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This book focuses on the relationship between the state and economy
in the development of cities. It reviews and reinterprets
fundamental theoretical models that explain how the operation of
markets in equilibrium shapes the scale and organization of the
commercial city in a mixed market economy within a liberal state.
These models link markets for the factors of production, markets
for investment and fixed capital formation, markets for
transportation, and markets for exports in equilibrium both within
the urban economy and the rest of the world. In each case, the
model explains the urban economy by revealing how assumptions about
causes and structures lead to predictions about scale and
organization outcomes. By simplifying and contrasting these models,
this book proposes another interpretation: that governance and the
urban economy are outcomes negotiated by political actors motivated
by competing notions of commonwealth and the individual desire for
wealth and power. The book grounds its analysis in economic
history, explaining the rise of commercial cities and the emergence
of the urban economy. It then turns to factors of production,
export, and factor markets, introducing and parsing the Mills
model, breaking it down into its component parts and creating a
series of simpler models that can better explain the significance
of each economic assumption. Simplified models are also presented
for real estate and fixed capital investment markets,
transportation, and land use planning. The book concludes with a
discussion of linear programming and the Herbert- Stevens and the
Ripper-Varaiya models. A fresh presentation of the theories behind
urban economics, this book emphasizes the links between state and
economy and challenges the reader to see its theories in a new
light. As such, this book will be of interest to scholars,
students, and practitioners of economics, public policy, public
administration, urban policy, and city and urban planning. >
This book focuses on the relationship between the state and economy
in the development of cities. It reviews and reinterprets
fundamental theoretical models that explain how the operation of
markets in equilibrium shapes the scale and organization of the
commercial city in a mixed market economy within a liberal state.
These models link markets for the factors of production, markets
for investment and fixed capital formation, markets for
transportation, and markets for exports in equilibrium both within
the urban economy and the rest of the world. In each case, the
model explains the urban economy by revealing how assumptions about
causes and structures lead to predictions about scale and
organization outcomes. By simplifying and contrasting these models,
this book proposes another interpretation: that governance and the
urban economy are outcomes negotiated by political actors motivated
by competing notions of commonwealth and the individual desire for
wealth and power. The book grounds its analysis in economic
history, explaining the rise of commercial cities and the emergence
of the urban economy. It then turns to factors of production,
export, and factor markets, introducing and parsing the Mills
model, breaking it down into its component parts and creating a
series of simpler models that can better explain the significance
of each economic assumption. Simplified models are also presented
for real estate and fixed capital investment markets,
transportation, and land use planning. The book concludes with a
discussion of linear programming and the Herbert- Stevens and the
Ripper-Varaiya models. A fresh presentation of the theories behind
urban economics, this book emphasizes the links between state and
economy and challenges the reader to see its theories in a new
light. As such, this book will be of interest to scholars,
students, and practitioners of economics, public policy, public
administration, urban policy, and city and urban planning. >
This book provides a comprehensive, up-to-date, and expert
synthesis of location theory. What are the impacts of a firm's
geographic location on the locations of customers, suppliers, and
competitors in a market economy? How, when, and why does this
result in the clustering of firms in space? When and how is society
made better or worse off as a result? This book uses dozens of
locational models to address aspects of these three questions.
Classical location problems considered include Greenhut-Manne,
Hitchcock-Koopmans, and Weber-Launhardt. The book reinterprets
competitive location theory, focusing on the linkages between
Walrasian price equilibrium and the localization of firms. It also
demonstrates that competitive location theory offers diverse ideas
about the nature of market equilibrium in geographic space and its
implications for a broad range of public policies, including free
trade, industrial policy, regional development, and investment in
infrastructure. With an extensive bibliography and fresh,
interdisciplinary approach, the book will be an invaluable
reference for academics and researchers with an interest in
regional science, economic geography, and urban planning, as well
as policy advisors, urban planners, and consultants.
Differential-geometric methods are gaining increasing importance in
the understanding of a wide range of fundamental natural phenomena.
