Subject: Re: Poincare Conjecture appears to be solved Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) Originator: israel@math.ubc.ca (Robert Israel) >According to a report in today's Boston Globe [...] >it looks like Perelman has solved the Poincare Conjecture, and >possibly the stronger Geometrization Conjecture. Specifically, Milnor refers to a remarkable trio of preprints and to further details to be provided in a fourth preprint. The Los Alamos ArXiv currently has four preprints by Perelman, but the fourth one seems to be on a different topic from the first three. Am I right in assuming that the fourth preprint that Milnor refers to is not yet available, and that therefore Perelman has not yet released his complete proof to the public? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: Poincare Conjecture appears to be solved Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) Content-Length: 1122 Originator: rusin@vesuvius >The Los Alamos ArXiv currently has four preprints by Perelman, but >the fourth one seems to be on a different topic from the first three. >Am I right in assuming that the fourth preprint that Milnor refers to >is not yet available, and that therefore Perelman has not yet released >his complete proof to the public? It has been pointed out to me that the preprint that I said seems to be on a different topic is in fact by a different author, whose last name also is Perelman. Sorry for the gaffe. Anyway, the question still stands...is it true that the complete proof has not yet been made available to the public? This would explain why there hasn't been a final verdict yet from the experts even though it's been such a long time since the result was announced. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: Poincare Conjecture appears to be solved Content-Length: 1731 Originator: rusin@vesuvius I don't think it is taking any longer than it did for Wiles's proof of Fermat's Last Theorem. Or for Hales's proof of the Kepler Conjecture. According to people I've consulted, Perelman's proof pushes the limits of what is known in geometry and topology. So it will take some time before people have worked through the proof carefully enough to say with confidence that the proof is correct and complete. The feedback I get is that there is nothing obviously wrong about Perelman's proof and that he has introduced new ideas that appear to overcome much and maybe all of the missing pieces to Hamilton's program for proving the geometrization conjecture. But that's still far from enough. Perelman's papers are relatively short, so it will take some time before people can decide whether they can fill in all the details. >>The Los Alamos ArXiv currently has four preprints by Perelman, but >>the fourth one seems to be on a different topic from the first three. >>Am I right in assuming that the fourth preprint that Milnor refers to >>is not yet available, and that therefore Perelman has not yet released >>his complete proof to the public? > It has been pointed out to me that the preprint that I said seems > to be on a different topic is in fact by a different author, whose > last name also is Perelman. Sorry for the gaffe. Anyway, the question > still stands...is it true that the complete proof has not yet been made > available to the public? This would explain why there hasn't been a > final verdict yet from the experts even though it's been such a long > time since the result was announced. === Subject: Hales's proof of Kepler Conjecture (was Re: Poincare Conjecture appears to be solved) Content-Length: 1232 Originator: rusin@vesuvius > I don't think it is taking any longer than it did for Wiles's proof of > Fermat's Last Theorem. Or for Hales's proof of the Kepler Conjecture. It sounds like you are saying that Hales's proof was verified? I'm very confused about the status of this proof since the last I heard, the Annals of Mathematics decided either not to publish it, being too hard to check, or will publish it with a disclaimer. It hasn't been published yet, so will it be published this year? (http://www.maa.org/devlin/devlin_06_03.html) which includes the following quote by MacPherson (editor of Annals): The news from the referees is bad, from my perspective. They have not been able to certify the correctness of the proof, and will not be able to certify it in the future, because they have run out of energy to devote to the problem. This is not what I had hoped for. the computational details of the proof, estimated to take 20 work years. Project homepage is http://www.math.pitt.edu/~thales/flyspeck/index.html === Subject: Re: Hales's proof of Kepler Conjecture (was Re: Poincare Conjecture appears to be solved) Content-Length: 1008 Originator: rusin@vesuvius > (http://www.maa.org/devlin/devlin_06_03.html) which includes the > following quote by MacPherson (editor of Annals): > The news from the referees is bad, from my perspective. They have not > been able to certify the correctness of the proof, and will not be able > to certify it in the future, because they have run out of energy to > devote to the problem. This is not what I had hoped for. > the computational details of the proof, estimated to take 20 work > years. IMHO, MacPherson has every right to reject the submission, and require the author to rewrite and resubmit. Annals is a prestigious journal, and should not feel initimidated into accepting unverified material, as a certain other well-known journal with a Latin name has done in the past. ZTW === Subject: Re: Hales's proof of Kepler Conjecture (was Re: Poincare Conjecture appears to be solved) Content-Length: 1570 Originator: rusin@vesuvius I definitely did not say that Hales's proof was verified. I just said that verification of Perelmen's proof is taking no longer that verification of Hales's proof. In fact, as you point out, verification of Hales's proof is probably going to take a long time. >>I don't think it is taking any longer than it did for Wiles's proof of >>Fermat's Last Theorem. Or for Hales's proof of the Kepler Conjecture. > It