Crowds in equations an introduction to the microscopic modeling of crowds 1st edition bertrand maury
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Advanced Textbooks in Mathematics
Print ISSN: 2059-769X
Online ISSN: 2059-7703
The Advanced Textbooks in Mathematics explores important topics for postgraduate students in pure and applied mathematics. Subjects covered within this textbook series cover key fields which appear on MSc, MRes, PhD and other multidisciplinary postgraduate courses which involve mathematics.
Written by senior academics and lecturers recognised for their teaching skills, these textbooks offer a precise, introductory approach to advanced mathematical theories and concepts, including probability theory, statistics and computational methods.
Published
Crowds in Equations: An Introduction to the Microscopic Modeling of Crowds by
Bertrand Maury and Sylvain Faure
The Wigner Transform by Maurice de Gosson
Periods and Special Functions in Transcendence by Paula B Tretkoff
Mathematics of Planet Earth: A Primer by Jochen Bröcker, Ben Calderhead, Davoud Cheraghi, Colin Cotter, Darryl Holm, Tobias Kuna, Beatrice Pelloni, Ted Shepherd and Hilary Weller edited by Dan Crisan
Forthcoming
Conformal Maps and Geometry by Dmitry Beliaev
Crowds in Equations
An Introduction to the Microscopic Modeling of Crowds
Bertrand Maury
Université Paris-Sud, France
Sylvain Faure
Université Paris-Sud & CNRS, France
Published by
World Scientific Publishing Europe Ltd.
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Library of Congress Cataloging-in-Publication Data
Title: Crowds in equations : an introduction to the microscopic modeling of crowds / by Bertrand Maury (Université Paris-Sud, France), Sylvain Faure (Université Paris-Sud & CNRS, France).
Description: New Jersey : World Scientific, 2018. | Series: Advanced textbooks in mathematics | Includes bibliographical references and index.
Identifiers: LCCN 2018013165 | ISBN 9781786345516 (hc : alk. paper)
Subjects: LCSH: Mathematical analysis. | Mathematical models. | Communication in mathematics.
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Chapter10isthemostexoticone,itcontainsfreemathematicaldevelopmentstoinvestigatethepossiblecauses(intheabstractsenseof model features)ofsomeobservedphenomenaincrowdmotions.
Weconsider N +1individualswalkingonastraightline.Theirrespective positionsaredenotedby
Themodelisbasedontheassumptionthattheinstantaneousvelocityof i dependsupon xi+1 xi only.
Model2.1(FTLmodel). Let ϕ : R+ → R+ beafunctionwhichassigns aspeedtoanynon-negativedistance.Themodelsread:
Weshalldesignateby linear thismodelwhenthespeedofindividual N +1 isprescribed,and periodic thecasewhere N +1isidentifiedto1(inwhich casethelengthoftheperiodicpathhastobespecified).Thesystemisthen autonomousinthelattersituation(noexplicitdependenceupontime),and non-autonomousinthelinearcase(because xN +1 (t)hastobeprescribed inthelattersituation).
Denotingby wm > 0(m for“minimal”)theinter-individualdistance whichcorrespondstofullpacking(peopleareincontact),itisnaturalto assumethat w → ϕ(w )vanishesat wm ,increaseswith w ,andtendstoa maximalvalue U when w becomeslarge.
Anexampleofsuchafunctionis
ϕ(w )= U (1 exp( (w wm )/ws ))for w ≥ wm , (2.3) and ϕ(w )=0otherwise,where ws isatypicaldistancebelowwhichthe modificationofthevelocityissignificant(seeFig.2.1),and U isthedesired speed.Typicalvaluesare U =1.25ms 1 , wm =0.3m, ws =0.9m(see Section9.4andFig.9.2therein).
presentcontext,thetermissomewhat improper,sincethemodelisrather basedonindividualswhohaveapersonalobjective(thatismovingahead atvelocity U ),andwhodecidetoreducetheirspeedtokeepareasonabledistancefromthepersoninfrontofthem,inordertoavoidcollision. Fromthisstandpointeachagentisnotreally led bytheagentahead,but rather disturbed or inhibited.
Underreasonableassumptionsonthebehaviorfunction ϕ,itcanbe provedthattheproblemiswell-posedinthelinearandperiodicsituations, whichmeansthat,forgiveninitialconditions(andprescribedvelocityof N +1inthelinearcase),existenceofa uniquesolutioncanbeestablished. Tobemoreprecise,ifthederivativeof ϕ at wm isnotinfinite,alldistances remainpositiveforalltimes(noaccident).Weputoffuntiltheendofthis sectionarigorousaccountofthoseproperties.