Very often, the starting point for such studies is a variational
problem formulated for a convenient Lagrangian. From a formal point
of view, a Lagrangian is a smooth real function defined on the
total space of the tangent bundle to a manifold satisfying some
regularity conditions. The main purpose of this book is to present:
(a) an extensive discussion of the geometry of the total space of a
vector bundle; (b) a detailed exposition of Lagrange geometry; and
(c) a description of the most important applications. New methods
are described for construction geometrical models for applications.
The various chapters consider topics such as fibre and vector
bundles, the Einstein equations, generalized Einstein--Yang--Mills
equations, the geometry of the total space of a tangent bundle,
Finsler and Lagrange spaces, relativistic geometrical optics, and
the geometry of time-dependent Lagrangians. Prerequisites for using
the book are a good foundation in general manifold theory and a
general background in geometrical models in physics. For
mathematical physicists and applied mathematicians interested in
the theory and applications of differential-geometric methods.
Asisknown,theLagrangeandHamiltongeometrieshaveappearedrelatively
recently [76, 86]. Since 1980thesegeometrieshave
beenintensivelystudied
bymathematiciansandphysicistsfromRomania,Canada,Germany,Japan,
Russia, Hungary,e.S.A. etc.
PrestigiousscientificmeetingsdevotedtoLagrangeandHamiltongeome-
tries and their applications have been organized in the above
mentioned countries and a number ofbooks and monographs have been
published by specialists in the field: R. Miron [94, 95], R.
Mironand M. Anastasiei [99, 100], R. Miron, D. Hrimiuc, H.
Shimadaand S.Sabau [115], P.L. Antonelli, R. Ingardenand
M.Matsumoto [7]. Finslerspaces,whichformasubclassof
theclassofLagrangespaces, havebeenthesubjectofsomeexcellentbooks,
forexampleby:Yl.Matsumoto[76], M.AbateandG.Patrizio[1],D.Bao,S.S.
Chernand Z.Shen [17]andA.BejancuandH.R.Farran [20]. Also, wewould
liketopointoutthemonographsofM. Crampin [34], O.Krupkova [72] and
D.Opri~,I.Butulescu [125],D.Saunders
[144],whichcontainpertinentappli-
cationsinanalyticalmechanicsandinthetheoryofpartialdifferentialequa-
tions. Applicationsinmechanics,
cosmology,theoreticalphysicsandbiology can be found in the well
known books ofP.L. Antonelliand T.Zawstaniak [11], G. S. Asanov
[14]' S. Ikeda [59], :VI. de LeoneandP.Rodrigues [73].
TheimportanceofLagrangeandHamiltongeometriesconsistsofthefact that
variational problems for important Lagrangiansor Hamiltonians have
numerous applicationsinvariousfields, such asmathematics,
thetheoryof dynamicalsystems, optimalcontrol, biology,andeconomy.
Inthisrespect, P.L. Antonelli'sremark isinteresting:
"ThereisnowstrongevidencethatthesymplecticgeometryofHamilto-
niandynamicalsystemsisdeeplyconnectedtoCartangeometry,thedualof
Finslergeometry", (seeV.I.Arnold,I.M.GelfandandV.S.Retach [13]).
The above mentioned applications have also imposed the introduction
x RaduMiron ofthe notionsofhigherorder Lagrangespacesand, ofcourse,
higherorder Hamilton spaces. The base manifolds ofthese spaces are
bundles ofaccel- erations ofsuperior order. The methods used in the
construction ofthese geometries are the natural extensions ofthe
classical methods used in the edification ofLagrange and Hamilton
geometries. These methods allow us to solvean old
problemofdifferentialgeometryformulated by Bianchiand Bompiani
[94]morethan 100yearsago,namelytheproblemofprolongation
ofaRiemannianstructure gdefinedonthebasemanifoldM,tothetangent k
bundleT M, k> 1. Bymeansofthissolutionofthe previousproblem, we
canconstruct, for thefirst time,goodexamplesofregularLagrangiansand
Hamiltoniansofhigherorder.