sounds like you are saying that Hales's proof was verified? I'm > very confused about the status of this proof since the last I heard, > the Annals of Mathematics decided either not to publish it, being too > hard to check, or will publish it with a disclaimer. It hasn't been > published yet, so will it be published this year? > (http://www.maa.org/devlin/devlin_06_03.html) which includes the > following quote by MacPherson (editor of Annals): > The news from the referees is bad, from my perspective. They have not > been able to certify the correctness of the proof, and will not be able > to certify it in the future, because they have run out of energy to > devote to the problem. This is not what I had hoped for. > the computational details of the proof, estimated to take 20 work > years. > Project homepage is > http://www.math.pitt.edu/~thales/flyspeck/index.html === Subject: Paper published by Algebraic and Geometric Topology Originator: israel@math.ubc.ca (Robert Israel) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-46.abs.html Title: Cell-like resolutions preserving cohomological dimensions Author(s): Michael Levin Abstract: We prove that for every compactum X with dim_Z X <= n >= 2 there is a cell-like resolution r: Z --> X from a compactum Z onto X such that dim Z <= n and for every integer k and every abelian group G such that dim_G X <= k >= 2 we have dim_G Z <=k. The latter property implies that for every simply connected CW-complex K such that e-dim X <= K we also have e-dim Z <= K. Keywords: Cohomological dimension, cell-like resolution Author(s) address(es): Department of Mathematics Ben Gurion University of the Negev P.O.B. 653 Be'er Sheva 84105, ISRAEL Email: mlevine@math.bgu.ac.il === Originator: israel@math.ubc.ca (Robert Israel) ILLC Scientific Publications ---------------------------- This document contains the titles of the reports that were published by the Institute for Logic, Language and Computation (ILLC) this year. All ILLC reports are available from the ILLC bureau: ILLC Bureau University of Amsterdam Plantage Muidergracht 24 NL-1018 TV Amsterdam The Netherlands Many reports are also electronically available, by WWW at http://www.illc.uva.nl/Publications and or FTP at ftp://ftp.science.uva.nl/pub/theory/illc/researchreports/ The ILLC bureau may be contacted by email, at illc@science.uva.nl Reports are numbered Series-Year-Number, where `Series' is one of PP = Prepublication Series MoL = Master of Logic Thesis ---------------------------------------------------------------------- Title: A Note on Modeling Theories Author: Johan van Benthem Title: Structural Properties of Dynamic Reasoning Author: Johan van Benthem Title: Is There Still Logic in Bolzano's Key? Author: Johan van Benthem Title: The Epistemic Logic of IF Games Author: Johan van Benthem Title: What Logic Games are Trying to Tell Us Author: Johan van Benthem Title: Rational Dynamics and Epistemic Logic in Games Author: Johan van Benthem Title: 'One is a Lonely Number': on the Logic of Communication Author: Johan van Benthem Title: Categorial Grammar at a Cross-Roads Author: Johan van Benthem Title: Conditional Probability and Update Logic Author: Johan van Benthem Title: Tableaux for Quantified Hybrid Logic Author: Patrick Blackburn, Maarten Marx Title: Silver Measurability and its Relation to other Regularity Properties Author: Jorg Brendle, Lorenz Halbeisen, Benedikt Lowe Title: The Pointwise View of Determinacy: Arboreal Forcings, Measurability and Weak Measurability Author: Benedikt Lowe Title: Canonical varieties with no canonical axiomatisation Author: Ian Hodkinson, Yde Venema Title: A Hierarchy of norms defined via Blackwell games Author: Benedikt Lowe Title: Stone Coalgebras Author: Clemens Kupke, Alexander Kurz, Yde Venema Title: The Pragmatic Dimension of Indefinites Author: Paul Dekker Title: Extending ILM with an operator for $Sigma_1$-ness Author: Evan Goris Title: The Simulation Technique and its Consequences for Infinitary Combinatorics under the Axiom of Blackwell Determinacy Author: Benedikt Lowe Title: Determinacy for infinite games with more than two players with preferences Author: Benedikt Lowe Title: The Categorial Fine-Structure of Natural Language Author: Johan van Benthem Title: Logic and the Dynamics of Information Author: Johan van Benthem Title: What One May Come to Know Author: Johan van Benthem Title: Optimal Interpolation in ALC Author: Stefan Schlobach Title: Monotonic Modal Logics Author: Helle Hvid Hansen Title: All normal extensions of S5-squared are finitely axiomatizable Author: Nick Bezhanishvili, Ian Hodkinson Title: Erd.9as graphs resolve Fine's canonicity problem Author: R. Goldblatt, I. Hodkinson, Y. Venema Title: Explaining New Phenomena in Terms of Previous Phenomena Author: Rens Bod Title: Some Intuitionistic Provability and Preservativity Logics (and their interrelations) Author: Chunlai Zhou Title: A Study of Stemming Effects on Information Retrieval in Bahasa Indonesia Author: Fadillah Tala Title: A Combined System for Update Logic and Belief Revision Author: Guillaume Aucher Title: Source Code Retrieval using Conceptual Graphs Author: Gilad Mishne === Subject: This Week's Finds in Mathematical Physics (Week 200) Originator: baez@math-cl-n01.math.ucr.edu (John Baez) Originator: israel@math.ubc.ca (Robert Israel) Also available at http://math.ucr.edu/home/baez/week200.html This Week's Finds in Mathematical Physics - Week 200 John Baez Happy New Year! I'm making some changes in my life. For many years I've dreamt of writing a book on higher-dimensional algebra that will explain n-categories and their applications to homotopy theory, representation theory, quantum physics, combinatorics, logic - you name it! It's an intimidating goal, because every time I learn something new about these subjects I want to put it in this imaginary book, so it keeps getting longer and longer in my mind! Actually writing it will require heroic acts of pruning. But, I want to get started. It'll be freely available online, and it'll show up here as