41. Atlas: Covering Morphisms In Topos Theory By Marta Bunge Covering morphisms in topos theory by Marta Bunge Department of Mathematicsand Statistics, McGill University, Montreal, Canada. http://atlas-conferences.com/c/a/j/f/21.htm

Atlas Document # cajf-21 Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories September 2328, 2002 Fields Institute Toronto, ON, Canada Organizers George Janelidze, Georgian Academy of Sciences, Bodo Pareigis, University of Munich, Walter Tholen, York University View Abstracts Conference Homepage Covering morphisms in topos theory by Marta Bunge Department of Mathematics and Statistics, McGill University, Montreal, Canada In the work of Janelidze in 1990, a formal notion of covering morphism arises from an abstract categorical framework given by a pair of adjoint functors. Associated with any such class of covering morphisms, there is a corresponding (pure) Galois theory, of which there are examples in different areas of mathematics. In topos theory, the class of all covering projections (local homeomorphisms determined by a locally constant object of the topos) that appears explicitly in the work of Barr and Diaconescu in 1981, has been shown already by Janelidze to be an instance of the above notion of covering morphism, but only under the conditions that either the base topos be Set (that is, for Grothendieck toposes), or else where the splitting cover in the topos is assumed connected. U (E) is defined by means of the pushout, in the 2-category Top

42. Atlas: Spectral Decomposition Of Ultrametric Spaces And Topos Theory By Alex J. Spectral Decomposition of Ultrametric Spaces and topos theory presentedby Alex J. Lemin Moscow State University of Civil Engineering http://atlas-conferences.com/c/a/c/l/81.htm

Atlas Document # cacl-81 1999 Summer Conference on Topology and its Applications August 4-7, 1999 C.W. Post Campus of Long Island University Brookville, NY 11548, USA Conference Organizers Sheldon Rothman and Ralph Kopperman View Abstracts Conference Homepage Spectral Decomposition of Ultrametric Spaces and Topos Theory presented by Alex J. Lemin Moscow State University of Civil Engineering We consider categories M E T R and M E T R c U L T R A M E T R and U L T R A M E T R c of ultrametric spaces and the same maps. Given a family of ultrametric spaces, we prove that sums and products, equalizer and co-equalizer, pull-back and push-out, limits of direct and inverse spectra, if exist, are ultrametric. A product and a limit of inverse spectrum of complete metric spaces are complete. A space (X,d) is uniformly discrete 'for all' x, y X. This is necessarily complete. Theorem Every complete ultrametric space is isometric to a limit of a countable inverse spectrum of uniformly discrete ultrametric spaces (and vise versa) (see [1]). Corollary 1 Every compact ultrametric space is isometric to a limit of inverse sequence of skeletons of finite dimensional isosceles simplexes lying in Euclidean spaces (see [2]).

43. Sketches Of An Elephant: A Topos Theory Compendiumm Vol. 1 (Oxford Logic Guides, Sketches of an Elephant A topos theory Compendiumm vol. Links to book stores whereyou can buy Sketches of an Elephant A topos theory Compendiumm vol. http://www.booksearch.nu/0198534256

Select your country: Information Help no books Register Sign in Search: Title Author ISBN Sketches of an Elephant: A Topos Theory Compendiumm vol. 1 (Oxford Logic Guides, 43) Author: Peter T. Johnstone Oxford University Press November, 2002 Hardcover ISBN: You have no books in your comparison cart. Bookstore Availability Delivery Time Shipping Charge Price Total Cost used Check website Standard Shipping 5- 7 Business Days Link to new Available to be ordered Check website Check website Link to new Check website Standard 10 à 12 jours more Link to new Available to be ordered Standard 12-15 days more Link to new Usually dispatched within 4 to 6 weeks Airmail 5-7 working days more Link to new Check website Standard Shipping 24-38 days Link to used Check website Standard Shipping 5-14 days Link to used Check website Check website Check website Link to new Usually Ships in 7-10 Business Days Standard Shipping 3 -7 business days more Link to used Check website Standard Shipping 5- 21 Business Days Link to used Check website Versand 5-21 Werktage Link to new Special Order Free Super Saver Shipping 6-12 days more Link to new Check website Standard Shipping 7- 21 Business Days Link to Total Delivery Time = Availability + Delivery Time Shipping charge vary, the amount quoted is an estimation.

