Optimal Aggregation Algorithms for Middleware
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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
Mathematics 144 Set Theory Fall 2012 Version
MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction -
A Taste of Set Theory for Philosophers
Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics. -
Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
Surviving Set Theory: a Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory
Surviving Set Theory: A Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Angela N. Ripley, M.M. Graduate Program in Music The Ohio State University 2015 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Johanna Devaney Copyright by Angela N. Ripley 2015 Abstract Undergraduate music students often experience a high learning curve when they first encounter pitch-class set theory, an analytical system very different from those they have studied previously. Students sometimes find the abstractions of integer notation and the mathematical orientation of set theory foreign or even frightening (Kleppinger 2010), and the dissonance of the atonal repertoire studied often engenders their resistance (Root 2010). Pedagogical games can help mitigate student resistance and trepidation. Table games like Bingo (Gillespie 2000) and Poker (Gingerich 1991) have been adapted to suit college-level classes in music theory. Familiar television shows provide another source of pedagogical games; for example, Berry (2008; 2015) adapts the show Survivor to frame a unit on theory fundamentals. However, none of these pedagogical games engage pitch- class set theory during a multi-week unit of study. In my dissertation, I adapt the show Survivor to frame a four-week unit on pitch- class set theory (introducing topics ranging from pitch-class sets to twelve-tone rows) during a sophomore-level theory course. As on the show, students of different achievement levels work together in small groups, or “tribes,” to complete worksheets called “challenges”; however, in an important modification to the structure of the show, no students are voted out of their tribes. -
IBM Research Report Proceedings of the IBM Phd Student Symposium at ICSOC 2006
RC24118 (W0611-165) November 28, 2006 Computer Science IBM Research Report Proceedings of the IBM PhD Student Symposium at ICSOC 2006 Ed. by: Andreas Hanemann (Leibniz Supercomputing Center, Germany) Benedikt Kratz (Tilburg University, The Netherlands) Nirmal Mukhi (IBM Research, USA) Tudor Dumitras (Carnegie Mellon University, USA) Research Division Almaden - Austin - Beijing - Haifa - India - T. J. Watson - Tokyo - Zurich Preface Service-Oriented Computing (SoC) is a dynamic new field of research, creating a paradigm shift in the way software applications are designed and delivered. SoC technologies, through the use of open middleware standards, enable collab- oration across organizational boundaries and are transforming the information- technology landscape. SoC builds on ideas and experiences from many different fields to produce the novel research needed to drive this paradigm shift. The IBM PhD Student Symposium at ICSOC provides a forum where doc- toral students conducting research in SoC can present their on-going dissertation work and receive feedback from a group of well-known experts. Each presentation is organized as a mock thesis-defense, with a committee of 4 mentors providing extensive feedback and advice for completing a successful PhD thesis. This for- mat is similar to the one adopted by the doctoral symposia associated with ICSE, OOPSLA, ECOOP, Middleware and ISWC. The closing session of the symposium is a panel discussion where the roles are reversed, and the mentors answer the students’ questions about research careers in industry and academia. The symposium agenda also contains a keynote address on writing a good PhD dissertation, delivered by Dr. Priya Narasimhan, Assistant Professor at Carnegie Mellon University and member of the ACM Doctoral Dissertation Committee. -
Basic Concepts of Set Theory, Functions and Relations 1. Basic
Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6 -
Applied Mathematics for Database Professionals
7451FM.qxd 5/17/07 10:41 AM Page i Applied Mathematics for Database Professionals Lex de Haan and Toon Koppelaars 7451FM.qxd 5/17/07 10:41 AM Page ii Applied Mathematics for Database Professionals Copyright © 2007 by Lex de Haan and Toon Koppelaars All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the copyright owner and the publisher. ISBN-13: 978-1-59059-745-3 ISBN-10: 1-59059-745-1 Printed and bound in the United States of America 9 8 7 6 5 4 3 2 1 Trademarked names may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, we use the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. Lead Editor: Jonathan Gennick Technical Reviewers: Chris Date, Cary Millsap Editorial Board: Steve Anglin, Ewan Buckingham, Gary Cornell, Jonathan Gennick, Jason Gilmore, Jonathan Hassell, Chris Mills, Matthew Moodie, Jeffrey Pepper, Ben Renow-Clarke, Dominic Shakeshaft, Matt Wade, Tom Welsh Project Manager: Tracy Brown Collins Copy Edit Manager: Nicole Flores Copy Editor: Susannah Davidson Pfalzer Assistant Production Director: Kari Brooks-Copony Production Editor: Kelly Winquist Compositor: Dina Quan Proofreader: April Eddy Indexer: Brenda Miller Artist: April Milne Cover Designer: Kurt Krames Manufacturing Director: Tom Debolski Distributed to the book trade worldwide by Springer-Verlag New York, Inc., 233 Spring Street, 6th Floor, New York, NY 10013. -