This monograph is mostly devoted to the problem of the geome-
trizing of Lagrangians which depend on higher order accelerations.
It naturally prolongs the theme of the monograph "The Geometry of
La- grange spaces: Theory and Applications", written together with
M. Anastasiei and published by Kluwer Academic Publishers in 1994.
The existence of Lagrangians of order k > 1 has been
contemplated by mechanicists and physicists for a long time.
Einstein had grasped their presence in connection with the Brownian
motion. They are also present in relativistic theories based on
metrics which depend on speeds and accelerations of particles or in
the Hamiltonian formulation of non- linear systems given by
Korteweg-de Vries equations. There resulted from here the methods
to be adopted in their theoretical treatment. One is based on the
variational problem involving the integral action of the
Lagrangian. A second one is derived from the axioms of Analytical
Mechanics involving the Poincare-Cartan forms. The geometrical
methods based on the study of the geometries of higher order could
invigorate the whole theory. This is the way adopted by us in
defining and studying the Lagrange spaces of higher order. The
problems raised by the geometrization of Lagrangians of order k
> 1 investigated by many scholars: Ch. Ehresmann, P. Libermann,
J. Pommaret; J.T. Synge, M. Crampin, P. Saunders; G.S. Asanov,
P.Aringazin; I. Kolar, D. Krupka; M. de Leon, W. Sarlet, P.
Cantrjin, H. Rund, W.M. Tulczyjew, A. Kawaguchi, K. Yano, K. Kondo,
D.
Since 1992 Finsler geometry, Lagrange geometry and their
applications to physics and biology, have been intensive1y studied
in the context of a 5-year program called "Memorandum
ofUnderstanding", between the University of Alberta and "AL.1.
CUZA" University in lasi, Romania. The conference, whose
proceedings appear in this collection, belongs to that program and
aims to provide a forum for an exchange of ideas and information on
recent advances in this field. Besides the Canadian and Romanian
researchers involved, the conference benefited from the
participation of many specialists from Greece, Hungary and Japan.
This proceedings is the second publication of our study group. The
first was Lagrange Geometry. Finsler spaces and Noise Applied in
Biology and Physics (1]. Lagrange geometry, which is concerned with
regular Lagrangians not necessarily homogeneous with respect to the
rate (i.e. velocities or production) variables, naturalIy extends
Finsler geometry to alIow the study of, for example, metrical
structures (i.e. energies) which are not homogeneous in these
rates. Most Lagrangians arising in physics falI into this class,
for example. Lagrange geometry and its applications in general
relativity, unified field theories and re1ativistic optics has been
developed mainly by R. Miron and his students and collaborators in
Romania, while P. Antonelli and his associates have developed
models in ecology, development and evolution and have rigorously
laid the foundations ofFinsler diffusion theory [1] .
This book provides a comprehensive, up-to-date, and expert
synthesis of location theory. What are the impacts of a firm's
geographic location on the locations of customers, suppliers, and
competitors in a market economy? How, when, and why does this
result in the clustering of firms in space? When and how is society
made better or worse off as a result? This book uses dozens of
locational models to address aspects of these three questions.
Classical location problems considered include Greenhut-Manne,
Hitchcock-Koopmans, and Weber-Launhardt. The book reinterprets
competitive location theory, focusing on the linkages between
Walrasian price equilibrium and the localization of firms. It also
demonstrates that competitive location theory offers diverse ideas
about the nature of market equilibrium in geographic space and its
implications for a broad range of public policies, including free
trade, industrial policy, regional development, and investment in
infrastructure. With an extensive bibliography and fresh,
interdisciplinary approach, the book will be an invaluable
reference for academics and researchers with an interest in
regional science, economic geography, and urban planning, as well
as policy advisors, urban planners, and consultants.
Asisknown,theLagrangeandHamiltongeometrieshaveappearedrelatively
recently [76, 86]. Since 1980thesegeometrieshave
beenintensivelystudied
bymathematiciansandphysicistsfromRomania,Canada,Germany,Japan,
Russia, Hungary,e.S.A. etc.
