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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
This volume is based on lectures delivered at the 2019 AMS Short
Course ""Sum of Squares: Theory and Applications'', held January
14-15, 2019, in Baltimore, Maryland. This book provides a concise
state-of-the-art overview of the theory and applications of
polynomials that are sums of squares. This is an exciting and
timely topic, with rich connections to many areas of mathematics,
including polynomial and semidefinite optimization, real and convex
algebraic geometry, and theoretical computer science. The six
chapters introduce and survey recent developments in this area;
specific topics include the algebraic and geometric aspects of sums
of squares and spectrahedra, lifted representations of convex sets,
and the algorithmic and computational implications of viewing sums
of squares as a meta algorithm. The book also showcases practical
applications of the techniques across a variety of areas, including
control theory, statistics, finance and machine learning.
In the study of the structure of substances in recent decades,
phenomena in the higher dimension was discovered that was
previously unknown. These include spontaneous zooming (scaling
processes), discovery of crystals with the absence of translational
symmetry in three-dimensional space, detection of the fractal
nature of matter, hierarchical filling of space with polytopes of
higher dimension, and the highest dimension of most molecules of
chemical compounds. This forces research to expand the formulation
of the question of constructing n-dimensional spaces, posed by
David Hilbert in 1900, and to abandon the methods of considering
the construction of spaces by geometric figures that do not take
into account the accumulated discoveries in the physics of the
structure of substances. There is a need for research that accounts
for the new paradigm of the discrete world and provides a solution
to Hilbert's 18th problem of constructing spaces of higher
dimension using congruent figures. Normal Partitions and
Hierarchical Fillings of N-Dimensional Spaces aims to consider the
construction of spaces of various dimensions from two to any finite
dimension n, taking into account the indicated conditions,
including zooming in on shapes, properties of geometric figures of
higher dimensions, which have no analogue in three-dimensional
space. This book considers the conditions of existence of polytopes
of higher dimension, clusters of chemical compounds as polytopes of
the highest dimension, higher dimensions in the theory of heredity,
the geometric structure of the product of polytopes, the products
of polytopes on clusters and molecules, parallelohedron and
stereohedron of Delaunay, parallelohedron of higher dimension and
partition of n-dimensional spaces, hierarchical filling of
n-dimensional spaces, joint normal partitions, and hierarchical
fillings of n-dimensional spaces. In addition, it pays considerable
attention to biological problems. This book is a valuable reference
tool for practitioners, stakeholders, researchers, academicians,
and students who are interested in learning more about the latest
research on normal partitions and hierarchical fillings of
n-dimensional spaces.
The notion of stability for algebraic vector bundles on curves was
originally introduced by Mumford, and moduli spaces of semi-stable
vector bundles were studied intensively by Indian mathematicians.
The notion of stability for algebraic sheaves was generalized to
higher dimensional varieties. The study of moduli spaces of
algebraic sheaves not only on curves but also on higher dimensional
algebraic varieties has attracted much interest for decades and its
importance has been increasing not only in algebraic geometry but
also in related fields as differential geometry, mathematical
physics.Masaki Maruyama is one of the pioneers in the theory of
algebraic vector bundles on higher dimensional algebraic varieties.
This book is a posthumous publication of his manuscript. It starts
with basic concepts such as stability of sheaves, Harder-Narasimhan
filtration and generalities on boundedness of sheaves. It then
presents fundamental theorems on semi-stable sheaves: restriction
theorems of semi-stable sheaves, boundedness of semi-stable
sheaves, tensor products of semi-stable sheaves. Finally, after
constructing quote-schemes, it explains the construction of the
moduli space of semi-stable sheaves. The theorems are stated in a
general setting and the proofs are rigorous.Published by
Mathematical Society of Japan and distributed by World Scientific
Publishing Co. for all markets
This new textbook is based upon the notes that accompany the
author's graduate course "Algebraic Geometry I" at Tsinghua
University. Much of commutative algebra owes its existence to
algebraic geometry and vice versa, and this is why there is no
clear border between the two. In learning algebraic geometry, you
not only learn more commutative algebra, but also develop a
geometrical way of thinking about it.
This comprehensive account of the Gross-Zagier formula on
Shimura curves over totally real fields relates the heights of
Heegner points on abelian varieties to the derivatives of L-series.
The formula will have new applications for the Birch and
Swinnerton-Dyer conjecture and Diophantine equations.
The book begins with a conceptual formulation of the
Gross-Zagier formula in terms of incoherent quaternion algebras and
incoherent automorphic representations with rational coefficients
attached naturally to abelian varieties parametrized by Shimura
curves. This is followed by a complete proof of its coherent
analogue: the Waldspurger formula, which relates the periods of
integrals and the special values of L-series by means of Weil
representations. The Gross-Zagier formula is then reformulated in
terms of incoherent Weil representations and Kudla's generating
series. Using Arakelov theory and the modularity of Kudla's
generating series, the proof of the Gross-Zagier formula is reduced
to local formulas.
"The Gross-Zagier Formula on Shimura Curves" will be of great
use to students wishing to enter this area and to those already
working in it.
