Colloquium Publications, Volume: 7
1927; 180 pp; softcover
ISBN-10: 0-8218-4601-9
ISBN-13: 978-0-8218-4601-8
Expected publication date is July 14, 2008.
The central topic of this book is the presentation of the author's principle
of arithmetical paraphrases, which won him the Bocher Prize in 1924. This
general principle served to unify and extend many isolated results in the
theory of numbers. The author successfully provides a systematic attempt
to find a unified theory for each of various classes of related important
problems in the theory of numbers, including its interrelations with algebra
and analysis. This book will be of interest to advanced students in various
branches of mathematics, including number theory, abstract algebra, elliptic
and theta functions, Bernoulli numbers and functions, and the foundations
of mathematics.
Graduate students and research mathematicians interested in number theory.
Introduction
Varieties of algebra useful in algebraic arithmetic
The algebra mathcal{B} of parity
The algebraic arithmetic of multiply periodic functions
Applications of the algebras mathcal{C, D}
Arithmetical structure
Index
Colloquium Publications, Volume: 10
1929; 282 pp; softcover
Reprint/Revision History:
reprinted 1961; fourth printing with corrections 1982; ninth printing 1986
ISBN-10: 0-8218-4602-7
ISBN-13: 978-0-8218-4602-5
Expected publication date is July 14, 2008.
This volume is an amplification of the Colloquium Lectures delivered under
the title The Determination of the Tritangent Planes of the Space Sextic
of Genus Four. In order to present clearly the state of that problem, a
comparison with the better known cases of genus two and genus three is
desirable. Preliminary chapters on algebraic geometry and theta functions
are incorporated in order to facilitate reading by recalling fundamental
ideas of these two subjects in such fashion as will be most helpful in
later applications.
Graduate students and research mathematicians interested in algebraic geometry.
Topics in algebraic geometry
Topics in theta functions
Geometric applications of the functions of genus two
Geometric applications of the functions of genus three
Geometric aspects of the abelian modular functions of genus four
Theta relations of genus four
References
Colloquium Publications, Volume: 12
1930; 413 pp; softcover
ISBN-10: 0-8218-4603-5
ISBN-13: 978-0-8218-4603-2
Expected publication date is July 14, 2008.
Lefschetz's Topology was written in the period in between the beginning of topology, by Poincare, and the establishment of algebraic topology as a well-formed subject, separate from point-set or geometric topology. At this time, Lefschetz had already proved his first fixed-point theorems. In some sense, the present book is a description of the broad subject of topology into which Lefschetz's theory of fixed points fits. Lefschetz takes the opportunity to describe some of the important applications of his theory, particularly in algebraic geometry, to problems such as counting intersections of algebraic varieties. He also gives applications to vector distributions, complex spaces, and Kronecker's characteristic theory.
Graduate students and research mathematicians interested in topology.
Elementary combinatorial theory of complexes
Topological invariance of the homology characters
Manifolds and their duality theorems
Intersections of chains on a manifold
Product complexes
Transformations of manifolds, their coincidences and fixed points
Infinite complexes and their applications
Applications to analytical and algebraic varieties
Bibliography
Addenda
Index
Colloquium Publications, Volume: 14
1932; 172 pp; softcover
ISBN-10: 0-8218-4605-1
ISBN-13: 978-0-8218-4605-6
Expected publication date is July 14, 2008.
This book can be viewed as a first attempt to systematically develop an
algebraic theory of nonlinear differential equations, both ordinary and
partial. The main goal of the author was to construct a theory of elimination,
which "will reduce the existence problem for a finite or infinite
system of algebraic differential equations to the application of the implicit
function theorem taken with Cauchy's theorem in the ordinary case and Riquier's
in the partial." In his 1934 review of the book, J. M. Thomas called
it "concise, readable, original, precise, and stimulating", and
his words still remain true.
A more fundamental and complete account of further developments of the algebraic approach to differential equations is given in Ritt's treatise Differential Algebra, written almost 20 years after the present work (Colloquium Publications, Vol. 33, American Mathematical Society, 1950).
Graduate students and research mathematicians interested in differential equations.
Decomposition of a system of ordinary algebraic differential equations into irreducible systems
General solutions and resolvents
First applications of the general theory
Systems of algebraic equations
Constructive methods
Constitution of an irreducible manifold
Analogue of the Hilbert-Netto theorem. Theoretical decomposition process
Analogue for form quotients of Luroth's theorem
Riquier's existence theorem for orthonomic systems
Systems of algebraic partial differential equations
Index
Colloquium Publications, Volume: 16
1933; 218 pp; softcover
ISBN-10: 0-8218-4607-8
ISBN-13: 978-0-8218-4607-0
Expected publication date is July 14, 2008.
This book, immediately striking for its conciseness, is one of the most remarkable works ever produced on the subject of algebraic functions and their integrals. The distinguishing feature of the book is its third chapter, on rational functions, which gives an extremely brief and clear account of the theory of divisors....
A very readable account is given of the topology of Riemann surfaces and of the general properties of abelian integrals. Abel's theorem is presented, with some simple applications. The inversion problem is studied for the cases of genus zero and genus unity. The chapter on the reduction of singularities is very noteworthy.... A final chapter illustrates the general theory with some examples. In particular, constructive methods are given for treating algebraic relations which are of the third degree in one of the variables.... The arithmetic theory of algebraic functions is a good thing. In making its study easy, Bliss has performed a service which will win him the gratitude of an ever increasing number of readers.
--Bulletin of the American Mathematical Society
Graduate students and research mathematicians interested in algebraic functions and their integrals.
Single-valued analytic functions
Algebraic functions and their expansions
Rational functions
The Riemann surface of an algebraic function
Integrals of rational functions
Abel's theorem
Birational transformations
The reduction of singularities by tranformation
Inversion of Abelian integrals
Examples
List of references