PrestigiousscientificmeetingsdevotedtoLagrangeandHamiltongeome-
tries and their applications have been organized in the above
mentioned countries and a number ofbooks and monographs have been
published by specialists in the field: R. Miron [94, 95], R.
Mironand M. Anastasiei [99, 100], R. Miron, D. Hrimiuc, H.
Shimadaand S.Sabau [115], P.L. Antonelli, R. Ingardenand
M.Matsumoto [7]. Finslerspaces,whichformasubclassof
theclassofLagrangespaces, havebeenthesubjectofsomeexcellentbooks,
forexampleby:Yl.Matsumoto[76], M.AbateandG.Patrizio[1],D.Bao,S.S.
Chernand Z.Shen [17]andA.BejancuandH.R.Farran [20]. Also, wewould
liketopointoutthemonographsofM. Crampin [34], O.Krupkova [72] and
D.Opri~,I.Butulescu [125],D.Saunders
[144],whichcontainpertinentappli-
cationsinanalyticalmechanicsandinthetheoryofpartialdifferentialequa-
tions. Applicationsinmechanics,
cosmology,theoreticalphysicsandbiology can be found in the well
known books ofP.L. Antonelliand T.Zawstaniak [11], G. S. Asanov
[14]' S. Ikeda [59], :VI. de LeoneandP.Rodrigues [73].
TheimportanceofLagrangeandHamiltongeometriesconsistsofthefact that
variational problems for important Lagrangiansor Hamiltonians have
numerous applicationsinvariousfields, such asmathematics,
thetheoryof dynamicalsystems, optimalcontrol, biology,andeconomy.
Inthisrespect, P.L. Antonelli'sremark isinteresting:
"ThereisnowstrongevidencethatthesymplecticgeometryofHamilto-
niandynamicalsystemsisdeeplyconnectedtoCartangeometry,thedualof
Finslergeometry", (seeV.I.Arnold,I.M.GelfandandV.S.Retach [13]).
The above mentioned applications have also imposed the introduction
x RaduMiron ofthe notionsofhigherorder Lagrangespacesand, ofcourse,
higherorder Hamilton spaces. The base manifolds ofthese spaces are
bundles ofaccel- erations ofsuperior order. The methods used in the
construction ofthese geometries are the natural extensions ofthe
classical methods used in the edification ofLagrange and Hamilton
geometries. These methods allow us to solvean old
problemofdifferentialgeometryformulated by Bianchiand Bompiani
[94]morethan 100yearsago,namelytheproblemofprolongation
ofaRiemannianstructure gdefinedonthebasemanifoldM,tothetangent k
bundleT M, k> 1. Bymeansofthissolutionofthe previousproblem, we
canconstruct, for thefirst time,goodexamplesofregularLagrangiansand
Hamiltoniansofhigherorder.
The title of this book is no surprise for people working in the
field of Analytical Mechanics. However, the geometric concepts of
Lagrange space and Hamilton space are completely new. The geometry
of Lagrange spaces, introduced and studied in [76],[96], was ext-
sively examined in the last two decades by geometers and physicists
from Canada, Germany, Hungary, Italy, Japan, Romania, Russia and
U.S.A. Many international conferences were devoted to debate this
subject, proceedings and monographs were published [10], [18],
[112], [113],... A large area of applicability of this geometry is
suggested by the connections to Biology, Mechanics, and Physics and
also by its general setting as a generalization of Finsler and
Riemannian geometries. The concept of Hamilton space, introduced in
[105], [101] was intensively studied in [63], [66], [97],... and it
has been successful, as a geometric theory of the Ham- tonian
function the fundamental entity in Mechanics and Physics. The
classical Legendre's duality makes possible a natural connection
between Lagrange and - miltonspaces. It reveals new concepts and
geometrical objects of Hamilton spaces that are dual to those which
are similar in Lagrange spaces. Following this duality Cartan
spaces introduced and studied in [98], [99],..., are, roughly
speaking, the Legendre duals of certain Finsler spaces [98], [66],
[67]. The above arguments make this monograph a continuation of
[106], [113], emphasizing the Hamilton geometry.