it materializes - but so far I've just got a tentative outline: 1) John Baez, Higher-Dimensional Algebra, http://math.ucr.edu/home/baez/hda.html Unfortunately, I'm very busy these days. As you get older, duties accumulate like barnacles on a whale if you're not careful! When I started writing This Week's Finds a bit more than ten years ago, I was lonely and bored with plenty of time to spare. My life is very different now: I've got someone to live with, a house and a garden that seem to need constant attention, a gaggle of grad students, and too many invitations to give talks all over the place. In short, the good news is I'm never bored and there's always something fun to do. The bad news is there's always TOO MUCH to do! So, a while ago I decided to shed some duties and make more time for things I consider really important: thinking, playing the piano, writing this book... and yes, writing This Week's Finds. First I quit working for all the journals I helped edit. Then I started job it's really fun to quit. But doing so didn't free up nearly enough time. So now I've also decided to stop moderating the newsgroup This is painful, because I've learned so much from this newsgroup over the last 10 years, met so many interesting people, and had such fun. I thank everyone on the group. I'll miss you! I'll probably be back whenever I get lonely or bored. Ahem. Before I get weepy and nostalgic, I should talk about some math. This November in Florence there was a conference in honor of the 40th anniversary of Bill Lawvere's Ph.D. thesis - a famous thesis called Functorial Semantics of Algebraic Theories, which explored the applications of category theory to algebra, logic and physics. There are videos of all the talks on the conference website: 2) Ramifications of Category Theory, http://ramcat.scform.unifi.it/ but right now this website seems to be down. This conference was organized and funded by Michael Wright, a businessman with a great love of mathematics and philosophy, so it was appropriate that it was held in the old city of Cosimo de Medici, Renaissance banker and patron of scholars. And since there were talks both by mathematicians and philosophers - especially Alberto Peruzzi, a philosopher at the University of Florence who helped run the show - I couldn't help but remember Cosimo's Platonic Academy, which spearheaded the rebirth of classical learning in Renaissance Italy. When not attending talks, I spent a lot of time roaming around twisty old streets, talking category theory at wonderful restaurants, reading The Rise and Fall of the House ofMedici, and desperately trying to soak up the overabundance of incredible art and architecture: the Ponte Vecchio, the Piazza del Duomo, the Santa Croce where everyone from Galileo to Dante to Machiavelli is buried.... Ahem. Math! What was Lawvere's thesis about? It's never been published, so I've never read it - though I hear it's going to be. So, my impression of its contents comes from gossip, rumors and later research that refers to his work. Lawvere started out as a student of Clifford Truesdell, working on continuum mechanics, which is the very practical branch of field theory that deals with fluids, elastic bodies and the like. In the process, Lawvere got very interested in the foundations of physics, particularly the notions of continuum and physical theory. Somehow he decided that only category theory could give him the tools to really make progress in understanding these notions. After all, this was the 1960s, and revolution was in the air. So, he somehow got himself sent to Columbia University to learn category theory from Sam Eilenberg, In my own education I was fortunate to have two teachers who used the term foundations in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplified by their work in Foundations of Algebraic Topology, published in 1952 by Eilenberg (with Steenrod), and The Mechanical Foundations of Elasticity and Fluid Mechanics, published in the same year by Truesdell. The orientation of these works seemed to be concentrate the essence of practice and in turn use the result to guide practice. It may seem like a big jump from the down-to-earth world of continuum mechanics to category theory, but to Lawvere the connection made perfect sense - and while I've always found his writings inpenetrable, after hearing him give four long lectures in Florence I think it makes sense to me too! Let's see if I can explain it. Lawvere first observes that in the traditional approach to physical theories, there are two key players. First, there are concrete particulars - like specific ways for a violin string to oscillate, or specific ways for the planets to move around the sun. Second, there are abstract generals: the physical laws that govern the motion of the violin string or the planets. In traditional logic, an abstract general is called a theory, while a concrete particular is called a model of this theory. A theory is usually presented by giving some mathematical language, some rules of deduction, and then some axioms. A model is typically some sort of map that sends everything in the theory to something in the world of sets and truth values, in such a way that all the axioms get mapped to true. Since theories involve playing around with symbols according to fixed rules, the study of theories is often called syntax. Since the meaning of a theory is revealed when you look at its models, the study of models is called semantics. The details vary a lot depending on what you want to do, and physicists rarely bother to formulate their theories axiomatically, but this general setup has been regarded as the ideal of rigor ever since the work of Bolzano, Frege, Peano and Russell around the turn of the 20th century. And this is what Lawvere wanted to overthrow! Actually, I'm sort of kidding. He didn't really want to overthrow this setup: he wanted to radically build on it. First, he wanted to free the notion of model from the chains of set theory. In other words, he wanted to consider models not just in the category of