44. Sketches Of An Elephant: A Topos Theory Compendium (Oxford Logic Guides, 43 & 44 Sketches of an Elephant A topos theory Compendium (Oxford Logic Guides, 43 44). Author Peter T. Johnstone Hardcover Oxford University Press June, 2003. http://www.booksearch.nu/019852496X

Select your country: Information Help no books Register Sign in Search: Title Author ISBN Author: Peter T. Johnstone Oxford University Press June, 2003 Hardcover ISBN: You have no books in your comparison cart. Bookstore Availability Delivery Time Shipping Charge Price Total Cost used Check website Standard Shipping 5- 7 Business Days Link to new Usually Ships in 7-10 Business Days Standard Shipping 3 -7 business days more Link to new Available to be ordered Standard 12-15 days more Link to new Check website Standard Shipping 24-38 days Link to new Check website Versand 10- 21 Werktage Link to new Usually dispatched within 4 to 6 weeks Airmail 5-7 working days more Link to new Check website Versand 7- 21 Werktage Link to new Check website International Shipping 5- 21 Business Days Link to new Special Order Free Super Saver Shipping 6-12 days more Link to new Check website Standard Delivery 3 to 8 business days more Link to new Available to be ordered Check website Check website Link to new Check website Standard Shipping 7- 21 Business Days Link to new Check website Standard 10 à 12 jours more Link to new Check website Check website Check website Link to Total Delivery Time = Availability + Delivery Time Shipping charges and Delivery time are calculated for shipping to: United States All prices on this page is in USD Custom information For International Shipping, your packages may be subject to the customs fees and import duties of the country to which you have your order shipped.

45. Online Encyclopedia - Background And Genesis Of Topos Theory Cpunks20020422 Re Quantum mechanics, England, and topos theoryRe Quantum mechanics, England, and topos theory. In reply to jamesd_at_echeque.com Re Quantum mechanics, England, and topos theory ; http://www.yourencyclopedia.net/Background_and_genesis_of_topos_theory

Encyclopedia Entry for Background and genesis of topos theory Dictionary Definition of Background and genesis of topos theory This page gives some very general background to the mathematical idea of topos . This is an aspect of category theory , and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. Table of contents showTocToggle("show","hide") 1 In the school of Grothendieck 2 From pure category theory to categorical logic 3 Position of topos theory 4 Summary In the school of Grothendieck During the latter part of the , the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of etale cohomology With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems, for a long time. The first was to do with its

46. C-punks20020422: Re: Quantum Mechanics, England, And Topos Theory Re Quantum mechanics, England, and topos theory. From Theory ; ReplyKen Brown Re Quantum mechanics, England, and topos theory ; http://archives.abditum.com/cypherpunks/C-punks20020422/0110.html

Re: Quantum mechanics, England, and Topos Theory From: georgemw_at_speakeasy.net Date: Wed Apr 24 2002 - 16:41:31 EDT Next message: Morlock Elloi: "Re: Two ideas for random number generation" Previous message: Mike Rosing: "Re: Sources for GBPS random bits" In reply to: Tim May: "Quantum mechanics, England, and Topos Theory" Next in thread: Ken Brown: "Re: Quantum mechanics, England, and Topos Theory" Reply: Ken Brown: "Re: Quantum mechanics, England, and Topos Theory" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... [ attachment ] On 23 Apr 2002 at 18:56, Tim May wrote: > sense. The sources of "divergence" (aka chaos, aka combinatorial I can explain why people might think it were. You could imagine that due to feedback mechanisms or statistical averaging, these small uncertainties tend to cancel each other out, provided you're confining your interest to macroscopic observables. For example, when a sheep dies you get more grass for the remaining sheep, which gets you more sheep again, so you can do a reasonable job of predicting sheep population without knowing anything about the fates of individual sheep.

