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1. JSTOR Combinatorial Set Theory Partition Relations For Cardinals.
Combinatorial set theory Partition relations for cardinals. Studies in logic and the foundations of mathematics, vol. 10o; Disquisitiones mathematicae<310:CSTPRF>2.0.CO;2-0

2. Jonsson-like Partition Relations And J V V
F. Galvin and K. Prikry Infinitary Jonsson algebras and Partition relations, Algebra Universalis, vol. 6 (1976), pp. 367376.
Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options previous :: next
Arthur W. Apter and Grigor Sargsyan Source: J. Symbolic Logic Volume 69, Issue 4 (2004), 1267-1281.
Primary Subjects: Keywords: Jonsson cardinals; partition relations; polarized partitions; elementary embeddings Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text Links and Identifiers Permanent link to this document: Digital Object Identifier: doi:10.2178/jsl/1102022223 Mathematical Reviews number (MathSciNet): back to Table of Contents
A. Apter

3. [math/0703647v1] A Note On Strong Negative Partition Relations
A note on strong negative Partition relations. Authors Todd Eisworth Comments 23 pages. Covers my talk at the AMS meeting at Miami University. math
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Mathematics > Logic
Title: A note on strong negative partition relations
Authors: Todd Eisworth (Submitted on 21 Mar 2007) Abstract: We analyze a natural function definable from a scale at a singular cardinal, and using this function we are able to obtain quite strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain club-guessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection. Comments: 23 pages. Covers my talk at the AMS meeting at Miami University. This is essentially in final form, and is more than twice the length of the previously circulated version because of many new results. Comments are welcome Subjects: Logic (math.LO)

4. Back To Zentralblatt MATH Pages
Trees and positive ordinary Partition relations (80104, 13-18), V. Negative o.p.r. s and the discussion of the finite case (105-157, 19-26), VI.
Back to Zentralblatt MATH Pages
CD-ROM published
on the occasion of the
ICM 1998 in Berlin
Zentralblatt MATH
Combinatorial set theory: Partition relations for cardinals. (In English)
Constitutive parts of the present book are: Preface (pp. 5-6), Contents (7-8), Chapter I. Introduction (9-33, sections 1-4), II. Preliminaries (34-51, 5-7), III. Fundamentals about partition relations (52-79, 8-12), IV. Trees and positive ordinary partition relations (80-104, 13-18), V. Negative o.p.r.'s and the discussion of the finite case (105-157, 19-26), VI. The canonization lemmas (158-167, 27-28), VII. Large cardinals (168-214, 29-34), VIII. Discussion of the o.p.r. with superscript 2 (215-232, 35-37), IX. Discussion of the o.p.r. with superscript (233-262, 38-42), X. Some applications of combinatorial methods (263-312, 43-48), XI. A brief survey of the square bracket relation (313-333, 49-55), Bibliography (335-340; 109 items), Author index (341-342), Subject index (343-347).
The Preface beginns like this: "Ramsey's classical theorem in its simplest form, published in 1930, says that if we put the edges of an infinite graph into two classes, then there will be an infinite subgraph all edges of which belong to the same class. The partition calculus has been developed as a collection of generalizations of this theorem". The o.p.r. (ordinary partition relation) (1)

5. Atlas: Ramsey Theory Of Open Covers: From Partition Relations To Games By Nadav
We describe a new approach that proves directly the game theoretic assertion from the Partition relation. This allows solving some open problems of Kocinac.
Atlas home Conferences Abstracts about Atlas III Workshop on Coverings, Selections and Games in Topology
April 25-29, 2007
Faculty of Sciences and Mthematics, Nis; Technical Faculty, Cacak
, Serbia Organizers
Scientific Committee: Cosimo Guido, Ljubisa Kocinac, Marion Scheepers, Boaz Tsaban Organizing Committee: Liljana Babinkostova, Dragan Djurcic, Ljubisa Kocinac, Malisa Zizovic View Abstracts
Conference Homepage
Ramsey Theory of open covers: From Partition Relations to Games
Nadav Samet
Weizmann Institute of Science
Coauthors: Boaz Tsaban The classical method (initiated by Scheepers and exploited further by Scheepers, Kocinac, and others) to show that selection hypotheses are equivalent to Ramsey theoretic partition relations can be roughly described as follows: Prove that the partition relation implies the selection hypothesis; then prove that the selection hypothesis is equivalent to its game-theoretic version; and finally show that the game-theoretic version implies the partition relation. The most difficult ingredient in this scheme is the second, which often requires close relations to a deep result of Pawlikowski. We describe a new approach that proves directly the game theoretic assertion from the partition relation. This allows solving some open problems of Kocinac.

