Gröbner Bases: Statistics and Software Systems
暫譯: Gröbner 基底:統計與軟體系統
- 出版商: Springer
- 出版日期: 2014-01-17
- 售價: $2,470
- 貴賓價: 9.5 折 $2,347
- 語言: 英文
- 頁數: 474
- 裝訂: Hardcover
- ISBN: 4431545735
- ISBN-13: 9784431545736
-
相關分類:
UML、機率統計學 Probability-and-statistics
海外代購書籍(需單獨結帳)
商品描述
The idea of the Gröbner basis first appeared in a 1927 paper by F. S. Macaulay, who succeeded in creating a combinatorial characterization of the Hilbert functions of homogeneous ideals of the polynomial ring. Later, the modern definition of the Gröbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gröbner basis by Hironaka and Buchberger, it was not actively pursued for 20 years. A breakthrough was made in the mid-1980s by David Bayer and Michael Stillman, who created the Macaulay computer algebra system with the help of the Gröbner basis. Since then, rapid development on the Gröbner basis has been achieved by many researchers, including Bernd Sturmfels.
This book serves as a standard bible of the Gröbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABC’s of the Gröbner basis, requiring no special knowledge to understand those basic points.
Starting from the introductory performance of the Gröbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gröbner basis (Chapter 3). These three chapters may be regarded as the first act of a mathematical play. The second act opens with topics on algebraic statistics (Chapter 4), a fascinating research area where the Gröbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gröbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gröbner basis of the ring of differential operators gives effective algorithms on holonomic functions (Chapter 6). The third act (Chapter 7) is a collection of concrete examples and problems for Chapters 4, 5 and 6 emphasizing computation by using various software systems.
商品描述(中文翻譯)
格羅賓納基底(Gröbner basis)的概念最早出現在F. S. Macaulay於1927年的一篇論文中,他成功地創造了多項式環中齊次理想的希爾伯特函數的組合特徵。後來,格羅賓納基底的現代定義分別由平野啟介(Heisuke Hironaka)於1964年和布魯諾·布赫伯格(Bruno Buchberger)於1965年獨立提出。然而,在平野和布赫伯格發現格羅賓納基底的概念後,這一領域在接下來的20年內並未受到積極追求。直到1980年代中期,戴維·貝耶(David Bayer)和邁克爾·斯蒂爾曼(Michael Stillman)創建了麥考利計算代數系統(Macaulay computer algebra system),並利用格羅賓納基底取得了突破。自那時以來,許多研究者,包括伯恩德·斯圖姆費爾斯(Bernd Sturmfels),在格羅賓納基底的研究上取得了快速發展。
本書作為格羅賓納基底的標準聖經,理論、應用和計算的和諧是不可或缺的。它提供了所有的基礎知識,讓研究生能夠學習格羅賓納基底的ABC,理解這些基本概念不需要特別的知識。
從格羅賓納基底的入門表現(第一章)開始,接著是數學軟體的巡禮(第二章)。然後深入討論如何計算格羅賓納基底(第三章)。這三章可以視為數學劇的第一幕。第二幕以代數統計(第四章)為主題展開,這是一個迷人的研究領域,其中圓錐理想的格羅賓納基底是馬可夫鏈蒙地卡羅方法的基本工具。此外,圓錐理想的格羅賓納基底對於凸多面體的研究(第五章)也有很大的影響。此外,微分算子的環的格羅賓納基底提供了對全能函數的有效算法(第六章)。第三幕(第七章)則是針對第四、第五和第六章的具體例子和問題的集合,強調使用各種軟體系統進行計算。