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This book gives a comprehensive treatment of the Grassmannian
varieties and their Schubert subvarieties, focusing on the
geometric and representation-theoretic aspects of Grassmannian
varieties. Research of Grassmannian varieties is centered at the
crossroads of commutative algebra, algebraic geometry,
representation theory, and combinatorics. Therefore, this text
uniquely presents an exciting playing field for graduate students
and researchers in mathematics, physics, and computer science, to
expand their knowledge in the field of algebraic geometry. The
standard monomial theory (SMT) for the Grassmannian varieties and
their Schubert subvarieties are introduced and the text presents
some important applications of SMT including the Cohen-Macaulay
property, normality, unique factoriality, Gorenstein property,
singular loci of Schubert varieties, toric degenerations of
Schubert varieties, and the relationship between Schubert varieties
and classical invariant theory. This text would serve well as a
reference book for a graduate work on Grassmannian varieties and
would be an excellent supplementary text for several courses
including those in geometry of spherical varieties, Schubert
varieties, advanced topics in geometric and differential topology,
representation theory of compact and reductive groups, Lie theory,
toric varieties, geometric representation theory, and singularity
theory. The reader should have some familiarity with commutative
algebra and algebraic geometry.
Bio-Instructive Scaffolds for Musculoskeletal Tissue Engineering
and Regenerative Medicine explores musculoskeletal tissue growth
and development across populations, ranging from elite athletes to
the elderly. The regeneration and reparation of musculoskeletal
tissues present the unique challenges of requiring both the need to
withstand distinct forces applied to the body and ability to
support cell populations. The book is separated into sections based
on tissue type, including bone, cartilage, ligament and tendon,
muscle, and musculoskeletal tissue interfaces. Within each tissue
type, the chapters are subcategorized into strategies focused on
cells, hydrogels, polymers, and other materials (i.e. ceramics and
metals) utilized in musculoskeletal tissue engineering
applications. In each chapter, the relationships that exist amongst
the strategy, stem cell differentiation and somatic cell
specialization at the intracellular level are emphasized. Examples
include intracellular signaling through growth factor delivery,
geometry sensing of the surrounding network, and cell signaling
that stems from altered population dynamics.
This book gives a comprehensive treatment of the Grassmannian
varieties and their Schubert subvarieties, focusing on the
geometric and representation-theoretic aspects of Grassmannian
varieties. Research of Grassmannian varieties is centered at the
crossroads of commutative algebra, algebraic geometry,
representation theory, and combinatorics. Therefore, this text
uniquely presents an exciting playing field for graduate students
and researchers in mathematics, physics, and computer science, to
expand their knowledge in the field of algebraic geometry. The
standard monomial theory (SMT) for the Grassmannian varieties and
their Schubert subvarieties are introduced and the text presents
some important applications of SMT including the Cohen-Macaulay
property, normality, unique factoriality, Gorenstein property,
singular loci of Schubert varieties, toric degenerations of
Schubert varieties, and the relationship between Schubert varieties
and classical invariant theory. This text would serve well as a
reference book for a graduate work on Grassmannian varieties and
would be an excellent supplementary text for several courses
including those in geometry of spherical varieties, Schubert
varieties, advanced topics in geometric and differential topology,
representation theory of compact and reductive groups, Lie theory,
toric varieties, geometric representation theory, and singularity
theory. The reader should have some familiarity with commutative
algebra and algebraic geometry.
Google is the most popular search engine ever created, but Google's
search capabilities are so powerful, they sometimes discover
content that no one ever intended to be publicly available on the
Web, including social security numbers, credit card numbers, trade
secrets, and federally classified documents. Google Hacking for
Penetration Testers, Third Edition, shows you how security
professionals and system administratord manipulate Google to find
this sensitive information and "self-police" their own
organizations. You will learn how Google Maps and Google Earth
provide pinpoint military accuracy, see how bad guys can manipulate
Google to create super worms, and see how they can "mash up" Google
with Facebook, LinkedIn, and more for passive reconnaissance. This
third edition includes completely updated content throughout and
all new hacks such as Google scripting and using Google hacking
with other search engines and APIs. Noted author Johnny Long,
founder of Hackers for Charity, gives you all the tools you need to
conduct the ultimate open source reconnaissance and penetration
testing.
The man who went around the United Kingdom on a G-String sets off
to cross the United States on a golfing odyssey. His goal is to
play golf with whoever happens to feature on the front page of the
newspaper in each town he visitsthose, that is, who havent been
shot or arrested.Along the way he encounters an alligator-hunting
New York deli owner, a clown wedding, a blind baseball commentator,
a 91-year-old beach queen, a 200-year-old cactus, the runner-up in
a New Orleans waiter race, and a Stevie Wonder impersonatorbut
would any of them play golf?
Flag varieties are important geometric objects and their study
involves an interplay of geometry, combinatorics, and
representation theory. This book is a detailed account of this
interplay. In the area of representation theory, the book presents
a discussion of complex semisimple Lie algebras and of semisimple
algebraic groups; in addition, the representation theory of
symmetric groups is also discussed. In the area of algebraic
geometry, the book gives a detailed account of Grassmann varieties,
flag varieties, and their Schubert subvarieties. Because of their
connections with root systems, many of the geometric results admit
elegant combinatorial description, a typical example being the
description of the singular locus of a Schubert variety. This is
shown to be a consequence of standard monomial theory (abbreviated
SMT). Thus the book includes SMT and some important applications -
singular loci of Schubert varieties, toric degenerations of
Schubert varieties, and the relationship between Schubert varieties
and classical invariant theory. In this second edition, two recent
results on Schubert varieties in the Grassmannian have been added,
and some errors in the first edition corrected.
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