The fifteen articles composing this volume focus on recent
developments in complex analysis. Written by well-known researchers
in complex analysis and related fields, they cover a wide spectrum
of research using the methods of partial differential equations as
well as differential and algebraic geometry. The topics include
invariants of manifolds, the complex Neumann problem, complex
dynamics, Ricci flows, the Abel-Radon transforms, the action of the
Ricci curvature operator, locally symmetric manifolds, the maximum
principle, very ampleness criterion, integrability of elliptic
systems, and contact geometry. Among the contributions are survey
articles, which are especially suitable for readers looking for a
comprehensive, well-presented introduction to the most recent
important developments in the field.
The contributors are R. Bott, M. Christ, J. P. D'Angelo, P.
Eyssidieux, C. Fefferman, J. E. Fornaess, H. Grauert, R. S.
Hamilton, G. M. Henkin, N. Mok, A. M. Nadel, L. Nirenberg, N.
Sibony, Y.-T. Siu, F. Treves, and S. M. Webster.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
Die vorliegende Arbeit ist ein Auszug aus einer ausfuhrlicheren
noch ungedruckten Einfuhrung in die Mechanik materieller
Punktsysteme und starrer Korper mit den Methoden der Grassmannschen
Punktrechnung. Sie ist hervorgegangen aus der Uberzeugung, dass die
Punktrechnung, welche in naturlichsterWeise samtliche Grundelemente
des Raumes gleich massig der Rechnung unterwirft und deren
Verknupfungen analytisch un mittelbar durch rechnerische
Grundoperationen wiedergibt, auch in der Mechanik eine weitgehende
Vereinfachung und Vereinheitlichung der Methoden und eine
naturgemassere Darstellungsweise ermoglichen wird. Mogen die
Ergebnisse dieser Arbeit weitere Kreise der Mathematiker und
Physiker von der Richtigkeit dieser Auffassung uberzeugen.
Stuttgart, im Fruhjahr 1921. A. Lotze. Inhalt. Seile Literatur . .
. . . . . . . . . IV Benennungen und Bezeichnungen . . V
Zerlegungsformeln . . . . . . . VI Einleitung: Kmematik des
einzelnen Punkts 1 I. Kinematik des starren Korpers 2 1. Endliche
Verruckung eines starren Korpers. 2 2. Kinematische Grundgleichung
des starren Korpers 4 3. Grundlegende Satze uber Grossen 2. Stufe .
. . . 5 4. Ebel)e Bewegung. Euler-Savarysche Gleichung 7 5.
Beschleuni ungszustand des bewegten starren Korpers 10 6.
Beschleumgung der Relativbewegung . . ., . . 12 11. Allgemeine
Dynamik materieller Punktsysteme . 14 1. Die Bewegungsgleichungen .
. . . . . . . . . . 14 2. Momente des Impulses J und der Dyname D.
. . . . 15 3. Invarianten der Bewegung eines "vollstandigen"
Systems 16 4. Potential. Energiesatz. . . . . . . . . . . . . . . .
17 5. Das Zweikorperproblem . . . . . . . . . . . . . . 1S 6. Das
Prinzip von d'Alembert. Lagranges Gleichungen 1. Art. 20 7. Das
Gausssehe Prinzip. . . . . 21 8. Hamiltons Prinzip . 22 9.
Lagranges Gleichungen 2. Art 23 III. Dynamik des starren Korpers 25
1. Die dynamische Grundgleichung des freien starren Korpers . 25 2.
Wucht und Arbeit am starren Korper 26 3. Tragheitsmomente . . ."
Linear Algebra: Concepts and Applications is designed to be used in
a first linear algebra course taken by mathematics and science
majors. It provides a complete coverage of core linear algebra
topics, including vectors and matrices, systems of linear
equations, general vector spaces, linear transformations,
eigenvalues, and eigenvectors. All results are carefully, clearly,
and rigorously proven. The exposition is very accessible. The
applications of linear algebra are extensive and
substantial-several of those recur throughout the text in different
contexts, including many that elucidate concepts from multivariable
calculus. Unusual features of the text include a pervasive emphasis
on the geometric interpretation and viewpoint as well as a very
complete treatment of the singular value decomposition. The book
includes over 800 exercises and numerous references to the author's
custom software Linear Algebra Toolkit.
This largely self-contained book on the theory of quantum
information focuses on precise mathematical formulations and proofs
of fundamental facts that form the foundation of the subject. It is
intended for graduate students and researchers in mathematics,
computer science, and theoretical physics seeking to develop a
thorough understanding of key results, proof techniques, and
methodologies that are relevant to a wide range of research topics
within the theory of quantum information and computation. The book
is accessible to readers with an understanding of basic
mathematics, including linear algebra, mathematical analysis, and
probability theory. An introductory chapter summarizes these
necessary mathematical prerequisites, and starting from this
foundation, the book includes clear and complete proofs of all
results it presents. Each subsequent chapter includes challenging
exercises intended to help readers to develop their own skills for
discovering proofs concerning the theory of quantum information.
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