sets, but in other categories as well. And to do this, he wanted a new way of describing theories, which is less tied up in the nitty-gritty details of syntax. To see what Lawvere did, we need to look at an example. But there are so many examples that first I should give you a vague sense of the *range* of examples. You see, in logic there are many levels of what you might call strength or expressive power, ranging from wimpy languages that don't let you say very much and deduction rules that don't let you prove very much, to ultra-powerful ones that let you do all sorts of marvelous things. Near the bottom of this hierarchy there's the propositional calculus where we only get to say things like ((P implies Q) and (not Q)) implies (not P) Further up there's the first-order predicate calculus, where we get to say things like for all x (for all y ((x = y and P(x)) implies P(y))) Even further up, there's the second-order predicate calculus where we get to quantify over predicates and say things like for all x (for all y (for all P (P(x) iff P(y)) implies x = y)) Etcetera... And, while you might think it's always best to use the most powerful form of logic you can afford, this turns out not to be true! One reason is that the more powerful your logic is, the fewer categories theories expressed in this logic can have models in. This point may sound esoteric, but the underlying principle should be familiar. Which is better: a hand-operated drill, an electric drill, or a drill press? A drill press is the most powerful. But I forgot to mention: you're using it to board up broken windows after a storm. You can't carry a drill press around, so now the electric drill sounds best. But another thing: this is in rural Ghana! With no electricity, now the hand-operated drill is your tool of choice. In short, there's a tradeoff between power and flexibility. Specialized tools can be powerful, but they only operate in a limited context. These days we're all painfully aware of this from using computers: fancy software only works in a fancy environment! Lawvere has even come up with a general theory of how this tradeoff works in mathematical logic... he called this the theory of doctrines. But I'm getting way ahead of myself! He came up with doctrines in 1969, and I'm still trying to explain his 1963 thesis. Just like traditional logic, Lawvere's new approach to logic has been studied at many different levels in the hierarchy of strength. He began fairly near the bottom, in a realm traditionally occupied by something called universal algebra, developed by Garrett Birkhoff in 1935. The idea here was that a bunch of basic mathematical gadgets can be defined using very simple axioms that only involve n-ary operations on some set and equations between different ways of composing these operations. A theory like this is called an algebraic theory. The axioms for an algebraic theory aren't even allowed to use words like and, or, not or implies. Just equations. Okay, now for an example. A good example is the algebraic theory of groups. A group is a set equipped with a binary operation called multiplication, a unary operation called inverse, and a nullary operation (that is, a constant) called the unit, satisfying these equational laws: (gh)k = g(hk) ASSOCIATIVITY 1g = g LEFT UNIT LAW g1 = g RIGHT UNIT LAW g^{-1}g = 1 LEFT INVERSE LAW gg^{-1} = 1 RIGHT INVERSE LAW Such a primitive gadget is robust enough to survive in very rugged environments... it's more like a stone tool than a drill press! Lawvere noticed that we can talk about models of these axioms not just in the category of sets, but in any category with finite products. The point is that to talk about an n-ary operation, we just need to be able to take the product of an object G with itself n times and consider a morphism f: G x ... x G -> G |- n times -| For example, the category of smooth manifolds has finite products, so we can talk about a group object in this category, which is just a *Lie group*. The category of topological spaces has finite products, so we can talk about a group object in this category too: it's a *topological group*. And so on. But Lawvere's really big idea was that there's a certain category with finite products whose only goal in life is to contain a group object. To build this category, first we put in an object G Since our category has finite products this automatically means it gets objects 1, G, G x G, G x G x G, and so on. Next, we put in a binary operation called multiplication, namely a morphism m: G x G -> G We also put in a unary operation called inverse: inv: G -> G and a nullary operation called the unit: i: 1 -> G And then we say a bunch of diagrams commute, which express all the axioms for a group listed above. Lawvere calls this category the theory of groups, Th(Grp). The object G is just like a group - but not any *particular* group, since its operations only satisfy those equations that hold in *every* group! By calling this category a theory, Lawvere is suggesting that like a theory of the traditional sort, it can have models - and indeed it can! A model of theory of groups in some category X with finite products is just a product-preserving functor F: Th(Grp) -> X By the way things are set up, this gives us an object F(G) in C, together with morphisms F(m): F(G) x F(G) -> F(G) F(inv): F(G) -> F(G) F(i): F(1) -> F(G) that serve as the multiplication, inverse and identity element for F(G)... all making a bunch of diagrams commute, that express the axioms for a group! So, a model of the theory of groups in X is just a group object in X. Whew. So far I've just explained the *title* of Lawvere's PhD thesis: Functorial Semantics of Algebraic Theories. In Lawvere's approach, an algebraic theory is given not by writing down a list of axioms, but by specifying a category C with finite products. And the semantics of such theories is all about product-preserving functors F: C -> X. Hence the term functorial semantics. Lawvere did a lot starting with these ideas. Let me just briefly summarize, and then move