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This book presents an introduction to the geometric theory of infinite dimensional dynamical systems. Many of the fundamental results are presented for asymptotically smooth dynamical systems that have applications to functional differential equations as well as classes of dissipative partial differential equations. However, as in the earlier edition, the major emphasis is on retarded functional differential equations. This updated version also contains much material on neutral functional differential equations. The results in the earlier edition on Morse-Smale systems for maps are extended to a class of semiflows, which include retarded functional differential equations and parabolic partial differential equations.
The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.
In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.
This book showcases cutting-edge research papers from the XIIth
international Milton Symposium hosted by the University of
Strasbourg, 17-21 June 2019. Strasbourg is home to Martin Bucer,
the Protestant reformer from whom Milton drew support for his
theory of divorce, and to Gustave Dore, the famous French
illustrator of Paradise Lost. The 26 essays gathered in the present
volume are by international scholars, including ones from countries
outside the Anglosphere, young or experienced. Opening with a
tribute to all Milton symposia organized since 1981, the book falls
into eight parts, covering all aspects of Milton studies. "Milton
and Materiality" starts with an essay by James G. Turner on
personal bodily reference in Milton. In "Milton's Style and
Language", the polemicist's use of satire is scrutinized and his
relation to enthusiasm is examined, while a new light is shed on
his sonnets. In "Milton's Prose", in a rare essay on Observations
upon the Articles of Peace (1649), David H. Sacks compares Milton's
view of Ireland with what he thought of Russia, delving into the
notions of "civilization" and "tyranny". Then the reader will find
six essays on Paradise Lost, including one by Hiroko Sano, followed
by three essays on his minor poems by promising scholars. The
debate on the authorship of De Doctrina Christiana is reopened,
with many stylometric tables and charts. A new track leads us to
Silesia. In "Reception Studies", two Brazilian contributors study
Milton through the lens of French philosophers, and the next essay
by Christophe Tournu focuses on the first French verse translation
of Paradise Lost. The concluding part, "Milton and his Audience",
considers Milton's relationship to his readers, music in Haydn's
Creation, while Beverley Sherry analyses portraits of Milton and
his works in stained glass.
Debate about the authorship of the manuscript known to us as De
Doctrina Christiana has bedevilled Milton studies over recent
years. In this book four leading scholars give an account of the
research project that demonstrated its Miltonic provenance beyond
reasonable doubt. But the authors do much more besides, locating
Milton's systematic theology in its broader European context,
picking open the stages and processes of its composition, and
analysing its Latinity.
Milton's poetry is one of the glories of the English language, and
yet it owes everything to Milton's widespread knowledge of other
languages: he knew ten, wrote in four, and translated from five. In
Milton's Languages, John K. Hale first examines Milton's
language-related arts in verse-composition, translations,
annotations of Greek poets, Latin prose and political polemic,
giving all relevant texts in the original and in translation. Hale
then traces the impact of Milton's multilingualism on his major
English poems. Many vexed questions of Milton studies are
illuminated by this approach, including his sense of vocation, his
attitude to print and publicity, the supposed blemish of Latinism
in his poetry, and his response to his literary predecessors.
Throughout this full-length study of Milton's use of languages,
Hale argues convincingly that it is only by understanding Milton's
choice among languages that we can grasp where Milton's own unique
English originated.
The present book builds upon an earlier work of J. Hale, "Theory of
Func tional Differential Equations" published in 1977. We have
tried to maintain the spirit of that book and have retained
approximately one-third of the material intact. One major change
was a complete new presentation of lin ear systems (Chapters 6 9)
for retarded and neutral functional differential equations. The
theory of dissipative systems (Chapter 4) and global at tractors
was completely revamped as well as the invariant manifold theory
(Chapter 10) near equilibrium points and periodic orbits. A more
complete theory of neutral equations is presented (see Chapters 1,
2, 3, 9, and 10). Chapter 12 is completely new and contains a guide
to active topics of re search. In the sections on supplementary
remarks, we have included many references to recent literature,
but, of course, not nearly all, because the subject is so
extensive. Jack K. Hale Sjoerd M. Verduyn Lunel Contents
Preface............................................................
v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1. Linear
differential difference equations . . . . . . . . . . . . . . 11 .
. . . . . 1.1 Differential and difference equations. . . . . . . .
. . . . . . . . . . . . 11 . . . . . . . . 1.2 Retarded
differential difference equations. . . . . . . . . . . . . . . . 13
. . . . . . . 1.3 Exponential estimates of x( cents, f) . . . . . .
. . . . . . . . . . . . . . . 15 . . . . . . . . . . 1.4 The
characteristic equation . . . . . . . . . . . . . . . . . . . . . .
. . 17 . . . . . . . . . . . . 1.5 The fundamental solution. . . .
. . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . .
. . 1.6 The variation-of-constants
formula............................. 23 1. 7 Neutral differential
difference equations . . . . . . . . . . . . . . . . . 25 . . . . .
. . 1.8 Supplementary remarks. . . . . . . . . . . . . . . . . . .
. . . . . . . . 34 . . . . . . . . . . . . . 2. Functional
differential equations: Basic theory . . . . . . . . 38 . . 2.1
Definition of a retarded equation. . . . . . . . . . . . . . . . .
. . . . . 38 . . . . . . . . . 2.2 Existence, uniqueness, and
continuous dependence . . . . . . . . . . 39 . . . 2.3 Continuation
of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 44
. . . . . . . . . . . ."
