can M only be a singleton, so the centre of ((V ) ∈ ) is well deﬁned as a single point. i i [k] When we want to understand the factorisation, we want a well-deﬁned way of deciding what label to take as the redundant integral in each of the tree factors. When we see how to do this systematically, we achieve the factorisation of the tree weights for the tilde generating function as well. In order to deﬁne the vertices over which we make our integrals independent, we deﬁne a digraph from the tree by orienting all edges away from the centre of the tree. In this case every vertex except the centre has a unique vertex pointing towards it. For each tree vertex outside of the centre, the intersection point neighbour that is the other endpoint of the edge pointing towards it is the vertex, which the integrand is taken independent of. Each intersection point outside of the centre ##LTLine##

text: S.

text: RAMAWADH

text: AND

text: S.

text: J.

text: TATE

text: We

text: wish

text: to

text: show

text: that

text: the

text: full

text: weight

text: a

text: tree

text: also

text: factorises

text: as

text: we

text: require.

text: The

text: full

text: weight

text: of

text: a

text: tree

text: is

text: given

text: by

text: ω˜(τ)

text: lim

text: dx

text: ω(τ).

text: Λ→Rd

text: Λ

text: Λ

text: i

text: i

text: V

text: (τ)

text: Since

text: we

text: are

text: assuming

text: that

text: the

text: pair

text: potentials

text: are

text: centred,

text: the

text: integrand

text: de-

text: pends

text: only

text: upon

text: the

text: distances

text: between

text: the

text: particles.

text: The

text: thermodynamic

text: limit

text: is

text: independent

text: of

text: boundary

text: conditions

text: and

text: so

text: we

text: consider

text: periodic

text: boundary

text: condi-

text: tions.

text: With

text: periodic

text: boundary

text: conditions,

text: we

text: may

text: make

text: the

text: change

text: of

text: variables

text: from

text: the

text: spatial

text: points

text: to

text: the

text: diﬀerences.

text: We

text: may

text: choose

text: the

text: change

text: of

text: variables

text: to

text: be

text: (x

text: x

text: (x

text: x

text: x

text: x

text: x

text: and

text: in

text: this

text: case

text: all

text: diﬀerences

text: can

text: N

text: −N

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: latter

text: N

text: variables

text: and

text: so

text: is

text: independent

text: of

text: x

text: integral.

text: The

text: key

text: point

text: is

text: the

text: periodic

text: boundary

text: conditions,

text: which

text: allows

text: for

text: the

text: change

text: in

text: coordinates

text: to

text: preserve

text: the

text: area

text: over

text: which

text: we

text: integrate.

text: The

text: factorisation

text: of

text: the

text: integrals

text: occurs

text: analogously

text: to

text: the

text: block

text: factorisation

text: M

text: of

text: connected

text: graph

text: weights

text: as

text: explained

text: in

text: The

text: graph

text: ((V

text: acts

text: in

text: i

text: i

text: [k]

text: the

text: same

text: way

text: as

text: the

text: block

text: cutpoint

text: tree,

text: where

text: the

text: vertices

text: corresponding

text: to

text: trees

text: bear

text: the

text: analogy

text: with

text: blocks

text: and

text: the

text: vertices

text: corresponding

text: to

text: the

text: common

text: label

text: bear

text: the

text: analogy

text: with

text: cutpoints.

text: First

text: we

text: give

text: the

text: deﬁnitions

text: required

text: to

text: rigorously

text: convey

text: how

text: to

text: treat

text: the

text: factorisation

text: of

text: these

text: integrals.

text: Deﬁnition

text: (The

text: Centre

text: of

text: a

text: Tree).

text: The

text: eccentricity

text: of

text: a

text: vertex

text: v

text: in

text: a

text: tree

text: τ

text: is:

text: ε(v)

text: max

text: d(v,

text: w)

text: w

text: V

text: (τ)

text: The

text: radius

text: of

text: a

text: tree

text: is:

text: radius(τ)

text: min

text: ε(v)

text: v

text: V

text: (τ)

text: The

text: centre

text: of

text: a

text: tree

text: is:

text: v

text: V

text: (τ)

text: ε(v)

text: radius(τ)

text: Remark

text: A

text: priori

text: the

text: centre

text: of

text: a

text: tree

text: may

text: be

text: a

text: set

text: of

text: cardinality

text: strictly

text: greater

text: M

text: than

text: We

text: are

text: working

text: with

text: a

text: special

text: class

text: of

text: trees

text: for

text: ((V

text: which

text: are

text: i

text: i

text: [k]

text: bipartite

text: and

text: importantly

text: have

text: all

text: their

text: leaves

text: in

text: a

text: single

text: set.