6. Combinatorial Set Theory Partition Relations For Cardinals - Elsevier
This work presents the most important combinatorial ideas in Partition calculus and discusses ordinary Partition relations for cardinals without the
Home Site map Elsevier websites Alerts ... Combinatorial Set Theory: Partition Relations for Cardinals Book information Product description Author information and services Ordering information Bibliographic and ordering information Conditions of sale Book-related information Submit your book proposal Other books in same subject area About Elsevier Select your view COMBINATORIAL SET THEORY: PARTITION RELATIONS FOR CARDINALS
A. Hajnal
R. Rado
Included in series

Studies in Logic and the Foundations of Mathematics, 106

This work presents the most important combinatorial ideas in partition calculus and discusses ordinary partition relations for cardinals without the assumption of the generalized continuum hypothesis. A separate section of the book describes the main partition symbols scattered in the literature. A chapter on the applications of the combinatorial methods in partition calculus includes a section on topology with Arhangel'skii's famous result that a first countable compact Hausdorff space has cardinality, at most continuum. Several sections on set mappings are included as well as an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis can not first fail at a singular cardinal of uncountable cofinality.
Hardbound, ISBN: 0-444-86157-2, 348 pages, publication date: 1984

7. Partition Relations For Countable Topological Spaces
Partition relations for countable topological spaces. Source, Journal of Combinatorial Theory Series A archive Volume 43 , Issue 2 (November 1986) table of

8. IngentaConnect Partition Relations For Kappa-normal Ideals On Pkappa(lambda)
Partition relations for kappa normal ideals on P kappa ( lambda ). Author Matet P. Source Annals of Pure and Applied Logic, Volume 121, Number 1,
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9. Using Boxes To Kill Squares And Other Results On Negative Partition Relations
We show equivalences under fairly weak conditions between some negative Partition relations. Then we generalize Jensen s boxes to give a parameter of
HOME DISSERTATIONS home search ... Columbia University Using boxes to kill squares and other results on negative partition relations
Richard Carr,
Faculty Advisor: Andras Hajnal
Date: 2000
See more information
Click here if you are not affiliated with Columbia University. Download the dissertation (PDF Format) Click here if you are a Columbia affiliate. Tell a colleague about this dissertation. Printing Tips Select "print as image" in the Acrobat print dialog if you have trouble printing. Abstract
Subject Area

10. Choodnowsky, Wolfsdorf: A Theorem On Polarised Partition Relations For Singular
A theorem on polarised Partition relations for singular cardinals. Bulletin de la Société Mathématique de France, 110 (1982), p. 349356

11. Akihiro Kanamori Bibliography
Negative Partition relations for Ultrafilters for Uncountable Cardinals. Regressive Partition relations for Infinite Cardinals.
UCI Department of Philosophy UCI Library
"The Empty Set, the Singleton, and the Ordered Pair" Abstract
Akihiro Kanamori
Department of Mathematics
Boston University
Tuesday, March 5, 2002
4:30 pm
SST 777
Akihiro Kanamory
A Bibliography
Compiled by
Eddie Yeghiayan
  • "Characterization of Nonregular Ultrafilters." Notices. American Mathematical Society
  • "P-Points in Betan." Notices. American Mathematical Society
  • "Ultrafilters over a Measurable Cardinal." Annals of Mathematical Logic
  • "Weakly Normal Filters and Irregular Ultrafilters." Transactions of the American Mathematical Society (June 1976), 220:393-399.
  • (with Robert M. Solovay and William N. Reinhardt.) "Strong Axioms of Infinity and Elementary Embeddings." Annals of Mathematical Logic
  • "Perfect Set Forcing for Uncountable Cardinals." Annals of Mathematical Logic
  • "On P-Points over a Measurable Cardinal." Journal of Symbolic Logic
  • "On Silver's and Related Principles." In D. Van Dalen, D. Lascar, T.J. Smiley, eds., Logic Colloquium '80
    Logic Colloquium, 1980, Prague, Czechoslovakia.

12. Albin L. Jones
A proof (involving elementary substructures) of a Partition relation . Polarized Partition relations Involving Weakly Compact Cardinals

13. Kafkoulis, George (1990-05-16) Homogeneous Sequences Of Cardinals For Ordinal De
Title, Homogeneous sequences of cardinals for ordinal definable Partition relations. Degree, PhD. Option, Mathematics. Advisory Committee
Caltech Library System
Browse Search Caltech Student Instructions
Kafkoulis, George (1990-05-16) Homogeneous sequences of cardinals for ordinal definable partition relations.
Type of Document Dissertation Author Kafkoulis, George URN etd-06132007-073731 Persistent URL Title Homogeneous sequences of cardinals for ordinal definable partition relations Degree PhD Option Mathematics Advisory Committee Advisor Name Title W. Hugh Woodin Committee Chair A. S. Kechris Committee Member Keywords
  • none
Date of Defense Availability restricted Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. < [...])([...]) (*), and we prove the consistency of this theory relative to the consistency of the existence of a supercompact cardinal and an inaccessible above it. If U is a normal measure on [...], then [...] denotes the Supercompact Prikry forcing induced by U. [...] is the partition relation [...] except that we consider only OD colorings of [...]. Theorems 1,2 are the main results of our thesis. Theorem 1. If there exists a model of ZFC in which [...] is a supercompact cardinal and [...] is an innaccessible above [...], then we can construct a model V of the same properties with the additional property that if U is a normal [...]-measure and G is [...] - generic over V, then V[G] does not satisfy the [...] partition property. [...]

