Steuding, Jorn
Value Distributions of L-Functions
Series: Lecture Notes in Mathematics, Vol. 1877
2005, Approx. 270 p., Softcover
ISBN: 3-540-26526-0
About this book
These notes present recent results in the value-distribution
theory of L-functions with emphasis on the phenomenon of
universality. In 1975, Voronin proved that any non-vanishing
analytic function can be approximated uniformly by certain shifts
of the Riemann zeta-function in the critical strip. This
spectacular universality property has a strong impact on the zero-distribution:
Riemannfs hypothesis is true if and only if the Riemann zeta-function
can approximate itself uniformly (in the sense of Voronin).
Meanwhile universality is proved for a large zoo of Dirichlet
series, and it is conjectured that all reasonable L-functions are
universal. In these notes we prove universality for polynomial
Euler products. Our approach follows mainly Bagchi's
probabilistic method. We further discuss related topics as, e.g.,
almost periodicity, density estimates, Nevanlinna theory, and
functional independence.
Table of contents
Introduction.- The Selberg class.- Mean-square Formulae.- Value-distribution
in the Complex Plane.- Interlude: Facts from Probability Theory.-
Limit Theorems.- Universality.- The Riemann Hypothesis.-
Effective Results.- Consequences of Universality.- Dirichlet
Series with Periodic Coefficients.- Appendix: A Short History of
Universality.- Bibliography.- Index.- Notation
Andersen, Kirsti
The Geometry of an Art
The History of the Mathematical Theory of Perspective from
Alberti to Monge
Series: Sources and Studies in the History of Mathematics and
Physical Sciences
2005, Approx. 445 p. 604 illus., Hardcover
ISBN: 0-387-25961-9
About this book
This book aims at giving a comprehensive review of literature on
perspective constructions from the Renaissance to the end of the
eighteenth century. Covering the work of some 175 authors, it
treats the emergence of the various methods of constructing
perspective, the development of the theories underlying the
constructions, the communication between mathematicians and
artisans in these developments, and the interactions between
these theories and various subjects in mathematical geometry. The
main protagonists are Peiro della Francesca, Guidobaldo del
Monte, Simon Stevin, Brook Taylor, and Johann Heinrich Lambert.
The book includes a comprehensive bibliography of books and
manuscripts on perspective.
Table of contents
Introduction.- Acknowledgements.- Notes to the reader.- The birth
of perspective.- Alberti and Piero della Francesca.- Leonardo da
Vinci.- Italy in cinquecento.- North of the Alps before sixteen
hundred.- The birth of the mathematical theory of perspective:
Guidobaldo and Stevin.- The Dutch development after Stevin.-
Italy after Guidobaldo.- France and the Southern Netherlands
after 1600.- Britain.- The German speaking areas after 1600.-
Lambert.- Monge closing a circle.- Summing up.- Appendix: On
ancient roots of perspective.- Appendix: The Appearance of a
rectangle a la Leonardo da Vinci.- Appendix: 'sGravesande taking
recourse to the infinitesimal calculus to draw a column base in
perspective.- Appendix: The perspective sources, listed
countrywise.- Bibliography.- Index.
Sepanski, Mark R
Compact Lie Groups
Series: Graduate Texts in Mathematics, Vol. 235
2006, Approx. 180 p., Hardcover
ISBN: 0-387-30263-8
About this textbook
Blending algebra, analysis, and topology, the study of compact
Lie groups is one of the most beautiful areas of mathematics and
a key stepping stone to the theory of general Lie groups.
Assuming no prior knowledge of Lie groups, this book covers the
structure and representation theory of compact Lie groups.
Included is the construction of the Spin groups, Schur
Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem,
the Maximal Torus Theorem, the Commutator Theorem, the Weyl
Integration and Character Formulas, the Highest Weight
Classification, and the Borel-Weil Theorem. The necessary Lie
algebra theory is also developed in the text with a streamlined
approach focusing on linear Lie groups.