This monograph is mostly devoted to the problem of the geome-
trizing of Lagrangians which depend on higher order accelerations.
It naturally prolongs the theme of the monograph "The Geometry of
La- grange spaces: Theory and Applications", written together with
M. Anastasiei and published by Kluwer Academic Publishers in 1994.
The existence of Lagrangians of order k > 1 has been
contemplated by mechanicists and physicists for a long time.
Einstein had grasped their presence in connection with the Brownian
motion. They are also present in relativistic theories based on
metrics which depend on speeds and accelerations of particles or in
the Hamiltonian formulation of non- linear systems given by
Korteweg-de Vries equations. There resulted from here the methods
to be adopted in their theoretical treatment. One is based on the
variational problem involving the integral action of the
Lagrangian. A second one is derived from the axioms of Analytical
Mechanics involving the Poincare-Cartan forms. The geometrical
methods based on the study of the geometries of higher order could
invigorate the whole theory. This is the way adopted by us in
defining and studying the Lagrange spaces of higher order. The
problems raised by the geometrization of Lagrangians of order k
> 1 investigated by many scholars: Ch. Ehresmann, P. Libermann,
J. Pommaret; J.T. Synge, M. Crampin, P. Saunders; G.S. Asanov,
P.Aringazin; I. Kolar, D. Krupka; M. de Leon, W. Sarlet, P.
Cantrjin, H. Rund, W.M. Tulczyjew, A. Kawaguchi, K. Yano, K. Kondo,
D.
Since 1992 Finsler geometry, Lagrange geometry and their
applications to physics and biology, have been intensive1y studied
in the context of a 5-year program called "Memorandum
ofUnderstanding", between the University of Alberta and "AL.1.
CUZA" University in lasi, Romania. The conference, whose
proceedings appear in this collection, belongs to that program and
aims to provide a forum for an exchange of ideas and information on
recent advances in this field. Besides the Canadian and Romanian
researchers involved, the conference benefited from the
participation of many specialists from Greece, Hungary and Japan.
This proceedings is the second publication of our study group. The
first was Lagrange Geometry. Finsler spaces and Noise Applied in
Biology and Physics (1]. Lagrange geometry, which is concerned with
regular Lagrangians not necessarily homogeneous with respect to the
rate (i.e. velocities or production) variables, naturalIy extends
Finsler geometry to alIow the study of, for example, metrical
structures (i.e. energies) which are not homogeneous in these
rates. Most Lagrangians arising in physics falI into this class,
for example. Lagrange geometry and its applications in general
relativity, unified field theories and re1ativistic optics has been
developed mainly by R. Miron and his students and collaborators in
Romania, while P. Antonelli and his associates have developed
models in ecology, development and evolution and have rigorously
laid the foundations ofFinsler diffusion theory [1] .
Differential-geometric methods are gaining increasing importance in
the understanding of a wide range of fundamental natural phenomena.
Very often, the starting point for such studies is a variational
problem formulated for a convenient Lagrangian. From a formal point
of view, a Lagrangian is a smooth real function defined on the
total space of the tangent bundle to a manifold satisfying some
regularity conditions. The main purpose of this book is to present:
(a) an extensive discussion of the geometry of the total space of a
vector bundle; (b) a detailed exposition of Lagrange geometry; and
(c) a description of the most important applications. New methods
are described for construction geometrical models for applications.
The various chapters consider topics such as fibre and vector
bundles, the Einstein equations, generalized Einstein--Yang--Mills
equations, the geometry of the total space of a tangent bundle,
Finsler and Lagrange spaces, relativistic geometrical optics, and
the geometry of time-dependent Lagrangians. Prerequisites for using
the book are a good foundation in general manifold theory and a
general background in geometrical models in physics. For
mathematical physicists and applied mathematicians interested in
the theory and applications of differential-geometric methods.
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