text: This

text: is

text: suﬃcient

text: to

text: get

text: a

text: unique

text: centre.

text: The

text: idea

text: is

text: that

text: the

text: vertex

text: w

text: for

text: which

text: the

text: eccentricity

text: of

text: a

text: vertex

text: is

text: attained

text: is

text: necessarily

text: a

text: leaf.

text: If

text: not

text: then

text: we

text: always

text: have

text: a

text: neighbour

text: away

text: from

text: the

text: original

text: vertex

text: which

text: allows

text: us

text: to

text: increase

text: eccentricity

text: by

text: So

text: any

text: intersection

text: point

text: has

text: odd

text: eccentricity

text: and

text: any

text: tree

text: vertex

text: has

text: even

text: eccentricity

text: and

text: so

text: the

text: centre

text: can

text: contain

text: only

text: vertices

text: from

text: one

text: of

text: the

text: two

text: sets

text: in

text: the

text: bipartite

text: graph.

text: The

text: centre

text: cannot

text: contain

text: two

text: points

text: of

text: distance

text: at

text: least

text: two

text: from

text: each

text: other

text: and

text: hence

text: can

text: M

text: only

text: be

text: a

text: singleton,

text: so

text: the

text: centre

text: of

text: ((V

text: is

text: well

text: deﬁned

text: as

text: a

text: single

text: point.

text: i

text: i

text: [k]

text: When

text: we

text: want

text: to

text: understand

text: the

text: factorisation,

text: we

text: want

text: a

text: well-deﬁned

text: way

text: of

text: deciding

text: what

text: label

text: to

text: take

text: as

text: the

text: redundant

text: integral

text: in

text: each

text: of

text: the

text: tree

text: factors.

text: When

text: we

text: see

text: how

text: to

text: do

text: this

text: systematically,

text: we

text: achieve

text: the

text: factorisation

text: of

text: the

text: tree

text: weights

text: for

text: the

text: tilde

text: generating

text: function

text: as

text: well.

text: In

text: order

text: to

text: deﬁne

text: the

text: vertices

text: over

text: which

text: we

text: make

text: our

text: integrals

text: independent,

text: we

text: deﬁne

text: a

text: digraph

text: from

text: the

text: tree

text: by

text: orienting

text: all

text: edges

text: away

text: from

text: the

text: centre

text: of

text: the

text: tree.

text: In

text: this

text: case

text: every

text: vertex

text: except

text: the

text: centre

text: has

text: a

text: unique

text: vertex

text: pointing

text: towards

text: it.

text: For

text: each

text: tree

text: vertex

text: outside

text: of

text: the

text: centre,

text: the

text: intersection

text: point

text: neighbour

text: that

text: is

text: the

text: other

text: endpoint

text: of

text: the

text: edge

text: pointing

text: towards

text: it

text: is

text: the

text: vertex,

text: which

text: the

text: integrand

text: is

text: taken

text: independent

text: of.