47. Sketches Of An Elephant: A Topos Theory Compendium, Vol. 2 Sketches of an Elephant A topos theory Compendium, vol. 2 Search for books atmathematicsbooks.org. Sketches of an Elephant A topos theory Compendium, vol. http://mathematicsbooks.org/0198515987.html

Home Search High Volume Orders Links ... Philosophy of Mathematics Additional Subjects Irish Litanies How to Make an American Quilt Death Dying History of Mathematics ... A Diplomatic History of the American People Sketches of an Elephant: A Topos Theory Compendium, vol. 2 Written by Peter T. Johnstone Published by Oxford Univ Pr (August 2002) ISBN 0198515987 Price $200.00 Customer Reviews Look for related books on other categories Algebra - Linear Mathematics Mathematical And Symbolic Logic Homology Theory ... Logic Still didn't find what you want? Try Amazon search Search: All Products Books Magazines Popular Music Classical Music Video DVD Baby Electronics Software Outdoor Living Wireless Phones Keywords: Or try to look for Sketches of an Elephant: A Topos Theory Compendium, vol. 2 at Fetch Used Books, at or at CampusI See Also Motel of the Mysteries Crinkleroot's Guide to Knowing the Birds American fiction Reader Allison Lassieur ... Calculus of a Single Variable Our Bookstores Cooking Books Games Books Computer Books Health Books ... Music Books Web Resources Amazon Australian Mathematics Trust European Mathematical Society (EMS) Kappa Mu Epsilon - Official Site ... Exchange links with us

48. Mathematics Toposes Sheaves in Geometry Logic A First Introduction to topos theory Sheaves in Geometry Logic A First Introduction to topos theory This book is written in the http://mathematicsbooks.org/Mathematics_Toposes.html

Home Search High Volume Orders Links ... Philosophy of Mathematics Additional Subjects Irish Litanies How to Make an American Quilt Death Dying History of Mathematics ... A Diplomatic History of the American People Featured Books This is a very readable introduction to the subject. Too bad it's out of print. Written by Michael Barr Charles Wells Published by Springer Verlag (April 1985) ISBN 0387961151 Price $82.95 This book is written in the best Mac Lane style, very clear and very well organized. It also benefits from Moerdijk's extensive work organizing the theory of Grothendieck toposes by elementary means. The reader should have basic graduate knowledge of algebra and topology. The book is long because it gives very explicit descriptions of many advanced topicsyou can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field. Written by S. Mac Lane I. Moerdijk Saunders MacLane Published by Springer Verlag (October 1994) ISBN 0387977104 Price $69.95

49. Re: The Relevance Of Category And Topos Theory http://www.mail-archive.com/everything-list@eskimo.com/msg03777.html

everything-list Chronological Find Thread Re: The relevance of category and topos theory From: Wei Dai Subject: Re: The relevance of category and topos theory Date: Thu, 18 Jul 2002 09:58:57 -0700 http://www.math.uu.se/~palmgren/topos-eng.html http://plato.stanford.edu/entries/logic-intuitionistic/ ). Does that mean I should learn something about intuitionistic logic and constructivism first before trying to tackle topos theory? I notice the book "Constructivism in Mathematics" by Troelstra and Dalen. Has anyone here read it, or can anyone recommend another book? Re: The relevance of category and topos theory Wei Dai Chronological Thread Reply via email to

50. Re: Some Books On Category And Topos Theory http://www.mail-archive.com/everything-list@eskimo.com/msg03764.html

everything-list Chronological Find Thread Re: Some books on category and topos theory From: Bruno Marchal Subject: Re: Some books on category and topos theory Date: Tue, 16 Jul 2002 01:25:34 -0700 Title: Re: Some books on category and topos theory At 12:24 -0700 9/07/2002, Tim May wrote: Whether knots are the key to physics, I can't say. Certainly there are suggestive notions that particles might be some kind of knots in spacetime (of some dimensionality)... Interesting! Chromosomes make knots, also. But my reading of Louis Kauffman, a great enthusiast and a great pedagogue, makes me believe that knots could be much more: -Knots could be "type" of multiverse! (multi-histories "skeleton"). Much more: -Invariant of knots could be quantum statistical abstract physical theories! As you know perhaps I get a proof that with the comp hyp, the mind-body problem is partially reduced into a derivation of physics from machine's psychology: poetically: physical realities is a web of infinite machine dreams(*). Machine's psychology can be approximated (at least) by the many possible intensionnal (modal) variant of the Godel Lob logic of Self-reference by the Universal Dovetailer when true)).