14. Research Papers
M. Scheepers, Rothberger s property and Partition relations, The Journal of Symbolic M. Scheepers, Large open covers and a polarized Partition relation,
Selection Principles in Mathematics
Seminar Meetings Literature Research Problems ... SPM bulletin Research Papers
  • Function spaces:
  • C p (X) C ... Various hyperspaces Classical Selection principles: S (A,B), S fin (A,B), U fin (A,B)
  • Open covers Borel Covers D ense sets ... Other selection properties
  • 15. DBLP: E. M. Kleinberg
    7, E. M. Kleinberg, Joel I. Seiferas Infinite Exponent Partition relations and WellOrdered Choice. J. Symb. Log. 38(2) 299-308 (1973)
    E. M. Kleinberg
    Eugene M. Kleinberg DBLP: [ Home Author Title Conferences ... Michael Ley ( Sat Dec 22 19:06:41 2007

    16. Richard A. Shore: Publications
    On large cardinals and Partition relations, Journal of Symbolic Logic 36 (1971), 305308 (with Weak compactness and square bracket Partition relations,
    Richard A. Shore : Publications
    Curriculum Vitae Most of the documents below with electronic versions have been compiled for optimum viewing in PDF format. However, some papers (for various reasons) look grainy as PDF files. All the papers with electronic versions are, however, also available in postscript and DVI format.
  • On large cardinals and partition relations, Journal of Symbolic Logic (1971), 305-308 (with E.M. Kleinberg).
  • Weak compactness and square bracket partition relations, Journal of Symbolic Logic (1972), 673-676 (with E.M. Kleinberg).
  • Square bracket partition relations in L Fundamenta Mathematica
  • Minimal alpha-degrees, Annals of Mathematical Logic
  • Cohesive sets: countable and uncountable, Proceedings of the American Mathematical Society
  • Sigma n sets which are Delta n -incomparable (uniformly), Journal of Symbolic Logic
  • Splitting an alpha-recursively enumerable set, Transactions of the American Mathematical Society
  • The recursively enumerable alpha-degrees are dense, Annals of Mathematical Logic
  • The irregular and non-hyperregular alpha-r.e. degrees
  • 17. Front: [math.LO/0407440] Positive Partition Relations For P_{kappa,lambda}
    Title Positive Partition relations for P_{kappa,lambda} Authors Pierre Matet, Saharon Shelah Categories math.LO Logic Report number Shelah MtSh804
    Front for the arXiv Mon, 24 Dec 2007
    math LO math.LO/0407440 search register submit
    ... iFAQ math.LO/0407440 Title:
    Pierre Matet , Saharon Shelah
    Categories: math.LO Logic
    Report number: Shelah [MtSh:804]
    Saharon Shelah's Office
    Version 1: Mon, 26 Jul 2004 16:26:25 GMT
    - for questions or comments about the Front
    arXiv contact page
    - for questions about downloading and submitting e-prints

    18. Citebase - Strong Partition Relations Below The Power Set: Consistency, Was Sier
    Strong Partition relations below the power set consistency, was Sierpinski right, II? Authors Shelah, Saharon. We continue here Sh276 but we do not

    19. Proceedings Of The American Mathematical Society
    A polarized Partition relation for cardinals of countable cofinality P. Erd s, A. Hajnal, and R. Rado, Partition relations for cardinal numbers,

    ISSN 1088-6826 (e) ISSN 0002-9939 (p) Previous issue Table of contents Next issue
    Articles in press
    Previous Article
    A polarized partition relation for cardinals of countable cofinality Author(s): Albin L. Jones
    Journal: Proc. Amer. Math. Soc.
    MSC (2000): Primary 03E05, 05D10; Secondary 05A18
    Posted: November 30, 2007
    Retrieve article in: PDF DVI PostScript Abstract ... Additional information Abstract: We prove that if and , then for all . This polarized partition relation holds if for every partition either there are and with or there are and with References:
    J. Baumgartner and A. Hajnal, A proof (involving Martin's axiom) of a partition relation , Fund. Math. (1973), no. 3, 193-203. MR
    Combinatorial properties of compact cardinals MR
    Partition relations for cardinal numbers , Acta Math. Acad. Sci. Hungar.
    A partition calculus in set theory , Bull. Amer. Math. Soc. MR
    Similar Articles: Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): Retrieve articles in all Journals with MSC (2000): Additional Information: Albin L. Jones