Inordertoinvestigatethestationarystatesofthesystem,itisconvenienttoexpressitintermsofdistances(whichcanbeexpectedtobeconstant,whereaspositionsalwaysmoveforward).Weintroduce wi = xi+1 xi and,sinceweshallbeinterestedinthesituationwherethevelocityof N +1 isconstantandequalto Ueq ,wedefine wN +1 as weq .Thesystemnowwrites
Equilibriumpoint,andstability
Let ϕ bedefinedby(2.3).Inthelinearsetting,weprescribethatthespeed of N +1isaconstant,equalto Ueq ∈]0,U [.Thesystemthenadmitsan
Proposition2.2. Weconsider N +1 individuals, initiallylocatedat x
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of Spenser's Faerie Queene, Vol. 2 (of 3)
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Original publication: Oxford: Clarendon Press, 1909
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The previous volume is available as Project Gutenberg ebook #70717
THE POETICAL WORKS OF EDMUND SPENSER
IN THREE VOLUMES
VOLUME III
SPENSER’S FAERIE QUEENE
EDITED BY
J. C. SMITH
VOLUME II: BOOKS IV-VII
OXFORD
AT THE CLARENDON PRESS
Oxford University Press, Amen House, London E.C.4
GLASGOW NEW
YORK TORONTO MELBOURNE WELLINGTON
BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA
CAPE TOWN SALISBURY NAIROBI IBADAN ACCRA
KUALA LUMPUR HONG KONG
FIRST PUBLISHED 1909
REPRINTED LITHOGRAPHICALLY IN GREAT BRITAIN AT THE UNIVERSITY PRESS, OXFORD FROM SHEETS OF THE FIRST IMPRESSION 1961, 1964
THE SECOND PART OF THE FAERIE QVEENE.
Containing
T F , F ,
S B .
By Ed. Spenser.
Imprinted at London for VVilliam Ponsonby. 1596.
THE FOVRTH BOOKE OF THE FAERIE QVEENE.
Containing
The Legend of C and T [1],
OR
OF FRIENDSHIP.
he rugged forhead that with graue foresight i Welds[2] kingdomes causes, and affaires of state, My looser rimes (I wote) doth sharply wite, For praising loue, as I haue done of late, And magnifying louers deare debate; By which fraile youth is oft to follie led, Through false allurement of that pleasing baite, That better were in vertues discipled, Then with vaine poemes weeds to haue their fancies fed.
Such ones ill iudge of loue, that cannot loue, ii Ne in their frosen hearts feele kindly flame: For thy they ought not thing vnknowne reproue, Ne naturall affection faultlesse blame,
For fault of few that haue abusd the same. For it of honor and all vertue is The roote, and brings forth glorious flowres of fame, That crowne true louers with immortall blis, The meed of them that loue, and do not liue amisse.
Which who so list looke backe to former ages, iii And call to count the things that then were donne, Shall find, that all the workes of those wise sages, And braue exploits which great Heroes wonne, In loue were either ended or begunne: Witnesse the father of Philosophie, Which to his Critias, shaded oft from sunne, Of loue full manie lessons did apply, The which these Stoicke censours cannot well deny.
To such therefore I do not sing at all, iv But to that sacred Saint my soueraigne Queene, In whose chast[3] breast all bountie naturall, And treasures of true loue enlocked beene, Boue all her sexe that euer yet was seene; To her I sing of loue, that loueth best, And best is lou’d of all aliue I weene:
To her this song most fitly is addrest, The Queene of loue, and Prince of peace from heauen blest.
Which that she may the better deigne to heare, v Do thou dred[4] infant, Venus dearling doue, From her high spirit chase imperious feare, And vse of awfull Maiestie remoue: In sted thereof[5] with drops of melting loue, Deawd with ambrosiall kisses, by thee gotten From thy sweete smyling mother from aboue, Sprinckle her heart, and haughtie courage soften, That she may hearke to loue, and reade this lesson often.
Twixt Scudamour and Blandamour: Their fight and warlike deedes.
Of louers sad calamities of old, i
Full many piteous stories doe remaine, But none more piteous euer was ytold, Then that of Amorets hart-binding chaine, And this of Florimels vnworthie paine: The deare compassion of whose bitter fit My softened heart so sorely doth constraine, That I with teares full oft doe pittie it, And oftentimes doe wish it neuer had bene writ.
For from the time that Scudamour her bought ii In perilous fight, she neuer ioyed day, A perilous fight when he with force her brought From twentie Knights, that did him all assay: Yet fairely well he did them all dismay: And with great glorie both the shield of loue, And eke the Ladie selfe he brought away, Whom hauing wedded as did him behoue,
A new vnknowen mischiefe did from him remoue.
For that same vile Enchauntour Busyran, iii
The very selfe same day that she was wedded, Amidst the bridale feast, whilest euery man
Surcharg’d with wine, were heedlesse and ill hedded, All bent to mirth before the bride was bedded, Brought in that mask of loue which late was showen: And there the Ladie ill of friends bestedded, By way of sport, as oft in maskes is knowen, Conueyed quite away to liuing wight vnknowen.
