The Second PRIMA Congress
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Questions About Boij-S\" Oderberg Theory
QUESTIONS ABOUT BOIJ–SODERBERG¨ THEORY DANIEL ERMAN AND STEVEN V SAM 1. Background on Boij–Soderberg¨ Theory Boij–S¨oderberg theory focuses on the properties and duality relationship between two types of numerical invariants. One side involves the Betti table of a graded free resolution over the polynomial ring. The other side involves the cohomology table of a coherent sheaf on projective space. The theory began with a conjectural description of the cone of Betti tables of finite length modules, given in [10]. Those conjectures were proven in [25], which also described the cone of cohomology tables of vector bundles and illustrated a sort of duality between Betti tables and cohomology tables. The theory itself has since expanded in many directions: allowing modules whose support has higher dimension, replacing vector bundles by coherent sheaves, working over rings other than the polynomial ring, and so on. But at its core, Boij–S¨oderberg theory involves: (1) A classification, up to scalar multiple, of the possible Betti tables of some class of objects (for example, free resolutions of finitely generated modules of dimension ≤ c). (2) A classification, up to scalar multiple, of the cohomology tables of some class of objects (for examples, coherent sheaves of dimension ≤ n − c). (3) Intersection theory-style duality results between Betti tables and cohomology tables. One motivation behind Boij and S¨oderberg’s conjectures was the observation that it would yield an immediate proof of the Cohen–Macaulay version of the Multiplicity Conjectures of Herzog–Huneke–Srinivasan [44]. Eisenbud and Schreyer’s [25] thus yielded an immediate proof of that conjecture, and the subsequent papers [11, 26] provided a proof of the Mul- tiplicity Conjecture for non-Cohen–Macaulay modules. -
Bibliography
Bibliography [1] Emil Artin. Galois Theory. Dover, second edition, 1964. [2] Michael Artin. Algebra. Prentice Hall, first edition, 1991. [3] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison Wesley, third edition, 1969. [4] Nicolas Bourbaki. Alg`ebre, Chapitres 1-3.El´ements de Math´ematiques. Hermann, 1970. [5] Nicolas Bourbaki. Alg`ebre, Chapitre 10.El´ements de Math´ematiques. Masson, 1980. [6] Nicolas Bourbaki. Alg`ebre, Chapitres 4-7.El´ements de Math´ematiques. Masson, 1981. [7] Nicolas Bourbaki. Alg`ebre Commutative, Chapitres 8-9.El´ements de Math´ematiques. Masson, 1983. [8] Nicolas Bourbaki. Elements of Mathematics. Commutative Algebra, Chapters 1-7. Springer–Verlag, 1989. [9] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton Math. Series, No. 19. Princeton University Press, 1956. [10] Jean Dieudonn´e. Panorama des mat´ematiques pures. Le choix bourbachique. Gauthiers-Villars, second edition, 1979. [11] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley, second edition, 1999. [12] Albert Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 17:891–921, 1905. [13] David Eisenbud. Commutative Algebra With A View Toward Algebraic Geometry. GTM No. 150. Springer–Verlag, first edition, 1995. [14] Jean-Pierre Escofier. Galois Theory. GTM No. 204. Springer Verlag, first edition, 2001. [15] Peter Freyd. Abelian Categories. An Introduction to the theory of functors. Harper and Row, first edition, 1964. [16] Sergei I. Gelfand and Yuri I. Manin. Homological Algebra. Springer, first edition, 1999. [17] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer, second edition, 2003. [18] Roger Godement. Topologie Alg´ebrique et Th´eorie des Faisceaux. -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
Finite Element Methods for the Simulation of Incompressible Flows
Weierstrass Institute for Applied Analysis and Stochastics Finite Element Methods for the Simulation of Incompressible Flows Volker John Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 Outline of the Lectures 1 The Navier–Stokes Equations as Model for Incompressible Flows 2 Function Spaces For Linear Saddle Point Problems 3 The Stokes Equations 4 The Oseen Equations 5 The Stationary Navier–Stokes Equations 6 The Time-Dependent Navier–Stokes Equations – Laminar Flows Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 2 (151) 1 A Model for Incompressible Flows • conservation laws ◦ conservation of linear momentum ◦ conservation of mass • flow variables ◦ r(t;x) : density [kg=m3] ◦ v(t;x) : velocity [m=s] ◦ P(t;x) : pressure [N=m2] assumed to be sufficiently smooth in • W ⊂ R3 • [0;T] Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 3 (151) 1 Conservation of Mass • change of fluid in arbitrary volume V ¶ Z Z Z − r dx = rv · n ds = ∇ · (rv) dx ¶t V ¶V V | {z } | {z } mass transport through bdry • V arbitrary =) continuity equation rt + ∇ · (rv) = 0 • incompressibility( r = const) ∇ · v = 0 Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 4 (151) • acceleration: using first order Taylor series expansion in time (board) dv (t;x) = ¶ v(t;x) + (v(t;x) · ∇)v(t;x) dt t movement of a particle 1 Newton’s Second Law of Motion • Newton’s second law of motion net force = mass × acceleration Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Shanghai, China Overview Introduction