    20. Peter Clote's Publications
    Weak Partition relations, finite games and independence results in Peano arithmetic , Model Theory of Algebra and Arithmetic, Springer Lecture Notes in
    Table of Contents
    Books Journal Articles Refereed Articles in Books Refereed Articles in Proceedings
  • Boolean Functions and Models of Computation , by P. Clote and E. Kranakis, 615 pages, Springer-Verlag, 2003, ISBN: 3-540-59436-1 (hardcover).
  • Computational Molecular Biology: An Introduction Japanese translation of Computational Molecular Biology: An Introduction , P. Clote and R. Backofen (2005).
  • Computer Science Logic , by P. Clote and H. Schwichtenberg (Eds.), 14th International Workshop, CSL 2000, Springer Lecture Notes in Computer Science , ISBN 3-540-67895-6, August 2000.
  • Arithmetic, Proof Theory and Computational Complexity , Oxford University Press, eds. P. Clote and J. Krajicek (1993).
  • Feasible Mathematics II
  • Research monograph translation, Theory of Relations Back to Table of Contents
    Journal Articles
  • Proteins: Structure, Function and Bioinformatics , to appear.
  • Nucleic Acids Res. , Web Server Issue (2006), in press.
  • DiANNA 1.1: An extension of the DiANNA web server for ternary cysteine classification, F. Ferre, P. Clote, Nucleic Acids Res.
  • 21. Math/0407440 Positive Partition Relations For P_{kappa,lambda}
    math/0407440 Positive Partition relations for P_{kappa,lambda}; Pierre Matet, Saharon Shelah.
    Username: Password: Main Archive Authors Personalization ... About astro-ph cond-mat cs gr-qc hep-ex hep-lat hep-ph hep-th math math-ph nlin nucl-ex nucl-th physics q-bio quant-ph stat January February March April May June July August September October November December Apr 07 -> astro-ph cond-mat cs gr-qc hep-ex hep-lat hep-ph hep-th math math-ph nlin nucl-ex nucl-th physics q-bio quant-ph stat Previous Next Mathematics (math) Abstract ... Citations
    author="Pierre Matet and Saharon Shelah",

    22. Powell's Books - Combinational Set Theory: Partition Relations For Cardinals By
    Powell s Books is the largest independent used and new bookstore in the world. We carry an extensive collection of out of print rare, and technical titles

    23. Hyperlinked List Of Shelah's Papers
    LO/9706224 Shelah, A polarized Partition relation and failure of GCH .. LO/9305206 Comfort+Kato+Shelah, Topological Partition relations to the Form
    Hyperlinked list of Shelah's papers
    Here is a list of papers that includes title, author, and some journal info. (If your connection is so slow that you cannot load the whole list in a reasonable time, try using one of these shorter lists: Those papers for which files are available are hyperlinked to the pdf files. The date next to the link [in year/month/day format] shows when the paper was last modified. We also have links to abstracts for some papers, and for those papers which appear in the e-print (formerly known as "xxx" or "Los Alamos archive"), we include a link to the relevant page at the Front for the Mathematics ArXiv If you find errors or have any problems with this list (such as links to nonexisting or read-protected files), please report them to us Prepared on 2007-07-07
    KShV:912 pdf ps
    Kennedy+Shelah+Vaananen, Regular Ultrafilters and Finite Square Principles
    GaSh:911 abstract
    Garti+Shelah, Depth of Boolean Algebras in the constructible universe
    Blass+Shelah, Basic Subgroups and Freeness, A Counterexample
    GhSh:909 pdf ps
    Sh:908 pdf ps abstract arXiv:0705.4130

    24. Polarized Partition Relations
    Polarized Partition relations. James E. Baumgartner, András Hajnal. Journal Title The Journal of Symbolic Logic. Date 2001. Volume 66. Issue 2

    25. Times Obituary
    Many of these developments were expounded in Combinatorial Set Theory Partition relations for Cardinals which Rado coauthored with three others and
    Richard Rado
    Advances in theoretical mathematics Professor Richard Rado, FRS, Emeritus Professor of Pure Mathematics in the University of Reading, who died on December 23 at the age of 83, made outstanding theoretical contributions to his subject over a period of more than 50 years. Rado was particularly known for his work in combinatorics, the study of the different ways certain operations can be performed a subject vastly developed over the last three decades. His name is often associated with Ramsey's Theory but Rado's paper, Studien Zur Kombinatorik , which appeared in 1933 is nowadays seen as a landmark and precursor of many subsequent developments. Many of these developments were expounded in Combinatorial Set Theory: Partition Relations for Cardinals In 1978, some felt belatedly, Rado was elected a Fellow of the Royal Society "for his work in combinatorics, including abstract independence structures, transversal theory and extensions of Ramsey's Theory (the partition calculus)". Perhaps the tribute which provided the most moving experience of his life, however, was his visit to the Free University of Berlin in October 1981 to lecture and receive an honorary doctorate. The London Mathematical Society, on whose council he had served from 1948 to 1957 and of which he was successively secretary and vice-president, awarded him the senior Berwick Prize in 1972 for his work on partition relations. Royalties received from