text: Each

text: intersection

text: point

text: outside

text: of

text: the

text: centre

text: ##LTLine##


text: VIRIAL EXPANSION BOUNDS THROUGH TREE PARTITION SCHEMES 13 has also precisely one incoming edge which marks the tree for which this point is integrated over. We now consider what happens at the centre. If the centre is an intersection point, then this is understood as the overall point which is not integrated over. If the centre is a tree point, then we may choose any of its neighbouring intersection points to be the independent point and we are left with this single independent point. Hence we have that the tilde weights factorise. (cid:18) (cid:19) (cid:18) (cid:19) (cid:88) (cid:88) (cid:89) By combining equations (6.1), (6.2), and(cid:81)(6.7) we ﬁnd: − n n n! n b ki β = k! i+1 n+1 (cid:88) (cid:18) k (cid:19) (cid:80) (cid:88)(cid:80) n k ! (cid:89)i! k=1 (k1,...,kn):ki≥0 (cid:81)i=1 i i=1 i ki=k, i iki=n − n n n n! = k! bki k(cid:88)=1 (cid:18) k (cid:19) (cid:80)(ik1k,i.=..(cid:88)k,k,(cid:80)n)i:kikii≥=0n (cid:81)ni=1 ki!(i!)ki i=1(cid:88)i+1 (cid:89) − n k n n! = k! ω˜(τ ), k=1 k (cid:80)(k1,...,k(cid:80)n):ki≥0 ni=1 ki!(i!)ki (τ1,...,τk) i=1 i i ki=k, i iki=n where the sum is over τ , which are spanning trees for the corresponding b in the i i product on the previous line. The next point is to write this product of the weights of tree graphs as the weight of an R-faithful merging. This involves assigning labels to the in(cid:88)div(cid:18)idual(cid:19)trees as (cid:88)described above. We then(cid:88)express it as: (cid:88) (cid:81) − n n n! 1 β = k! · · · ω˜(τ), n+1 k=1 k (cid:80)(k1,...,k(cid:80)n):ki≥0 ni=1 ki!(i!)ki (τ1,···,τk) N(τ1, , τk) τ∈T [n+1] i ki=k, i iki=n (6.9) · · · · · · where in this case τ is some R-faithful merging of (τ , , τ ) and N(τ , , τ ) 1 k 1 k is the number of such R-faithful mergings. T We want to express the latter sum as a sum on [n + 1] and drop the condition that the tree graph is an R-faithful merging of a given set of k smaller tree graphs. ∈ T Note that this can be done, since any τ [n + 1] can be made its own R-faithful merging by choosing a suitable labeling on its vertices. However, if a tree graph { } { } τ has a splitting τ , . . . , τ , then it also has splittings τ˜ , . . . , τ˜ with j < k 1 k 1 j and with the τ˜ being mergings of distinct τ , meaning that some trees will appear i j multiple times in the last sum in the right-hand side. We can overcome this problem by rewriting this sum as follows. We call a tree graph τ l-splittable if τ has a splitting into l parts, but not into l + 1 parts. The number of l-splittable tree graphs is a combinatorial factor which depends on the partition scheme being used. Combining this factor with the other combinatorial factors yields: (cid:88) (cid:88) n β = C(n + 1, l) ω˜(τ). (6.10) n+1 ∈T l=1 τ [n+1]: τ is l-splittable ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VIRIAL

text: EXPANSION

text: BOUNDS

text: THROUGH

text: TREE

text: PARTITION

text: SCHEMES

text: has

text: also

text: precisely

text: one

text: incoming

text: edge

text: which

text: marks

text: the

text: tree

text: for

text: which

text: this

text: point

text: is

text: integrated

text: over.

text: We

text: now

text: consider

text: what

text: happens

text: at

text: the

text: centre.

text: If

text: the

text: centre

text: is

text: an

text: intersection

text: point,

text: then

text: this

text: is

text: understood

text: as

text: the

text: overall

text: point

text: which

text: is

text: not

text: integrated

text: over.

text: If

text: the

text: centre

text: is

text: a

text: tree

text: point,

text: then

text: we

text: may

text: choose

text: any

text: of

text: its

text: neighbouring

text: intersection

text: points

text: to

text: be

text: the

text: independent

text: point

text: and

text: we

text: are

text: left

text: with

text: this

text: single

text: independent

text: point.

text: Hence

text: we

text: have

text: that

text: the

text: tilde

text: weights

text: factorise.

text: By

text: combining

text: equations

text: we

text: ﬁnd:

text: n

text: n

text: n!

text: n

text: b

text: ki

text: β

text: k!

text: k

text: n

text: k

text: i

text: i

text: ki=k,

text: i

text: iki=n

text: n

text: n

text: n

text: n!

text: k!

text: bki

text: k

text: ki!(i!)ki

text: n

text: k

text: n

text: n!

text: k!

text: ω˜(τ

text: k

text: ki!(i!)ki

text: i

text: i

text: ki=k,

text: i

text: iki=n

text: where

text: the

text: sum

text: is

text: over

text: τ

text: which

text: are

text: spanning

text: trees

text: for

text: the

text: corresponding

text: b

text: in

text: the

text: i

text: i

text: product

text: on

text: the

text: previous

text: line.

text: The

text: next

text: point

text: is

text: to

text: write

text: this

text: product

text: of

text: the

text: weights

text: of

text: tree

text: graphs

text: as

text: the

text: weight

text: of

text: an

text: R-faithful

text: merging.

text: This

text: involves

text: assigning

text: labels

text: to

text: the

text: as

text: above.

text: We

text: it

text: as:

text: n

text: n

text: n!

text: β

text: k!

text: ω˜(τ),

text: k

text: ki!(i!)ki

text: τk)

text: τ∈T

text: i

text: ki=k,

text: i

text: iki=n

text: where

text: in

text: this

text: case

text: τ

text: is

text: some

text: R-faithful

text: merging

text: of

text: (τ

text: τ

text: and

text: N(τ

text: τ

text: k

text: k

text: is

text: the

text: number

text: of

text: such

text: R-faithful

text: mergings.