51. Sheaves In Geometry And Logic: A First Introduction To Topos Theory (Universitex Sheaves in Geometry and Logic A First Introduction to topos theory (Universitext).Sheaves in Geometry and Logic A First Introduction http://www.sciencesbookreview.com/Sheaves_in_Geometry_and_Logic_A_First_Introduc

Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) by Authors: S. Mac Lane , I. Moerdijk , Saunders MacLane Released: October, 1994 ISBN: 0387977104 Paperback Sales Rank: List price: Our price: You save: Book > Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) > Customer Reviews: Average Customer Rating: Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) > Customer Review #1: Clear explicit descriptions This book is written in the best Mac Lane style, very clear and very well organized. It also benefits from Moerdijks extensive work organizing the theory of Grothendieck toposes by elementary means. The reader should have basic graduate knowledge of algebra and topology. The book is long because it gives very explicit descriptions of many advanced topicsyou can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field. Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) > Related Products Categories for the Working Mathematician (2nd Ed)(Graduate Texts in Mathematics, 5)

52. FOM: Topos Theory FOM topos theory. I wrote that the topos theory way to ground real analysisis not inspiring in the sense that it does not inspire me. http://www.cs.nyu.edu/pipermail/fom/1998-January/000852.html

FOM: topos theory Kanovei kanovei@wminf2.math.uni-wuppertal.de Fri, 16 Jan 98 18:12:10 +0100 Previous message: FOM: topos theory qua f.o.m.; a quote from Mac Lane Next message: FOM: the cumulative hierarchy versus topos theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] From: "Michael Thayer" < mthayer@ix.netcom.com Date: Fri, 16 Jan 1998 08:11:02 -0600 I am trying to see exactly what is the nature of the problem Vladimir and Steve have with Colin's statement that you can do ZFC kinds of things in the appropriate topos. Would you say that KF does NOT "ground and support" analysis because of the underlying ambiguity in the nature of "set" in KF? (After all, NF and Z(FC) have somewhat different notions of set, don't they?) I am not an expert on KF, so I would refrain from answering your direct question. Vladimir Kanovei Previous message: FOM: topos theory qua f.o.m.; a quote from Mac Lane Next message: FOM: the cumulative hierarchy versus topos theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

53. FOM: Topos Theory Qua F.o.m.; Topos Theory Qua Pure Math] FOM topos theory qua fom; topos theory qua pure math. Maybe topos theoryis to be viewed as simply a tool or technique in pure mathematics. http://www.cs.nyu.edu/pipermail/fom/1998-January/000831.html

FOM: topos theory qua f.o.m.; topos theory qua pure math] Colin Mclarty cxm7@po.cwru.edu (Colin Mclarty) Thu, 15 Jan 1998 17:29:09 -0500 (EST) Previous message: FOM: Realism/Philosophy Next message: FOM: topos theory qua f.o.m.; topos theory qua pure math Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Reply to message from simpson@math.psu.edu Let me try to summarize the current state of the discussion regarding topos theory qua f.o.m. (= foundations of mathematics). I started the discussion by asking about real analysis in topos theory. McLarty claimed that there is no problem about this. After a lot of back and forth, it turned out that the basis of McLarty's claim is that the topos axioms plus two additional axioms give a theory that is easily intertranslatable with Zermelo set theory with bounded comprehension and choice. No, not at all. The basis of my claim was that you can do real analysis in any topos with a natural number object. In that generality the results are far weaker than in ZF (even without the axiom of choice)and allow many variant extensions with various uses. If you want to copy the ZFC case pretty nearly (so closely that only a logican could tell the difference) then you will want a topos with numbers and choice. To copy the ZFC case exactly you want a topos with numbers, choice, and replacement. Now some people will say "See, topos foundations just copy set theory!". A more apt conclusion would be "If you ask for copies of set theory, then yestopos foundations can give them". >