on to his work on topos theory and mathematical physics. Wise mathematicians are interested not just in models, but also the homomorphisms between these. So, given an algebraic theory C, Lawvere defined its category of models in X, say Mod(C,X), to have product-preserving functors F: C -> X as objects and natural transformations between these as morphisms. For example, taking C to be the theory of groups and X to be the category of sets, we get the usual category of groups: Mod(Th(Grp),Set) = Grp That's reassuring, and that's how it always works. What's less obvious, though, is that one can always recover C from Mod(C,Set) together with its forgetful functor to the category of sets. In other words: not only can we get the models from the theory, but we can also get back the theory from its category of models! I explained how this works in week136 so I won't do so again here. This result actually generalizes an old theorem of Birkhoff on universal algebra. But fans of the Tannaka-Krein reconstruction theorem for quantum groups will recognize this duality between theories and their category of models as just another face of the duality between algebras and their category of representations - the classic example being the Fourier transform and inverse Fourier transform! And this gives me an excuse to explain another bit of Lawvere's jargon: while a theory is an abstract general, and particular model of it is a concrete particular, he calls the category of *all* its models in some category a concrete general. For example, Th(Grp) is an abstract general, and any particular group is a concrete particular, but Grp is a concrete general. I mention this mainly because Lawvere flings around this trio of terms quite a bit, and some people find them off-putting. There are lots of reasons to find his work daunting, but this need not be one. In short, we have this kind of setup: ABSTRACT GENERAL CONCRETE GENERAL theory models syntax semantics and a precise duality between the two columns! I would love to dig deeper in this direction - I've really just scratched the surface so far, and I'm afraid the experts will be disappointed... but I'm even more afraid that if I went further, the rest of you readers would drop like flies. So instead, let me say a bit about Lawvere's work on topos theory and physics. Most practical physics makes use of logic that's considerably stronger than that of algebraic theories, but still considerably weaker than what most of us have been brainwashed into accepting as our default setting, namely Zermelo-Fraenkel set theory with the axiom of choice. So if we want, we can do physics in a context less general than an arbitrary category with finite products, while still not restricting ourselves to the category of sets. This is where topoi come in - they're a lot like the category of sets, but vastly more general. Topos theory was born when Grothendieck decided to completely rewrite algebraic geometry as part of a massive plan to prove the Weil conjectures. Grothendieck was another revolutionary of the early 1960s, and he arrived at his concept of topos sometime around 1962. In 1963, Lawvere and Myles Tierney took this concept - now called a Grothendieck topos - and made it both simpler and more general, arriving at the present definition. Briefly put, a topos is a category with finite limits, exponentials, and a subobject classifier. But instead of saying what these words mean, I'll just say that this lets you do most of what you normally want to do in mathematics, but without the law of excluded middle or the axiom of choice. One of the many reasons this middle ground is so attractive is that it lets you do calculus with infinitesimals the way physicists enjoy doing it! Lawvere started doing this in 1967 - he called it synthetic differential geometry. Basically, he cooked up some axioms on a topos that let you do calculus and differential geometry with infinitesimals. The most famous topos like this is the topos of schemes - algebraic geometers use this one a lot. The usual category of smooth manifolds is not even a topos, but there are topoi that can serve as a substitute, which have infinitesimals. I won't list the axioms of synthetic differential geometry, but the main idea is that our topos needs to contain an object T called the infinitesimal arrow. This is a rigorous version of those little arrows physicists like to draw when talking about vectors: -----> The usual problem with these little arrows is that they need to be really tiny, but still point somewhere. In other words, the head can't be at a finite distance from the tail - but they can't be at the same place, either! This seems like a paradox, but one can neatly sidestep it by dropping the law of excluded middle - or in technical jargon, working with a non-Boolean topos. That sounds like a drastic solution - a cure worse than the disease, perhaps! - but it's really not so bad. Indeed, algebraic geometers are perfectly comfortable with the topos of schemes, and they don't even raise an eyebrow over the fact that this topos is non-Boolean - mainly because you're allowed to use ordinary logic to reason *about* a topos, even if its internal logic is funny. But enough logic! Let's do some geometry! Let's say we're in some topos with an infinitesimal arrow object, T. I'll call the objects of this topos smooth spaces and the morphisms smooth maps. How does geometry work in here? It's very nice. The first nice thing is that given any smooth space X, a tangent vector in X is just a smooth map f: T -> X that is, a way of drawing an infinitesimal arrow in X. In general, the maps from any object A of a topos to any other object B form an object called B^A - this is part of what we mean when we say a topos has exponentials. So, the space of all tangent vectors in X is X^T. And this is what people usually call the tangent bundle of X! So, the tangent bundle is pathetically simple in this setup: it's just a space of maps. This means we can compose a tangent vector f: T -> X with any smooth map g: X -> Y to get