    26. Union College Math Department (Publications): Publications (Alan D. Taylor)
    Negative Partition relations for ultrafilters on uncountable cardinals (with A. Kanamori), Proceedings of the American Mathematical Society 92 (1984) 83–89.
    Math Department (Publications) jsMath.Setup.Script("plugins/tex2math.js")
    Publications (Alan D. Taylor)
  • Structural Properties of Ideals , (with J. Baumgartner and S. Wagon), Warszawa, Poland: Dissertationes Mathematicae (1982) ISBN 83-01-01506-3
  • Mathematics and Politics: Strategy, Voting, Power, and Proof , New York: Springer-Verlag (1995) ISBN 0-387-94391-9 and 0-387-94500-8
  • Fair Division: From Cake-Cutting to Dispute Resolution (with S. Brams), Cambridge, England: Cambridge University Press (1996) ISBN 0-521-55390-3 and 0-521-55644-9
  • Simple Games: Desirability Relations, Trading, and Pseudoweightings (with W. Zwicker), Princeton: Princeton University Press (1999) ISBN 0-691-00120-0
  • The Win-Win Solution: Guaranteeing Fair Shares to Everybody (with S. Brams), New York: Norton (1999) ISBN 0-393-04729-6 Translations have appeared in China, Japan, Russia, Brazil, Korea, and Portugal.
  • Social Choice and the Mathematics of Manipulation , Cambridge England: Cambridge University Press (2005) ISBN 0-521-81052-3 and 0-521-00883-2
  • Journal of Combinatorial Theory (A)
  • On splitting stationary subsets of large cardinals (with J. Baumgartner and S. Wagon)
  • 27. Federalism, Not Partition - Council On Foreign Relations
    Sen. Joe Biden (DDE) and CFR s Leslie Gelb in the Washington Post say the US should support federalism, not Partition, in Iraq.
    @import url(/css/main.css); Why does this page look this way? It appears that you are using either an older, classic Web browser or a hand-held device that allows you to view our content but may not work with every feature of our site. If you are using an older browser, please upgrade for the best experience. Welcome to CFR. Skip to section navigation Skip to content Home Site Index ... Podcast Navigation

    28. E-Notes: India-Pakistan Relations - FPRI
    Without understanding the significance of Partition, it’s impossible to understand the evolution of IndoPakistani relations. The experience of Partition
    Home About Us News Research Programs ... Transcripts
    India-Pakistan Relations
    by Sumit Ganguly April 2006 Sumit Ganguly of Indiana University is the author of Fearful Symmetry: India and Pakistan under the Shadow of Nuclear Weapons (Oxford and University of Washington, 2005) and India Since 1980 (Cambridge, forthcoming). This talk was given at the History Institute for Teachers Teaching India Marvin Wachman Fund for International Education , cosponsored by the University of Tennessee at Chattanooga Asia Program and the South Asia Center of the University of Pennsylvania, and made possible by a grant from the Annenberg Foundation. Unbiased at least he was when he arrived on his mission,
    Having never set eyes on the land he was called to partition
    Between two peoples fanatically at odds,
    With their different diets and incompatible gods.
    For mutual reconciliation or rational debate:
    Factors Animating the Conflict
    Three closely related factors animate this conflict. First, the conflict is really about competing notions of state-building in South Asia. Pakistan was created as the putative homeland of the Muslims of South Asia when the British left the subcontinent. The leaders of the Pakistani nationalist movement argued that with the departure of the British, despite the secular professions of the Congress Party that had brought India its independence, for all practical purposes this would be a Hindu-dominated state, and consequently Muslims would not be treated as equal citizens of India.

    29. ASL Info Main Page
    In his thesis, he introduced new techniques to solve a problem of Erdös on ordinal Partition relations which had been open for about 30 years.
    ASL Information About the ASL
    Contact Us


    Prizes and Awards

    Committees, Reports

    Member Search Sacks Prize Recipients 2006 Sacks Prize

    Matteo Viale, University of Torino and the University of Paris 7 Viale received his Ph.D. in 2006 from the University of Torino and the University of Paris 7, under the supervision of Alessandro Andretta and Boban Velickovic. The Committee's citation reads: "Viale's thesis makes fundamental contributions to our understanding of the consequences of forcing axioms in the combinatorics of singular cardinals. In particular, it solves a well-known problem, by showing that the Proper Forcing Axiom implies the Singular Cardinals Hypothesis.'' 2005 Sacks Prize
    Montalbán received his Ph.D. in 2005 from Cornell University, under the supervision of Richard Shore. The Committee on Prizes and Awards' citation reads: "The thesis, entitled Beyond the Arithmetic