Shanghai, China Overview Introduction The name Shanghai still conjures images of romance, mystery and adventure, but for decades it was an austere backwater. After the success of Mao Zedong's communist revolution in 1949, the authorities clamped down hard on Shanghai, castigating China's second city for its prewar status as a playground of gangsters and colonial adventurers. And so it was. In its heyday, the 1920s and '30s, cosmopolitan Shanghai was a dynamic melting pot for people, ideas and money from all over the planet. Business boomed, fortunes were made, and everything seemed possible. It was a time of breakneck industrial progress, swaggering confidence and smoky jazz venues. Thanks to economic reforms implemented in the 1980s by Deng Xiaoping, Shanghai's commercial potential has reemerged and is flourishing again. Stand today on the historic Bund and look across the Huangpu River. The soaring 1,614-ft/492-m Shanghai World Financial Center tower looms over the ambitious skyline of the Pudong financial district. Alongside it are other key landmarks: the glittering, 88- story Jinmao Building; the rocket-shaped Oriental Pearl TV Tower; and the Shanghai Stock Exchange. The 128-story Shanghai Tower is the tallest building in China (and, after the Burj Khalifa in Dubai, the second-tallest in the world). Glass-and-steel skyscrapers reach for the clouds, Mercedes sedans cruise the neon-lit streets, luxury- brand boutiques stock all the stylish trappings available in New York, and the restaurant, bar and clubbing scene pulsates with an energy all its own. Perhaps more than any other city in Asia, Shanghai has the confidence and sheer determination to forge a glittering future as one of the world's most important commercial centers. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Mechanical Engineering Department Nazarbayev University
Nazarbayev University, School of Engineering Bachelor of Mechanical Engineering OPTIMUM 2D GEOMETRIES THAT MINIMIZE DRAG FOR LOW REYNOLDS NUMBER FLOW Final Capstone Project Report By: Ilya Lutsenko Manarbek Serikbay Principal Supervisor: Dr.Eng. Konstantinos Kostas Dr.Eng. Marios Fyrillas April 2017 MECHANICAL ENGINEERING DEPARTMENT NAZARBAYEV UNIVERSITY Declaration Form We hereby declare that this report entitled “OPTIMUM 2D GEOMETRIES THAT MINIMIZE DRAG FOR LOW REYNOLDS NUMBER FLOW” is the result of our own project work except for quotations and citations which have been duly acknowledged. We also declare that it has not been previously or concurrently submitted for any other degree at Nazarbayev University. ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Name: Date: ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Name: Date: II MECHANICAL ENGINEERING DEPARTMENT NAZARBAYEV UNIVERSITY Abstract In this work, a two-dimensional Oseen’s approximation for Navier-Stokes equations is to be studied. As the theory applies only to low Reynolds numbers, the results are focused in this region of the flow regime, which is of interest in flows appearing in bioengineering applications. The investigation is performed using a boundary element formulation of the Oseen’s equation implemented in Matlab software package. The results are compared with simulations performed in COMSOL software package using a finite element approach for the full Navier-Stokes equations under the assumption of laminar and steady flow. Furthermore, experimental and numerical data from pertinent literature for the flow over a cylinder are used to verify the obtained results. The second part of the project employs the boundary element code in an optimization procedure that aims at drag minimization via body-shape modification with specific area constraints. The optimization results are validated with the aid of finite element simulations in COMSOL. -
An Experimental Test for the Existence of the Euler Wake Velocity, Validating Eulerlet Theory by Using a Bluff Body in a Low-Speed Wind Tunnel