The Axiom of Choice and Its Implications
THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial. -
Design and Architectures for Signal and Image Processing
EURASIP Journal on Embedded Systems Design and Architectures for Signal and Image Processing Guest Editors: Markus Rupp, Dragomir Milojevic, and Guy Gogniat Design and Architectures for Signal and Image Processing EURASIP Journal on Embedded Systems Design and Architectures for Signal and Image Processing Guest Editors: Markus Rupp, Dragomir Milojevic, and Guy Gogniat Copyright © 2008 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in volume 2008 of “EURASIP Journal on Embedded Systems.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editor-in-Chief Zoran Salcic, University of Auckland, New Zealand Associate Editors Sandro Bartolini, Italy Thomas Kaiser, Germany S. Ramesh, India Neil Bergmann, Australia Bart Kienhuis, The Netherlands Partha S. Roop, New Zealand Shuvra Bhattacharyya, USA Chong-Min Kyung, Korea Markus Rupp, Austria Ed Brinksma, The Netherlands Miriam Leeser, USA Asim Smailagic, USA Paul Caspi, France John McAllister, UK Leonel Sousa, Portugal Liang-Gee Chen, Taiwan Koji Nakano, Japan Jarmo Henrik Takala, Finland Dietmar Dietrich, Austria Antonio Nunez, Spain Jean-Pierre Talpin, France Stephen A. Edwards, USA Sri Parameswaran, Australia Jurgen¨ Teich, Germany Alain Girault, France Zebo Peng, Sweden Dongsheng Wang, China Rajesh K. Gupta, USA Marco Platzner, Germany Susumu Horiguchi, Japan Marc Pouzet, France Contents -
Middleware-Based Database Replication: the Gaps Between Theory and Practice
Appears in Proceedings of the ACM SIGMOD Conference, Vancouver, Canada (June 2008) Middleware-based Database Replication: The Gaps Between Theory and Practice Emmanuel Cecchet George Candea Anastasia Ailamaki EPFL EPFL & Aster Data Systems EPFL & Carnegie Mellon University Lausanne, Switzerland Lausanne, Switzerland Lausanne, Switzerland [email protected] [email protected] [email protected] ABSTRACT There exist replication “solutions” for every major DBMS, from Oracle RAC™, Streams™ and DataGuard™ to Slony-I for The need for high availability and performance in data Postgres, MySQL replication and cluster, and everything in- management systems has been fueling a long running interest in between. The naïve observer may conclude that such variety of database replication from both academia and industry. However, replication systems indicates a solved problem; the reality, academic groups often attack replication problems in isolation, however, is the exact opposite. Replication still falls short of overlooking the need for completeness in their solutions, while customer expectations, which explains the continued interest in developing new approaches, resulting in a dazzling variety of commercial teams take a holistic approach that often misses offerings. opportunities for fundamental innovation. This has created over time a gap between academic research and industrial practice. Even the “simple” cases are challenging at large scale. We deployed a replication system for a large travel ticket brokering This paper aims to characterize the gap along three axes: system at a Fortune-500 company faced with a workload where performance, availability, and administration. We build on our 95% of transactions were read-only. Still, the 5% write workload own experience developing and deploying replication systems in resulted in thousands of update requests per second, which commercial and academic settings, as well as on a large body of implied that a system using 2-phase-commit, or any other form of prior related work. -
Scheduling Many-Task Workloads on Supercomputers: Dealing with Trailing Tasks
Scheduling Many-Task Workloads on Supercomputers: Dealing with Trailing Tasks Timothy G. Armstrong, Zhao Zhang Daniel S. Katz, Michael Wilde, Ian T. Foster Department of Computer Science Computation Institute University of Chicago University of Chicago & Argonne National Laboratory [email protected], [email protected] [email protected], [email protected], [email protected] Abstract—In order for many-task applications to be attrac- as a worker and allocate one task per node. If tasks are tive candidates for running on high-end supercomputers, they single-threaded, each core or virtual thread can be treated must be able to benefit from the additional compute, I/O, as a worker. and communication performance provided by high-end HPC hardware relative to clusters, grids, or clouds. Typically this The second feature of many-task applications is an empha- means that the application should use the HPC resource in sis on high performance. The many tasks that make up the such a way that it can reduce time to solution beyond what application effectively collaborate to produce some result, is possible otherwise. Furthermore, it is necessary to make and in many cases it is important to get the results quickly. efficient use of the computational resources, achieving high This feature motivates the development of techniques to levels of utilization. Satisfying these twin goals is not trivial, because while the efficiently run many-task applications on HPC hardware. It parallelism in many task computations can vary over time, allows people to design and develop performance-critical on many large machines the allocation policy requires that applications in a many-task style and enables the scaling worker CPUs be provisioned and also relinquished in large up of existing many-task applications to run on much larger blocks rather than individually.