The 1986 NATO Advanced Study Insti tute on Dynamics of Infini te
Dimensional Systems was held at the Instituto Superior Tecnico.
Lisbon. Portugal. In recent years. there have been several research
workers who have been considering partial differential equations
and functional differential equations as dynamical systems on
function spaces. Such approaches have led to the formulation of
more theoretical problems that need to be investigated. In the
applications. the theoretical ideas have contributed significantly
to a better understanding of phenomena that have been
experimentally and computationally observed. The investigators of
this development come wi th several different backgrounds - some
from classical partial differential equations. some from classical
ordinary differential equations and some interested in specific
applications. Each group has special ideas and often these ideas
have not been transmitted from one group to another. The purpose of
this NATO Workshop was to bring together research workers from
these various areas. It provided asoundboard for the impact of the
ideas of each respective discipline. We believe that goal was
accomplished. but time will be a better judge. We have included the
list of participants at the workshop. with most of these giving a
presentation. Although the proceedings do not include all of the
presentations. it is a good representative sampie. We wish to
express our gratitude to NATO. and.to Dr. M. di Lullo of NATO. who
unfortunately did not live to see the completion of this project.
In recent years, due primarily to the proliferation of computers,
dynamical systems has again returned to its roots in applications.
It is the aim of this book to provide undergraduate and beginning
graduate students in mathematics or science and engineering with a
modest foundation of knowledge. Equations in dimensions one and two
constitute the majority of the text, and in particular it is
demonstrated that the basic notion of stability and bifurcations of
vector fields are easily explained for scalar autonomous equations.
Further, the authors investigate the dynamics of planar autonomous
equations where new dynamical behavior, such as periodic and
homoclinic orbits appears.
Since the publication of my lecture notes, Functional Differential
Equations in the Applied Mathematical Sciences series, many new
developments have occurred. As a consequence, it was decided not to
make a few corrections and additions for a second edition of those
notes, but to present a more compre hensive theory. The present
work attempts to consolidate those elements of the theory which
have stabilized and also to include recent directions of research.
The following chapters were not discussed in my original notes.
Chapter 1 is an elementary presentation of linear differential
difference equations with constant coefficients of retarded and
neutral type. Chapter 4 develops the recent theory of dissipative
systems. Chapter 9 is a new chapter on perturbed systems. Chapter
11 is a new presentation incorporating recent results on the
existence of periodic solutions of autonomous equations. Chapter 12
is devoted entirely to neutral equations. Chapter 13 gives an
introduction to the global and generic theory. There is also an
appendix on the location of the zeros of characteristic
polynomials. The remainder of the material has been completely
revised and updated with the most significant changes occurring in
Chapter 3 on the properties of solutions, Chapter 5 on stability,
and Chapter lOon behavior near a periodic orbit."
State-of-the-art in qualitative theory of functional differential
equations; Most of the new material has never appeared in book form
and some not even in papers; Second edition updated with new topics
and results; Methods discussed will apply to other equations and
applications
Milton's poetry is one of the glories of the English language, and
yet it owes everything to Milton's widespread knowledge of other
languages: he knew ten, wrote in four, and translated from five. In
Milton's Languages, John K. Hale first examines Milton's
language-related arts in verse-composition, translations,
annotations of Greek poets, Latin prose and political polemic,
giving all relevant texts in the original and in translation. Hale
then traces the impact of Milton's multilingualism on his major
English poems. Many vexed questions of Milton studies are
illuminated by this approach, including his sense of vocation, his
attitude to print and publicity, the supposed blemish of Latinism
in his poetry, and his response to his literary predecessors.
Throughout this full-length study of Milton's use of languages,
Hale argues convincingly that it is only by understanding Milton's
choice among languages that we can grasp where Milton's own unique
English originated.
The second of eleven volumes of Milton's Complete Works to be
published contains his systematic theology, De Doctrina Christiana.
It is his longest work and was, Milton said, his dearest
possession. In it, he works out his religious beliefs from
Scripture; what Scripture does not mention, such as the Trinity, he
energetically refutes. The work exists in manuscript and was
written in Latin for European as well as home consumption. Its
chapters are conceived and arranged according to the binarizing
logic devised by the Protestant martyr Ramus.
De Doctrina Christiana first appeared in print nearly two hundred
years ago but the previous editions are now overdue for
replacement. For this ground-breaking edition, the manuscript has
been freshly transcribed, with fuller textual apparatus and
commentary than in any of its few predecessors. The edition aims
above all at accuracy, clarity, and completeness, presenting Latin
and English on facing pages, amplifying the Biblical citations
where necessary, and adding extensive annotations not only on the
text and its transcription but also on the content and context of
Milton's ideas. The provenance and history of the work are expertly
narrated, enabling readers to get closer than ever before to its
composition. Milton's Latin is examined in unprecedented detail,
and the translation aims to reproduce the nuances and changes of
register which characterize his Latin in all its individuality -
from the high-flown rhetoric of his arguments in favour of divorce
and polygamy, and against tithing, to the plainer style of those
sections where he states his main points more dispassionately but
bolsters them with strong and wide-ranging Biblical support. The
structure of this massive edifice is clarified by the addition of
charts which show the Ramist scheme he followed, whereby the
primary division between faith (Book One) and worship (Book Two) is
mirrored by smaller and smaller subdivisions whose relationship to
the whole can be seen at a glance.
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