An experimental test for the existence of the Euler wake velocity, validating Eulerlet theory by using a bluff body in a low-speed wind tunnel By Kiran Chalasani (B. Eng., M.Sc.) Ph.D. Thesis 2020 An experimental test for the existence of the Euler wake velocity, validating Eulerlet theory by using a bluff body in a low-speed wind tunnel Kiran Chalasani School of Computing, Science, and Engineering University of Salford, Manchester, UK Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy, July 2020 ii Abstract The application of manoeuvrability problems in aerodynamics mainly works for Reynolds average Navier-Stokes equations particularly for a uniform, steady flow past a fixed body. Further, consider large Reynolds number such that the flow is not turbulent, and the boundary layer is negligibly small and the impermeability boundary condition holds. Instead of using standard techniques and theory for describing the problem, a new method is employed based upon the concept of matching two different Green’s integral representations over a common boundary, one given by approximations valid in the near-field and the other by approximations in the far-field such that, a near-field Euler flow is matched to a far-field Oseen flow. Far from the body, linearise the velocity to the uniform stream yielding Oseen flow to leading order and match the near-field Euler and far-field Oseen flow on a common matching boundary. In particular, match the Green’s integral representations that use Green’s functions which are point force solutions. This gives new Green’s functions which we call Eulerlets and are obtained by collapsing the diffuse wake of the corresponding Oseenlets onto a wake line represented by Heaviside and delta functions. -
MSRI Celebrates Its Twentieth Birthday, Volume 50, Number 3
MSRI Celebrates Its Twentieth Birthday The past twenty years have seen a great prolifera- renewed support. Since then, the NSF has launched tion in mathematics institutes worldwide. An in- four more institutes: the Institute for Pure and spiration for many of them has been the Applied Mathematics at the University of California, Mathematical Sciences Research Institute (MSRI), Los Angeles; the AIM Research Conference Center founded in Berkeley, California, in 1982. An es- at the American Institute of Mathematics (AIM) in tablished center for mathematical activity that Palo Alto, California; the Mathematical Biosciences draws researchers from all over the world, MSRI has Institute at the Ohio State University; and the distinguished itself for its programs in both pure Statistical and Applied Mathematical Sciences and applied areas and for its wide range of outreach Institute, which is a partnership of Duke University, activities. MSRI’s success has allowed it to attract North Carolina State University, the University of many donations toward financing the construc- North Carolina at Chapel Hill, and the National tion of a new extension to its building. In October Institute of Statistical Sciences. 2002 MSRI celebrated its twentieth year with a Shiing-Shen Chern, Calvin C. Moore, and I. M. series of special events that exemplified what MSRI Singer, all on the mathematics faculty at the Uni- has become—a focal point for mathematical culture versity of California, Berkeley, initiated the original in all its forms, with the discovery and delight of proposal for MSRI; Chern served as the founding new mathematical knowledge the top priority. director, and Moore was the deputy director. -
[Pdf] Kbc 2015
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Regal Shanghai East Asia Hotel
Regal Shanghai East Asia Hotel Preferred Family Certification Ensures exceptional hotel experiences for the whole family. Regal Shanghai East Asia Hotel Preferred Family Certified Member of Sterling Hotels Regal Shanghai East Asia Hotel is Certified for 800 Ling Ling Road the Following Age Groups Shanghai, XU Hui District • 0-2 Infants and Toddlers 200030, China • 3-4 Preschoolers • 5-8 School-Age Children • 9-12 Preteens For reservations, please call: • 13-17 Teens + 1 800 323 7500 Overview Local Family Fun For Your Family’s Information For Your Family’s Adventure Set in the heart of the downtown Xu Jia Hui entertainment Regal Shanghai East Asia is within the grounds of the district, the four-star Regal Shanghai East Asia Hotel Shanghai Indoor Stadium in the center of Xu Jia Hui consistently remains a popular choice for leisure travelers. commercial and shopping center, and a short distance Families are greeted with warm smiles, attention to detail, from Yu Yuan Garden, Shanghai Museum, People’s and a total commitment to providing the best in service. Square, and the Bund. Pudong Int’l Airport: 31 mi/50 km, 45 minutes Hongqiao Airport: 6 mi/14 km, 15 minutes Accommodations Amenities and Services For Your Family’s Comfort For Your Family’s Convenience • Suites • In-room minibars or refrigerators • No extra fees for children 12 and younger • Highchairs sharing guestroom with adults • Cribs or cots with appropriate bedding • Wheelchair and stroller accessible • Roll-away beds • In-room safety kits Learn more on the next page > Information has been provided by hotel and is subject to change without notice.