    30. Topics In Discovering Modern Set Theory. II.
    Some negative Partition relations and the Negative SteppingUp Lemma. Characterizations of weakly compact cardinals. Arrow relations with infinite
    Discovering Modern Set Theory
    Winfried Just and Martin Weese
    Topics covered in Volume II
    From the Preface:
    "Our aim is to present the most important set-theoretic techniques that have found applications outside of set theory. We think of Volume II as a natural continuation of Volume I of the same text, but it is sufficiently self-contained to be studied separately. The main prerequisite is a knowledge of basic naive and axiomatic set theory. Moreover, some knowledge of mathematical logic and general topology is indispensible for reading this volume. A minicourse in mathematical logic was given in Chapters 5 and 6 of Volume I, and we include an appendix on general topology at the end of this volume. Our terminology is fairly standard. For the benefit of those readers who learned their basic set theory from a different source than our Volume I we include a short section on somewhat idiosyncratic notations introduced in Volume I. In particular, some of the material on mathematical logic covered in Chapter 5 is briefly reviewed. The book can be used as a text in the classroom as well as for self-study."
    Chapter 13. Filters and Ideals in Partial Orders

    31. Pakistan
    Since Partition, relations between Pakistan and India have been characterized by rivalry and suspicion. Although many issues divide the two countries,
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      Pakistan, along with parts of western India, contains the archeological remains of an urban civilization dating back 4,500 years. Alexander the Great included the Indus Valley in his empire in 326 B.C., and his successors founded the Indo-Greek kingdom of Bactria based in what is today Afghanistan and extending to Peshawar. Following the rise of the Central Asian Kushan Empire in later centuries, the Buddhist culture of Afghanistan and Pakistan, centered on the city of Taxila just west of Islamabad, experienced a cultural renaissance known as the Gandhara period. Pakistan emerged from an extended period of agitation by Muslims in the subcontinent to express their national identity free from British colonial domination as well as domination by what they perceived as a Hindu-controlled Indian National Congress. Muslim anti-colonial leaders formed the All-India Muslim League in 1906. Initially, the League adopted the same objective as the Congressself-government for India within the British Empirebut Congress and the League were unable to agree on a formula that would ensure the protection of Muslim religious, economic, and political rights.

    32. Review And Survey Of Subject Area [End Of First Year Report]
    A hash algorithm is used to Partition relations into disjoint buckets, so that all tuples with the same join attribute value(s) are in the same bucket.
    Chapter 2. Review and survey of subject area
    The purpose of this chapter is to describe the applicability of previous work on database machine (DBM) architecture, Distributed Databases (DDB) , and Data Dictionaries (DD) , to the DRAT architecture. Although it is assumed that the reader is familiar with the basic concepts of relational databases, parallel processing, and Flynn's taxonomy, some definitions are given, in order that no misapprehensions may arise.
    Traditional general purpose computers are better suited to numerical applications than to the kind of tasks required by the users of databases, which are qualitatively different. (There is no suggestion here that faster 'number-crunchers' are not needed). One problem with using a traditional architecture is that of speed as perceived by the user (if users were satisfied with performance there would be little incentive for developing faster machines). Langdon stated, "programs running on the central processor tend to be the bottleneck to system throughput. The database machine (DBM) is the result of an architectural approach which distributes processing power closer to the devices on which data are stored. Another impetus for the approach is due to LSI (sic) technology.

    33. Background Notes Archive - South Asia
    India Since Partition, relations between Pakistan and India have been characterized by rivalry and suspicion. Although many issues divide the two countries,
    Return to South Asia Background Notes Archive
    Return to Background Notes Archive Homepage
    Return to Electronic Research Collection Homepage

    34. 03E: Set Theory
    03E02 Partition relations; 03E04 Ordered sets and their cofinalities; pcf theory; 03E05 Other combinatorial set theory; 03E10 Ordinal and cardinal
    Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields
    POINTERS: Texts Software Web links Selected topics here
    03E: Set theory
    Naive set theory considers elementary properties of the union and intersection operators Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Indeed, one may "construct" the natural numbers, real numbers, and so on in this framework. However, situations such as Russell's paradox show that some care must be taken to define what, precisely, is a set. However, results in mathematical logic imply it is impossible to determine whether or not these axioms are consistent using only proofs expressed in this language. Assuming they are indeed consistent, there are also statements whose truth or falsity cannot be determined from them. These statements (or their negations!) can be taken as axioms for set theory as well. For example, Cohen's technique of forcing showed that the Axiom of Choice is independent of the other axioms of ZF. (That axiom states that for every collection of nonempty sets, there is a set containing one element from each set in the collection.) This axiom is equivalent to a number of other statements (e.g. Zorn's Lemma) whose assumption allows the proof of surprising even paradoxical results such as the Banach-Tarski sphere decomposition. Thus, some authors are careful to distinguish results which depend on this or other non-ZF axioms; most assume it (that is, they work in ZFC Set Theory).