a tangent vector gf: T -> Y. This is what people usually call pushing forward tangent vectors. This trick gives a smooth map between tangent bundles, the differential of g, which it makes sense to call g^T: X^T -> Y^T Moreover, it's pathetically easy to check the chain rule: (gh)^T = g^T h^T And so far we haven't used *any* axioms about the object T - just basic stuff about how maps work! We can also define higher derivatives using T. For second derivatives we start with T x T, which looks like an infinitesimal square. Then we mod out by the map S_{T,T}: T x T -> T x T that switches the two factors. You should visualize this map as reflection across the diagonal. When we mod out by it, we get a quotient space that deserves the name T^2/2! and if we now use some axioms about T, it turns out that a smooth map f: T^2/2! -> X picks out what's called a second-order jet in X. This is a concept familiar from traditional geometry, but not as familiar as it should be. The information in a second-order jet consists of a point in X, the first derivative of a curve through X, and also the *second* derivative of a curve through X. Or in physics lingo: position, velocity and acceleration! We can go ahead and define nth-order jets using T^n/n! in a perfectly analogous way, and the visual resemblance to Taylor's theorem is by no means an accident... but let me stick to second derivatives, since I'm trying to get to Newton's good old F = ma. Just as the space of all tangent vectors in X is the tangent bundle X^T, the space of all 2nd-order jets in X is the 2nd-order jet bundle X^{T^2/2!} Using some axioms about T, we can show there is a smooth map T^2/2! -> T which throws out the second-order infinitesimal data and just keeps the first-order part. This gives a smooth map p_X: X^{T^2/2!} -> X^T from the 2nd-order jet bundle to the tangent bundle. Intuitively you can think of this as sending any position-velocity-acceleration triple, say (q,q',q), to the pair (q,q'). Now for the fun part: Lawvere defines a dynamical law to be a smooth map going the other way: s_X: X^T -> X^{T^2/2!} such that s_X followed by p_X is the identity. In other words, it's a way of mapping any position-velocity pair (q,q') to a triple (q,q',q). So, it's a formula for acceleration in terms of position and velocity! There is a category where an object is a smooth space equipped with a dynamical law and a morphism is a lawful motion: that is, a smooth map f: X -> Y that makes the obvious diagram commute: s_X X^T -------------> X^{T^2/2!} | | | | | | f^T | | f^{T^2/2!} | | | | | | V s_Y V Y^T -------------> Y^{T^2/2!} In particular, if we take R to be the real numbers - time - and equip it with the law saying q = 0 meaning that time ticks at an unchanging rate, then a lawful motion f: R -> X is precisely a trajectory in X that follows the law, meaning that the acceleration of the trajectory is the desired function of position and velocity. This example is a setup for the classical mechanics by replacing R by a higher-dimensional space. I'm sure many of you have the same impression that I had when seeing this stuff, namely that it's a bit quixotic for a high-powered mathematician to be reformulating the foundations of classical mechanics here at the turn of the 21st century, instead of working on something cutting-edge like string theory. Even if Lawvere's approach is better, one can't help but wonder if it gives truly *new* insights, or just a clearer formulation of existing ones. And either way, one can't help wonder: does he actually expect enough people to learn this stuff to make a difference? Does he really think topos theory can break the Microsoft-like grip that ordinary set theory has on mathematics? (Note the software analogy raising its ugly head again. Zermelo-Fraenkel set theory is a bit like the Windows operating system: once you're locked into it, it's hard to imagine breaking out. You use it because everyone else does and you're too lazy to do anything about it. Topos theory is more like the open source movement: you're welcome and even expected to keep tinkering with the code.) I have some sense of the answer to these questions. First of all, Lawvere wants to do math the right way regardless of whether it's popular. But secondly, he's been hard at work trying to make the subject accessible to beginners. He's recently written a couple of textbooks you don't need a degree in math to read: 3) F. William Lawvere and Steve Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge U. Press, Cambridge, 1997. 4) F. William Lawvere and Robert Rosebrugh, Sets for Mathematics, Cambridge U. Press, Cambridge, 2002. And third, the great thing about topos theory is that you don't need to accept it to profit from it. In math, what really matters is not believing the axioms but coming up with good ideas. Topos theory is full of good ideas, and these are bound to propagate. I'll finish off with some references to help you learn more about this stuff. Alas, I believe Lawvere's thesis is still lurking in the stacks at Columbia University: 5) F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963. and so far he's only gotten around to publishing a brief summary: 6) F. William Lawvere, Functorial semantics of algebraic theories, Proceedings, National Academy of Sciences, U.S.A. 50 (1963), 869-872. But, you can find expositions of his work on algebraic theories here and there. Here's a gentle one geared towards computer scientists: 7) Roy L. Crole, Categories for Types, Cambridge U. Press, Cambridge, 1993. A considerably more macho one is available free online: 8) Michael Barr and Charles Wells, Toposes, Triples and Theories. Springer-Verlag, New York, 1983. Available for free electronically at http://www.cwru.edu/artsci/math/wells/pub/ttt.html This book also talks about sketches, which are a way of syntactically presenting a category with finite products. It also serves as an introduction to topoi... umm, or at least toposes. I used to find it fearsomely difficult and dry. Now I don't, which is sort of scary. A really beautiful more advanced treatment of algebraic theories and also essentially algebraic theories can be found here: 9) Maria Cristina Pedicchio, Algebraic Theories, in Textos de Matematica: School on Category Theory and Applications, Coimbra, July 13-17, 1999, pp. 101-159. Someone should urge her to make this available online - it's already in TeX, and it deserves to be easier to get! Shortly after his thesis, Lawvere tackled topoi in this paper: 10) F. William Lawvere, Elementary theory of the category of sets, Proceedings of the National Academy of Science 52 (1964), 1506-1511. the like: 11) F. William Lawvere, Algebraic theories, algebraic categories, and algebraic functors, in Theory of Models, North-Holland, Amsterdam (1965), 413-418. 