Seuen moneths he so her kept in bitter smart, iv Because his sinfull lust she would not serue, Vntill such time as noble Britomart Released her, that else was like to sterue, Through cruell knife that her deare heart did kerue. And now she is with her vpon the way, Marching in louely wise, that could deserue No spot of blame, though spite did oft assay To blot her with dishonor of so faire a pray.
Yet should it be a pleasant tale, to tell v The diuerse vsage and demeanure daint, That each to other made, as oft befell.
For Amoret right fearefull was and faint, Lest she with blame her honor should attaint, That euerie word did tremble as she spake, And euerie looke was coy, and wondrous quaint, And euerie limbe that touched her did quake: Yet could she not but curteous countenance to her make.
For well she wist, as true it was indeed, vi That her liues Lord and patrone of her health
Right well deserued as his duefull meed, Her loue, her seruice, and her vtmost wealth. All is his iustly, that all freely dealth: Nathlesse her honor dearer then her life,
She sought to saue, as thing reseru’d from stealth; Die had she leuer with Enchanters knife, Then to be false in loue, profest a virgine wife.
Thereto her feare was made so much the greater vii Through fine abusion of that Briton mayd: Who for to hide her fained sex the better, And maske her wounded mind, both did and sayd Full many things so doubtfull to be wayd, That well she wist not what by them to gesse[6] , For other whiles to her she purpos made Of loue, and otherwhiles of lustfulnesse, That much she feard his mind would grow to some excesse.
His will she feard; for him she surely thought viii To be a man, such as indeed he seemed, And much the more, by that he lately wrought, When her from deadly thraldome he redeemed, For which no seruice she too much esteemed, Yet dread of shame, and doubt of fowle dishonor Made her not yeeld so much, as due she deemed. Yet Britomart attended duly on her, As well became a knight, and did to her all honor.
It so befell one euening, that they came ix Vnto a Castell, lodged there to bee, Where many a knight, and many a louely Dame Was then assembled, deeds of armes to see: Amongst all which was none more faire then shee, That many of them mou’d to eye her sore. The custome of that place was such, that hee Which had no loue nor lemman there in store, Should either winne him one, or lye without the dore.
Amongst the rest there was a iolly knight, x Who being asked for his loue, auow’d That fairest Amoret was his by right,
And offred that to iustifie alowd. The warlike virgine seeing his so prowd And boastfull chalenge, wexed inlie wroth, But for the present did her anger shrowd; And sayd, her loue to lose she was full loth, But either he should neither of them haue, or both.
So foorth they went, and both together giusted; xi But that same younker soone was ouerthrowne, And made repent, that he had rashly lusted For thing vnlawfull, that was not his owne: Yet since[7] he seemed valiant, though vnknowne, She that no lesse was courteous then[8] stout, Cast how to salue, that both the custome showne Were kept, and yet that Knight not locked out, That seem’d full hard t’accord two things so far in dout.
The Seneschall was cal’d to deeme the right, xii Whom she requir’d, that first fayre Amoret Might be to her allow’d, as to a Knight, That did her win and free from chalenge set: Which straight to her was yeelded without let. Then since that strange Knights loue from him was quitted, She claim’d that to her selfe, as Ladies det, He as a Knight might iustly be admitted; So none should be out shut, sith all of loues were fitted.
With that her glistring helmet she vnlaced; xiii Which doft, her golden lockes, that were vp bound Still in a knot, vnto her heeles downe traced, And like a silken veile in compasse round About her backe and all her bodie wound: Like as the shining skie in summers night, What time the dayes with scorching heat abound, Is creasted all with lines of firie light, That it prodigious seemes in common peoples sight.
Such when those Knights and Ladies all about xiv
Beheld her, all were with amazement smit, And euery one gan grow in secret dout
Of this and that, according to each wit: Some thought that some enchantment faygned it; Some, that Bellona in that warlike wise
To them appear’d, with shield and armour fit; Some, that it was a maske of strange disguise: So diuersely each one did sundrie doubts deuise.
But that young Knight, which through her gentle deed xv Was to that goodly fellowship restor’d, Ten thousand thankes did yeeld her for her meed, And doubly ouercommen, her ador’d: So did they all their former strife accord; And eke fayre Amoret now freed from feare, More franke affection did to her afford, And to her bed, which she was wont forbeare, Now freely drew, and found right safe assurance theare.
Where all that night they of their loues did treat, xvi
And hard aduentures twixt themselues alone, That each the other gan with passion great, And griefull[9] pittie priuately bemone.
The morow next so soone as Titan shone, They both vprose, and to their waies them dight: Long wandred they, yet neuer met with none[10] , That to their willes could them direct aright, Or to them tydings tell, that mote their harts delight.
Lo thus they rode, till at the last they spide xvii
Two armed Knights, that toward them did pace, And ech of them had ryding by his side
A Ladie, seeming in so farre a space, But Ladies none they were, albee in face And outward shew faire semblance they did beare; For vnder maske of beautie and good grace,