    35. Equivalence Relations
    Each equivalence relation on a set Partitions the set into its equivalence classes but also for each Partition of the set there is an equivalence relation
    Next: Congruence modulo n Up: NOTES ON RELATIONS Previous: Relations on a set
    Equivalence relations
    A relation on A is an equivalence relation if it is reflexive, symmetric and transitive. An example of such is equality on a set. One might think of equivalence as a way to glob together elements that can be considered the same relative to a property. That is elements become indistinguishable relative to the relation. For example in arithmetic we don't think twice about 1/2 and 2/4 as having the same value but they are different objects. In geometry, similarity of triangles is an equivalence relation. A right angled triangle with legs of length 3 and 4 and hypotenuse of length 5 is not the same as one with lengths 6, 8 and 10. Yet, we think of them as equivalent because the ratios of the corresponding sides are the same. For each , we define the equivalency class containing a to be the set of those elements which are equivalent to a . We denote this set by [ a ] although you should be aware that there is no standard notation for this. Some authors use a . Two equivalence classes are either equal or disjoint. In fact

    36. Partition Of A Set - Wikipedia, The Free Encyclopedia
    If an equivalence relation is given on the set X, then the set of all equivalence classes forms a Partition of X. Conversely, if a Partition P is given on X
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    Partition of a set
    From Wikipedia, the free encyclopedia
    Jump to: navigation search A partition of a set into 6 parts: an Euler diagram representation. In mathematics , a partition of a set X is a division of X into non-overlapping " parts " or " blocks " or " cells " that cover all of X . More formally, these "cells" are both collectively exhaustive and mutually exclusive with respect to the set being partitioned.
    edit Definition
    A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets. Equivalently, a set P of nonempty subsets of X , is a partition of X if
  • The union of the elements of P is equal to X . (We say the elements of P cover X The intersection of any two elements of P is empty. (We say the elements of P are pairwise disjoint
  • The elements of P are sometimes called the blocks or parts of the partition.

    37. Geometric Functional Analysis Probability Seminar
    These simplified selection principles can be characterized by the BaumgartnerTaylor Partition relation, which is a generalized form of the ordinary Ramsey
    Organizers (2005/6-2007/8): Itai Benjamini, Gady Kozma, Gideon Schechtman, and Boaz Tsaban. Remark. This page was maintained by Boaz during the academic years 2005/6-2007/8. It is kept here for documentational purposes only. For talks in later years, visit Gideon Schechtman's GFPA webpage Description: This is/was an interdisciplinary seminar which deals mainly with topics in geometric functional analysis and probability theory, but occasionally also with combinatorics and other fields of mathematics of wide mathematical interest and accessibility. The talks in the seminar are often not about results of the speakers but rather expositions of interesting works done by others. To allow providing full definitions and at least some of the details of the proofs at a convenient pace, each talk usually lasts from two hours (one meeting) to two meetings. The special emphasis on accessibility of the talks makes this seminar suitable for students and working mathematicians with interest in either functional analysis, probability, or combinatorics.
    List of talks
    The following list is sorted backwards 20 June 2007: Emil Saucan (Technion)

    38. Laver: A Saturation Property On Ideals
    In this paper we consider a saturation property of ideals on K which implies this Partition relation, as well as generalizations to cases where K 2e .

    39. Equivalence Relation
    The power of an equivalence relation lies in its ability to Partition a set into the disjoint . The Correspondence of Equivalence Relation and Partition
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    Equivalence Relation by Jimmy Tseng
    Equivalence relation, a mathematical concept, is a type of relation on a given set that provides a way for elements of that set to be identified with (meaning considered equivalent to for some present purpose) other elements of that set. Equivalence relation is defined in a branch of mathematics called set theory, a vital branch underpinning all branches of mathematics and those fields that use mathematics. The power of an equivalence relation lies in its ability to partition a set into the disjoint union of subsets called equivalence class es. Because of its power to partition a set, an equivalence relation is one of the most used and pervasive tools in mathematics.
    measured in hours. How can the mathematician express the idea that 12 is equivalent to, or as mathematicians also say identified with, 24 or 48? In general, take any non-negative integer. Divide this integer by 12 and keep the remainder. Any non-negative number that gives the same remainder in this way is equivalent to any other such number. Hence, 2 is equivalent to 14, 26, 38 …. By positing that any non-negative number is equivalent to its remainder after division by 12, the mathematician succeeds in gaining a new perspective on the set of non-negative integers. The definition of equivalence relation is based on this simple idea of considering some elements to be equivalent to others under the equivalence relation.