12) F. William Lawvere, Functorial semantics of elementary theories, Journal of Symbolic Logic, Abstract, 31 (1966), 294-295. 13) F. William Lawvere, The category of categories as a foundation for mathematics, in La Jolla Conference on Categorical Algebra, Springer, Berlin 1966, pp. 1-20. 14) F. William Lawvere, Some algebraic problems in the context of functorial semantics of algebraic theories, in Reports of the Midwest Category Seminar, eds. Jean Benabou et al, Springer Lecture Notes in Mathematics No. 61, Springer, Berlin 1968, pp. 41-61. Then came his work on doctrines, which I vaguely alluded to a while back: 15) F. William Lawvere, Ordinal sums and equational doctrines, Springer Lecture Notes in Mathematics No. 80, Springer, Berlin, 1969, pp. 141-155. I think he first published on synthetic differential geometry in Lawvere started publishing his ideas on mathematical physics in the late 1970s, though he must have been thinking about them all along: 17) F. William Lawvere, Categorical dynamics, in Proceedings of Aarhus May 1978 Open House on Topos Theoretic Methods in Geometry, Aarhus/Denmark (1979). 18) F. William Lawvere, Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body, Cahiers de Topologie et Geometrie Differentielle Categorique 21 (1980), 337-392. In 1981, Anders Kock came out with a textbook on synthetic differential geometry: 19) Anders Kock, Synthetic Differential Geometry, Cambridge U. Press, Cambridge, 1981. More recently, Lawvere came out with a book on applications of category theory to physics: 19) F. William Lawvere and S. Schanuel, editors, Categories in Continuum Physics, Springer Lecture Notes in Mathematics No. 1174, Springer, Berlin, 1986. The quote about Lawvere's teachers is from: 20) F. William Lawvere, Foundations and applications: axiomatization and at http://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps and this gives a good overview of his ideas, though not easy to read! Finally, Colin McLarty - whom I was delighted to meet in Florence - has a nice quick introduction to synthetic differential geometry in his textbook on categories and topos theory: 21) Colin McLarty, Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1995. Along with Lawvere's books Conceptual Mathematics and Sets for Mathematics, this is the one reference that's really good for beginners! Okay... now that everyone is gone except the people who are absolutely nuts about category theory, let me say a bit more about doctrines and theory-model duality. The nuts who are still reading are probably disappointed that I kept everything very gentle and expository and didn't drop any mind-blowing bombshells of abstraction, which is what they like about category theory! So, let's turn up the abstraction a few notches. What's a doctrine? Well, in week89 I described a monad in an arbitrary 2-category. But most of the time when people talk about monads they mean monads in Cat, the 2-category of all categories. These are the most important monads - but I've never really said what they're good for! I need to come clean and explain this now, since a doctrine is a categorified version of a monad. What monads are good for is to describe how objects in one category can be regarded as objects of some other category equipped with extra structure. This theme pervades mathematics, and is of the utmost importance. For example: groups are sets equipped with extra structure, abelian groups are groups equipped with extra structure, rings are abelian groups equipped with extra structure, and so on. We keep building up fancier gadgets from simpler ones. And pretty much whenever we do, there's a monad lurking in the background, running the show! Suppose we've got two categories C and D, and the objects of D are objects of C equipped with extra structure. Then we get a pair of adjoint functors: R: D -> C L: C -> D The right adjoint R sends each D-object to its underlying C-object, and the left adjoint L sends each C-object to the free D-object on it. Often R is called a forgetful functor. For example, if C = Set and D = Grp then we can take the underlying set of any group, and the free group on any set. We get a monad on C by letting T = LR: C -> C Then, we can use facts about adjoint functors to get natural transformations called multiplication m: TT => T and the unit i: 1_C => T Using more facts about adjoint functors, we can check that these satisfy associativity and the left and right unit laws. I did all this in week92 so I won't do it again here. The upshot is that T is a lot like a monoid - which is why Mac Lane dubbed it a monad. Now, monoids like to *act* on things, and the same is true for monads. It turns out that a monad T on C can act on any object of C. When this happens, we call that object an algebra of T, or a T-algebra for short. And when our monad comes from a pair of adjoint functors as above, the main way we get T-algebras is from objects of D. And in nice cases, T-algebras are the *same* as objects of D. So, for example, we can describe groups as T-algebras where T is some monad on the category of sets. And we can describe abelian groups as T-algebras where T is some monad on the category of groups. And we can describe rings as T-algebras where T is some monad on the category of abelian groups. And so on! To really see how this