This book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.
This brief presents a solution to the interpolation problem for arithmetically Cohen-Macaulay (ACM) sets of points in the multiprojective space P^1 x P^1. It collects the various current threads in the literature on this topic with the aim of providing a self-contained, unified introduction while also advancing some new ideas. The relevant constructions related to multiprojective spaces are reviewed first, followed by the basic properties of points in P^1 x P^1, the bigraded Hilbert function, and ACM sets of points. The authors then show how, using a combinatorial description of ACM points in P^1 x P^1, the bigraded Hilbert function can be computed and, as a result, solve the interpolation problem. In subsequent chapters, they consider fat points and double points in P^1 x P^1 and demonstrate how to use their results to answer questions and problems of interest in commutative algebra. Throughout the book, chapters end with a brief historical overview, citations of related results, and, where relevant, open questions that may inspire future research. Graduate students and researchers working in algebraic geometry and commutative algebra will find this book to be a valuable contribution to the literature.
For the first time the complete financial history of Berkshire Hathaway is available under one cover in chronological format. Beginning at the origins of the predecessor companies in the textile industry, the reader can examine the development of the modern-day conglomerate year-by-year and decade-by-decade, watching as the struggling textile company morphs into what it has become today. This comprehensive analysis distils over 10,000 pages of research material, including Buffett’s Chairman’s letters, Berkshire Hathaway annual reports and SEC filings, annual meeting transcripts, subsidiary financials, and more. The analysis of each year is supplemented with Buffett’s own commentary where relevant, and examines all important acquisitions, investments, and other capital allocation decisions. The appendices contain balance sheets, income statements, statements of cash flows, and key ratios dating back to the 1930s, materials brought together for the first time. The structure of the book allows the new student to follow the logic, reasoning, and capital allocation decisions made by Warren Buffett and Charlie Munger from the very beginning. Existing Berkshire shareholders and long-time observers will find new information and refreshing analysis, and a convenient reference guide to the decades of financial moves that built the modern-day respected enterprise that is Berkshire Hathaway.
The Buyer's Guide to Law Schools "What makes "The Complete Book of Law Schools" the leading law school guide? All the information one needs to make a crucial decision The Complete Book of Law Schools advises applicants of the attributes and possible drawbacks of all U.S. ABA-accredited schools, plus select California Bar Association (CBA)-accredited schools and Canadian schools. It also provides all the practical information readers need to apply: What is the student/faculty ratio? What are the average GPA and LSAT scores for students who are accepted? What is the job placement rate for graduates? How much financial aid is available? What is the bar passage rate for each school? Plus: campus and e-mail addresses, telephone numbers, admissions deadlines, tuition, and more "Includes: Complete and up-to-date information on 202 accredited law schools All the tools one needs to choose and apply: addresses, websites, deadlines, tuition, employment profiles, bar-exam pass rates, and more Tips on how to succeed in law school: special advice from Wentworth Miller Special tips to help readers crack the LSAT
WE KNOW THE LSAT The experts at The Princeton Review take the LSAT and other standardized tests several times a year to make sure you get the most up-to-date, thoroughly researched books possible. WE KNOW STUDENTS Each year we help more than two million students score high with our courses, bestselling books, and award-winning software. WE GET RESULTS Students who take our six-week LSAT course have an average score increase of 70 points (verified by International Communications Research). The proven techniques we teach in our course are in this book. AND IF IT'S ON THE LSAT, IT'S IN THIS BOOK We dont' try to teach you everything there is to know about reading comprehension or analytic thinking. We just tell you what you'll need to know to score high on the LSAT. There's a big difference. In Cracking the LSAT, we'll teach you how to think like the test makers and *Eliminate answer choices that look right but are planted to fool you *Crack complex argument problems by zeroing in on the conclusion *Use powerful methods of diagramming to solve games problems *Ace the reading-comprehension sections by "mapping out" the passages *Improve your writing sample by knowing what they're really looking for This book includes two full-length sample tests. The questions in the test are the same kind of problems you'll see on teh actual LSAT, and we fully explain every solution.
Proven techniques for scoring high from the world's #1 test-prep company Four complete sample tests on CD-ROM The Book That Gets You Results Earn more points by eliminating obviously wrong answers and guessing intelligently Dodge the traps and pitfalls that cost you points Practice your skills on two full-length sample tests in the book and four on CD-ROM LSAT TEST DATES: 1998: Feb. 7, June 15, Sept. 28, Dec. 5 LSAT (Law School Admission Test) is a registered trademark of the Law School Admission Council, Inc., which does not endorse this book.
This book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.
This brief presents a solution to the interpolation problem for arithmetically Cohen-Macaulay (ACM) sets of points in the multiprojective space P^1 x P^1. It collects the various current threads in the literature on this topic with the aim of providing a self-contained, unified introduction while also advancing some new ideas. The relevant constructions related to multiprojective spaces are reviewed first, followed by the basic properties of points in P^1 x P^1, the bigraded Hilbert function, and ACM sets of points. The authors then show how, using a combinatorial description of ACM points in P^1 x P^1, the bigraded Hilbert function can be computed and, as a result, solve the interpolation problem. In subsequent chapters, they consider fat points and double points in P^1 x P^1 and demonstrate how to use their results to answer questions and problems of interest in commutative algebra. Throughout the book, chapters end with a brief historical overview, citations of related results, and, where relevant, open questions that may inspire future research. Graduate students and researchers working in algebraic geometry and commutative algebra will find this book to be a valuable contribution to the literature.
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