54. PLT Online (link). Wesley Phoa. An introduction to fibrations, topos theory, the effectivetopos and modest sets. (gzipped PS). Andrew M. Pitts. Lecture Notes on Types. http://www.cs.uu.nl/people/franka/ref

PLT Online Programming language theory texts online This is a collection of programming language theory texts and resources, all of which are freely available over the Internet. Many valuable reference texts on programming language theory, previously only available in paper form, have in recent years become publicly accessible from the net. I list here the ones I know of; below that you will also find a much broader list of lecture notes and tutorials other interesting reading , plus a collection of related resources Newest additions 13 May 2004: Posted to Resources University of Cambridge Computer Laboratory Theory Mini-Courses 13 May 2004: Posted to Books Kees Doets and Jan van Eijck The Haskell Road to Logic, Maths and Programming. link 13 May 2004: Posted to Books E.C.R. Hehner A Practical Theory of Programming. link 13 May 2004: Posted to Books Eric G. Wagner Universal Algebra for Computer Scientists. link 26 Feb 2004: Posted to Notes Peter D. Mosses Fundamental Concepts and Formal Semantics of Programming Languages: An Introductory Course. link 26 Feb 2004: Posted to Books Kenneth Slonneger and Barry L. Kurtz.

55. From Baez@galaxy.ucr.edu (John Baez) Subject Topos Theory For From baez@galaxy.ucr.edu (John Baez) Subject topos theory for physicists Date28 Dec 2000 191247 GMT Newsgroups sci.physics.research Summary missing http://www.math.niu.edu/~rusin/known-math/00_incoming/topos

From: baez@galaxy.ucr.edu (John Baez) Subject: Topos theory for physicists Date: 28 Dec 2000 19:12:47 GMT Newsgroups: sci.physics.research Summary: [missing] In article

56. Research Areas Research Areas Category theory, topos theory, topology. It has natural connectionswith topos theory. See the Seminar of the Rough Set Technology Lab. http://www.math.uregina.ca/~funk/research.html

Research Areas Category theory, topos theory, topology Current Research Toposes and rough set theory: The idea of a rough set comes from computer science. It has natural connections with topos theory. See the Seminar of the Rough Set Technology Lab Inverse semigroups and etendue: Joint with David Cowan. Braid group orderings: Patrick Dehornoy has discovered that an Artin braid group carries a left-invariant linear ordering. In my article ``The Hurwitz action and braid group orderings,'' Theory and Applications of Categories , Vol. 9 (2001), No. 7, pp 121-150, a ramified covering space is used to find a linear ordering of a countably generated free group, which is not invariant under the product in the free group, and an action of the countably generated Artin braid group in the free group that preserves the ordering. Cosheaf spaces and ramified covers: A theory of branched covers in topos theory, which is based on the ideas of Ralph Fox, is developed in ``On branched covers in topos theory,'' Theory and Applications of Categories , Vol. 7 (2000), pp 1-22. This approach uses complete spread geometric morphisms.

57. Home_page_workshop There were three short courses on Set Theory (Philip Welch), topos theory(Ieke Moerdijk) and Constructive Type Theory (Giovanni Sambin). http://www.science.unitn.it/~baratell/ftm2.html