text: T

text: We

text: want

text: to

text: express

text: the

text: latter

text: sum

text: as

text: a

text: sum

text: on

text: [n

text: and

text: drop

text: the

text: condition

text: that

text: the

text: tree

text: graph

text: is

text: an

text: R-faithful

text: merging

text: of

text: a

text: given

text: set

text: of

text: k

text: smaller

text: tree

text: graphs.

text: T

text: Note

text: that

text: this

text: can

text: be

text: done,

text: since

text: any

text: τ

text: [n

text: can

text: be

text: made

text: its

text: own

text: R-faithful

text: merging

text: by

text: choosing

text: a

text: suitable

text: labeling

text: on

text: its

text: vertices.

text: However,

text: if

text: a

text: tree

text: graph

text: τ

text: has

text: a

text: splitting

text: τ

text: τ

text: then

text: it

text: also

text: has

text: splittings

text: τ˜

text: τ˜

text: with

text: j

text: k

text: k

text: j

text: and

text: with

text: the

text: τ˜

text: being

text: mergings

text: of

text: distinct

text: τ

text: meaning

text: that

text: some

text: trees

text: will

text: appear

text: i

text: j

text: multiple

text: times

text: in

text: the

text: last

text: sum

text: in

text: the

text: right-hand

text: side.

text: We

text: can

text: overcome

text: this

text: problem

text: by

text: rewriting

text: this

text: sum

text: as

text: follows.

text: We

text: call

text: a

text: tree

text: graph

text: τ

text: l-splittable

text: if

text: τ

text: has

text: a

text: splitting

text: into

text: l

text: parts,

text: but

text: not

text: into

text: l

text: parts.

text: The

text: number

text: of

text: l-splittable

text: tree

text: graphs

text: is

text: a

text: combinatorial

text: factor

text: which

text: depends

text: on

text: the

text: partition

text: scheme

text: being

text: used.

text: Combining

text: this

text: factor

text: with

text: the

text: other

text: combinatorial

text: factors

text: yields:

text: n

text: β

text: C(n

text: l)

text: ω˜(τ).

text: ∈T

text: τ

text: τ

text: is

text: l-splittable

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: VIRIAL EXPANSION BOUNDS THROUGH TREE PARTITION SCHEMES 5 It is possible to relate the virial coeﬃcients β to the well understood cluster n coeﬃcients b . This is done by inverting (2.11), so as to obtain an expression for n the fugacity as a power series in terms of the density, and then substituting this (cid:34)(cid:18) (cid:19) (cid:35) power series into (2.9). Lagrange inversion [6] gives the following: − ρ(z) n β = n![zn] , (2.17) n+1 z Using this expression, we can ﬁnd a nice combinatorial expression of the β in n terms of the b . n ∈ R Deﬁne, for any x and any nonnegative integer(cid:89)k, the falling factorial as: − − · · · − k 1 − (x)k := x(x 1) (x k + 1) = (x i). (cid:0) (cid:1)(2.18) i=0 If x is a positive integer, we may also write (x)k = −x! and (x)k = x . We (x k)! k! k ∈ R use this last equality to extend th(cid:18)e d(cid:19)eﬁnition of binomial coeﬃcients. For x and any nonnegative integer k, we deﬁne x (x)k := . (2.19) k k! Let x , x , . . . be an inﬁnite number of variables, then the partial Bell polynomials 1 2(cid:32) (cid:33) (cid:32) (cid:33) B (x , x , . . .(cid:88), x ) are deﬁned by(cid:88): (cid:88) n,k 1 2 n ∞ ∞ tm tn n exp u x = 1 + ukB (x , . . . , x ) . (2.20) m m! n! n,k 1 n m=1 n=1 k=1 There exists an explicit expression for(cid:88)the partial Bell p(cid:89)olyn(cid:16)om(cid:17)ials: (cid:81) n B (x , x , . . . , x ) = n! xi ki . (2.21) n,k 1 2 n (cid:80) (cid:80) n k ! i! (k1,...,kn):ki≥0 i=1 i i=1 i ki=k, i iki=n For more known results on Bell polynomials we refer the reader to [6]. Lastly, we deﬁne the potential polynomials. Let x = 1, let x , x , . . . be an inﬁnite sequence 0 1 2 of numbers and let r be any number. The potential polynomials P(r) (x , . . . , x ) (cid:32) (cid:33) n 1 n are deﬁned by: (cid:88) (cid:88) ∞ r ∞ tn tn x = 1 + P(r) (x , . . . , x ) . (2.22) n n! n 1 n n! n=0 n=1 There is also the following explicit e(cid:88)xpression for the potential polynomials: n P(r) (x , . . . , x ) =(cid:80) (r)kB (x , . . (cid:80). , x ) . (2.23) n 1 n n,k 1 n k=1 ∞ − ∞ We return to (2.17). Since ρ(zz) = n=1 (nb−n1)!zn 1 = n=0 bn+1 znn! and b1 = 1, it follows from (2.22) and (2.23) that: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VIRIAL