Table of contents
Compact Lie Groups.- Repersentations.- Harmonic Analysis.- Lie
Algebras.- Albenian Lie Subgroups and Structure.- Roots and
Associated Structures.- Highest Weight Theory.- Index.-
Bibliography
Rautenberg, Wolfgang
A Concise Introduction to Mathematical Logic
Series: Universitext
2006, Approx. 255 p. 8 illus., Softcover
ISBN: 0-387-30294-8
About this textbook
Traditional logic as a part of philosophy is one of the oldest
scientific disciplines. Mathematical logic, however, is a
relatively young discipline and arose from the endeavors of
Peano, Frege, Russell and others to create a logistic foundation
for mathematics. It steadily developed during the 20th century
into a broad discipline with several sub-areas and numerous
applications in mathematics, informatics, linguistics and
philosophy. While there are already several well-known textbooks
on mathematical logic, this book is unique in that it is much
more concise than most others, and the material is treated in a
streamlined fashion which allows the professor to cover many
important topics in a one semester course.
Although the book is intended for use as a graduate text, the
first three chapters could be understood by undergraduates
interested in mathematical logic. These initial chapters cover
just the material for an introductory course on mathematical
logic combined with the necessary material from set theory. This
material is of a descriptive nature, providing a view towards
decision problems, automated theorem proving, non-standard models
and other subjects. The remaining chapters contain material on
logic programming for computer scientists, model theory,
recursion theory, Godelfs Incompleteness Theorems, and
applications of mathematical logic. Philosophical and
foundational problems of mathematics are discussed throughout the
text. The author has provided exercises for each chapter, as well
as hints to selected exercises.
Table of contents
Foreword.- Preface.- Introduction.- Notation.- Propositional
Logic.- Predicate Logic.- Godelfs Completeness Theorem.- The
Foundations of Logic Programming.- Elements of Model Theory.-
Incompleteness and Undecidability.- On the Theory of Self-Reference.-
Hints to the Exercises.- Literature.- Index of Terms and Names.-
Index of Symbols.
Sernesi, Edoardo
Deformations of Algebraic Schemes
Series: Grundlehren der mathematischen Wissenschaften, Vol.
334
2006, Approx. 300 p., Hardcover
ISBN: 3-540-30608-0
About this book
The study of small and local deformations of algebraic varieties
originates in the classical work of Kodaira and Spencer and its
formalization by Grothendieck in the late 1950's. It has become
increasingly important in algebraic geometry in every context
where variational phenomena come into play, and in classification
theory. Today deformation theory is highly formalized and has
ramified widely. This self-contained account of deformation
theory in classical algebraic geometry (over an algebraically
closed field) brings together for the first time some results
previously scattered in the literature, with relatively little
known proofs, yet of everyday relevance to algebraic geometers.
It also includes applications to the construction and properties
of Severi varieties of families of plane nodal curves, space
curves, deformations of quotient singularities, Hilbert schemes
of points, local Picard functors, etc. The exposition, amenable
at graduate student level. includes many examples.
Table of contents
Introduction.- Infinitesimal Deformations: Extensions. Locally
Trivial deformations.- Formal Deformation Theory: Obstructions.
Extensions of Schemes. Functors of Artin Rings. The Theorem of
Schlessinger. The Local Moduli Functors.- Formal Versus Algebraic
Deformations. Automorphisms and Prorepresentability.- Examples of
Deformation Functors: Affine Schemes. Closed Subschemes.
Invertible Sheaves. Morphisms.- Hilbert and Quot Schemes:
Castelnuovo-Mumford Regularity. Flatness in the Projective Case.
Hilbert Schemes. Quot Schemes. Flag Hilbert Schemes. Examples and
Applications. Plane Curves.- Appendices: Flatness. Differentials.
Smoothness. Complete Intersections. Functorial Language.- List of
Symbols.- Bibliography