Mini-Workshop on Foundational Theories in Mathematics Department of Mathematics University of Trento September 1214, 2002 We invite you to participate in the mini-workshop on Foundational Theories in Mathematics that will be held in Trento, Italy, September 1214, 2002 The workshop will take place at the Department of Mathematics of the University of Trento. In September 2001 a workshop on "Foundational Theories in Mathematics" took place at the Department of Mathematics of the University of Trento. There were three short courses on Set Theory (Philip Welch), Topos Theory (Ieke Moerdijk) and Constructive Type Theory (Giovanni Sambin). During that workshop our sensation was confirmed that an essential part was missing: the one of tying the different theoretical threads of Set Theory, Topos Theory and Constructive Type Theory even more together. For this reason we have invited John Bell to lecture on a Synthesis of the three different foundational theories. We have also invited three leading specialists: Keith Devlin (Set Theory - to be confirmed), Giuseppe Rosolini (Topos Theory) and Jan Smith (Constructive Type Theory) to participate with John Bell in two panel discussions and to share with the participants their views on the foundations of Mathematics. This second workshop on "Foundational Theories in Mathematics" does not presuppose participation in the first one; it can be attended quite independently of the latter.

58. Music And Mathematics From Topos Theory To Music Software Music and Mathematics From topos theory To Music Software . Prof.Guerino Mazzola. Universität Zürich ­ Duebendorf Institut für http://www.di.fc.ul.pt/~llf/etaps98/mazzola.html

"Music and Mathematics: From Topos Theory To Music Software" Prof. Guerino Mazzola We give an overview of mathematical music theory and first describe and exemplify topoi of denotators. These are special set-valued functors on the category of modules; they realize a general concept of music objects which meets requirements of mathematics and of data base management systems. We then outline the theory of transformation processes of symbolic music objects, as they are codified in traditional scores, into objects of physical reality, as they appear in artistic performance. Such processes rely on performance vector fields and involve mathematical music analyses of the given symbolic data, as well as performance grammars to shape performance fields as an expression of analytical facts. We discuss the theory, known results, and related problems. We conclude with an outlook on future research in mathematical music theory.

59. Abstract:010801bunge In this lecture I intend to introduce the notion of a (Fox complete)spread in topology and then in topos theory. This requires http://www.maths.usyd.edu.au:8000/u/stevel/auscat/abstracts/010808bunge.html

Spreads and their completions Marta Bunge (1/8/01) The notion of a (complete) spread was introduced by R.H. Fox [6] in topology in order to give a common generalization of two different types of coverings with singularities (branched and folded). A different notion of a (proper) spread was given by E. Michael [10] in connection with topological cuts. In both cases, the basic ideas is that of a spread , meaning a continuous map p:Y>X , with Y locally connected, satisfying the property that the connected components (or more generally, the clopen subsets) of the p (U) , for U the opens of X , form a base for the topology of Y A topos-theoretic version of the notion of a spread was given by J. Funk and myself [3] as that of a geometric morphism p:F>E between toposes bounded over a base topos S , with F locally connected, for which there is a generating family a:f>p (E) of F over E which is an S-definable The two types of completions (Fox [6] and Micheal [10]) have a topos-theoretic counterpart, which, unlike the notion of a spread, are far from obvious. We deal with the Fox-like completion in [3] and with the Michael-type completion in [5]. An instance of complete spreads are the branched coverings, introduced in this context in [4] and (with minor variations) in [8]. On the other hand, the notion of folded cover has not been defined for toposes.

60. Peter Johnstone Peter Johnstone, Open/compact duality in topos theory Abstract Itis well known that, whilst the closed subsets of a topological http://www.mimuw.edu.pl/TARSKI/abstracts/johnstone.html

Peter Johnstone, Open/compact duality in topos theory Abstract: It is well known that, whilst the closed subsets of a topological space may be regarded as the formal duals of its open subsets, when one considers continuous mappings between spaces the correct dual of the class of open maps is (not the class of closed maps, but) the class of `relatively compact maps', commonly known as proper maps. In the late 1940's Alfred Tarski, in collaboration with J.C.C. McKinsey, showed that much of general topology could be reduced to algebra by the use of a notion of `formal space' based on lattices of closed sets; this was an important forerunner of the modern theory of locales (or frames), although workers in locale theory traditionally take open-set lattices as the primitive notion. In 1994, Japie Vermeulen in a ground-breaking paper `rediscovered the closed-set lattices', and showed that one could use them as the basis for a completely formal duality between open and proper maps of locales, whereby results proved for one class could be easily translated into results about the other. More recently, Vermeulen

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