text: EXPANSION

text: BOUNDS

text: THROUGH

text: TREE

text: PARTITION

text: SCHEMES

text: It

text: is

text: possible

text: to

text: relate

text: the

text: virial

text: coeﬃcients

text: β

text: to

text: the

text: well

text: understood

text: cluster

text: n

text: coeﬃcients

text: b

text: This

text: is

text: done

text: by

text: inverting

text: so

text: as

text: to

text: obtain

text: an

text: expression

text: for

text: n

text: the

text: fugacity

text: as

text: a

text: power

text: series

text: in

text: terms

text: of

text: the

text: density,

text: and

text: then

text: substituting

text: this

text: power

text: series

text: into

text: Lagrange

text: inversion

text: gives

text: the

text: following:

text: ρ(z)

text: n

text: β

text: n![zn]

text: z

text: Using

text: this

text: expression,

text: we

text: can

text: ﬁnd

text: a

text: nice

text: combinatorial

text: expression

text: of

text: the

text: β

text: in

text: n

text: terms

text: of

text: the

text: b

text: n

text: R

text: Deﬁne,

text: for

text: any

text: x

text: and

text: any

text: nonnegative

text: the

text: falling

text: factorial

text: as:

text: k

text: (x)k

text: x(x

text: (x

text: k

text: (x

text: i).

text: If

text: x

text: is

text: a

text: positive

text: integer,

text: we

text: may

text: also

text: write

text: (x)k

text: −x!

text: and

text: (x)k

text: x

text: We

text: (x

text: k)!

text: k!

text: k

text: R

text: use

text: this

text: last

text: equality

text: to

text: extend

text: of

text: binomial

text: coeﬃcients.

text: For

text: x

text: and

text: any

text: nonnegative

text: integer

text: k,

text: we

text: deﬁne

text: x

text: (x)k

text: k

text: k!

text: Let

text: x

text: x

text: be

text: an

text: inﬁnite

text: number

text: of

text: variables,

text: then

text: the

text: partial

text: Bell

text: polynomials

text: B

text: (x

text: x

text: x

text: are

text: deﬁned

text: n,k

text: n

text: tm

text: tn

text: n

text: exp

text: u

text: x

text: ukB

text: (x

text: x

text: m

text: m!

text: n!

text: n,k

text: n

text: There

text: exists

text: an

text: explicit

text: expression

text: partial

text: Bell

text: n

text: B

text: (x

text: x

text: x

text: n!

text: xi

text: ki

text: n,k

text: n

text: n

text: k

text: i!

text: i

text: i

text: ki=k,

text: i

text: iki=n

text: For

text: more

text: known

text: results

text: on

text: Bell

text: polynomials

text: we

text: refer

text: the

text: reader

text: to

text: Lastly,

text: we

text: deﬁne

text: the

text: potential

text: polynomials.

text: Let

text: x

text: let

text: x

text: x

text: be

text: an

text: inﬁnite

text: sequence

text: of

text: numbers

text: and

text: let

text: r

text: be

text: any

text: number.

text: The

text: potential

text: polynomials

text: P(r)

text: (x

text: x

text: n

text: n

text: are

text: deﬁned

text: by:

text: r

text: tn

text: tn

text: x

text: P(r)

text: (x

text: x

text: n

text: n!

text: n

text: n

text: n!

text: There

text: is

text: also

text: the

text: following

text: explicit

text: for

text: the

text: potential

text: polynomials:

text: n

text: P(r)

text: (x

text: x

text: (r)kB

text: (x

text: x

text: n

text: n

text: n,k

text: n

text: We

text: return

text: to

text: Since

text: ρ(zz)

text: znn!

text: and

text: it

text: follows

text: from

text: and

text: that:

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##

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text: ##LTLine##