    40. The Partition Function And Relation To Thermodynamics
    The Partition function and relation to thermodynamics.
    Next: Pressure and work virial Up: Isothermal-Isobaric ensemble Previous: Basic Thermodynamics
    The partition function and relation to thermodynamics
    In principle, we should derive the isothermal-isobaric partition function by coupling our system to an infinite thermal reservoir as was done for the canonical ensemble and also subject the system to the action of a movable piston under the influence of an external pressure P . In this case, both the temperature of the system and its pressure will be controlled, and the energy and volume will fluctuate accordingly. However, we saw that the transformation from E to T between the microcanonical and canonical ensembles turned into a Laplace transform relation between the partition functions. The same result holds for the transformation from V to T . The relevant ``energy'' quantity to transform is the work done by the system against the external pressure P in changing its volume from V =0 to V , which will be PV . Thus, the isothermal-isobaric partition function can be expressed in terms of the canonical partition function by the Laplace transform: where is a constant that has units of volume. Thus

    41. Relation Subclass-Partition In Theory Frame-Ontology
    So I could have made the subclassPartition relation take a variable number of arguments. I decided not to use a sequence variable because that is not a
    A subclass-partition of a class C is a set of subclasses of C that are mutually disjoint.
    Domain Class
    Range ... Class-partition
    Subclass-Partition ?C ?Class-Partition) (And ( Class ?C) ( Class-Partition Member ?Subclass ?Class-Partition) ( Subclass-Of ?Subclass ?C)))))
    • Why is the second argument a set, rather than a sequence variable? Interesting design choice. The ``notation'' question here is not new syntax for a language, it's just the definition of a particular relation called subclass-partition . In KIF you can define relations that take an arbitrary number of arguments, using a a subclass-partition relation take a variable number of arguments. I decided not to use a sequence variable because that is not a minimal ontological commitment; it imposes an extra logical constraint for the sake of syntactic convenience. A sequence (list) imposes an order. But a class-partition requires no order among the classes. And I wanted the second argument to

    42. Equivalence Relation
    Theorem 2 Let {A1, , An} be a Partition of a set A. Define a binary relation R on A as follows a, b R if and only if a Ai and b Ai for some i,
    Equivalence Relation
    Subjects to be Learned
    • equivalence relation
    • equivalence class
    • partition
    On the face of most clocks, hours are represented by integers between 1 and 12. However, since a day has 24 hours after 12 hours, a clock goes back to hour 1, and starts all over again from there. Thus each pair of hours such as 1 and 13, 2 and 14, etc. share one number 1, 2, ...etc., respectively. The same applies when we are interested in more than 24 hours. 25th hour is 1, so are 37th, 49th etc. What we are doing here essentially is that we consider the numbers in each group such as 1, 13, 25, ..., equivalent in the sense that they all are represented by one number (they are congruent modulo 12). Being representable by one number such as we see on clocks is a binary relation on the set of natural numbers and it is an example of equivalence relation we are going to study here.
    The concept of equivalence relation is characterized by three properties as follows:
    Definition(equivalence relation): A binary relation R on a set A is an equivalence relation if and only if
    R is reflexive
    R is symmetric, and

    43. PlanetMath: Partition Is Equivalent To An Equivalence Relation
    Corollary 1 For every Partition $ P=\lbrace P_i\mid i\in I\rbrace$ of $ S$ , $ E= \bigcup \lbrace P_i^2\mid i\in I is an equivalence relation.
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    Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About partition is equivalent to an equivalence relation (Derivation) Proposition There is a one-to-one correspondence between the set of partitions of and the set of equivalence relations on Proof . Let be a set. Suppose is a partition of . Form . Given any for some . Then . If , then for some , so , whence . Finally, suppose . Then and , or and . Since since is a partition. So , or Conversely, suppose is an equivalence relation on . For each , define Then each since is reflexive . If , then or as is symmetric . So as well. Next, pick any . Then . But . So since is transitive . Therefore , which implies . Collect all the information we have so far, this implies . Therefore forms a partitition of is being interpreted as a set, not a

    44. BBC NEWS | India Pakistan | Timeline
    Ever since the Partition of the subcontinent in 1947, when Britain dismantled its Indian empire, India and Pakistan have been arch rivals.


    ... Kashmir flashpoint Intro Click on the years above Next Introduction
    Ever since the partition of the sub-continent in 1947, when Britain dismantled its Indian empire, India and Pakistan have been arch rivals. The animosity has its roots in religion and history, and is epitomised by the long-running conflict over the state of Jammu and Kashmir. This has recently escalated into a dangerous nuclear arms race. Click on the above dates for a history of the tensions between the two countries.

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