works, we'd need to look at a few examples. I remember when James Dolan was first teaching me this stuff in a little coffeeshop here in Riverside, which has since gone out of business. I considered monads too abstract and dug my heels in like a stubborn mule, refusing to learn about them - until I went through a bunch of examples and saw that *yes*, this monad business really *does* capture the essence of what it means to build up fancy gadgets from simple ones by adding extra structure! And by now I'm completely sold on it. One reason is the relation to topology, which I explained in part N of week118, and also week174. But alas, I'm too eager to get to the *really* cool stuff to work through examples right now. So if you're a complete novice at monads, you'll have to work out some examples yourself. Right now, I'll just say a bit of fancier stuff to fill in a couple gaps for the semi-experts. First, when I said in nice cases, I really meant that the category of T-algebras is equivalent to D when the forgetful functor R: D -> C is monadic. A bit more precisely: for any monad T on C there's a category of T-algebras, which is usually called C^T for some silly reason. And, whenever we have a pair of adjoint functors R: D -> C and L: C -> D, we get a monad T = LR and a functor from D to C^T. This is just a careful way of saying that any D-object gives us a T-algebra. And finally, we say that R is monadic if this functor from D to C^T is an equivalence of categories. There's a theorem by Beck that says how to tell when a functor is monadic, just by looking at it. Second, to make the analogy between monoids and monads precise, we just need to realize that a monad on C is a monoid object in the monoidal category hom(C,C). I already explained this in week92, in even greater generality than we need here, but we need this now because I'm about to categorify monads and get doctrines. Okay: so, monads are good for describing objects equipped with extra structure and properties. But now suppose we want to describe *categories* equipped with extra structure and properties! For example, the categories with finite products that I was talking about earlier, or topoi. There are LOTS of different interesting kinds of categories equipped with extra structure and properties, and each of them gives a different kind of *logic*: the logic that works inside this kind of category! The more structure and properties our category has, the more powerful logic we can use inside it. This is what gives the hierarchy of expressive power I was talking about. So, it pays to have a good general way to describe categories equipped with extra structure and properties. And this is what Lawvere's doctrines do! I've said how monads on a category C are good for describing objects of C equipped with extra structure and properties. But there's a certain category called Cat whose objects are categories! So, let's take C = Cat! A monad on Cat will describe categories equipped with extra structure and properties. And this is the simplest definition of doctrine: a monad on Cat. However, those of you familiar with n-categories will realize that it's odd to talk about the category of all categories. Not because of Russell's paradox - though that's a problem too, forcing us to talk about the category of *small* categories - but because what's really important is the 2-CATEGORY of all categories. It's best to think of Cat as a 2-category. But this suggests that we should work with a categorified, *weakened* version of monad when defining doctrines. For this, we need to categorify and weaken the concept of monad. People have done this, and the result is sometimes called a pseudomonad, but I prefer to call it a weak 2-monad, since I have dreams of categorifying further, and I don't want my notation to become ridiculous. I'd rather talk about weak 3-monads than pseudopseudomonads, wouldn't you? Furthermore, if you look up pseudomonad in the dictionary you'll get this: PSEUDOMONAD: bacterium usually producing greenish fluorescent water-soluble pigment; some pathogenic for plants and animals. Yuck! So, let's be very general and sketch how to define a weak 2-monad in any weak 3-category (aka tricategory). Given a weak 3-category C and an object c of C, a weak 2-monad on c is just a weak monoidal category object in hom(c,c). Huh? Well, hom(c,c) is a weak monoidal 2-category, which is precisely the right environment in which to define a weak monoidal category object, and that's what we're doing here. Start with the usual definition of a weak monoidal category, which is a gadget living in Cat. Cat is an example of a weak monoidal 2-category, and we can write down the same definition in *any* weak monoidal 2-category X, getting the concept of weak monoidal category object in X. Then, take X = hom(c,c). (Of course I'm lying slightly here: Cat is more strict than your average weak monoidal 2-category, so it may not be immediately obvious how to generalize the concept of weak monoidal category as I'm suggesting. Still, I claim it's not hard if you know about this stuff.) Now that you know how to define a weak 2-monad on any object c of a 3-category C, you can take c to be Cat and C to be 2Cat... and this is what we really should call a doctrine. Unsurprisingly, people often consider stricter versions of the concept of 2-monad and doctrine. For example, most people define their pseudomonads not in a weak 3-category but just a semistrict one, also known as a Gray-category - since 2Cat is one of these. For more details, try these papers: 22) R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, Jour. Pure Appl. Algebra 59 (1989), 1-41. 23) Brian Day and Ross Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997) 99-157. 24) F. Marmolejo, Doctrines whose structure forms a fully faithful adjoint string, Theory and Applications of Categories 3 (1997), 23-44. Available at http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html 23) S. Lack, A coherent approach to pseudomonads, Adv. Math. 152 (2000), 179-202. Also available at http://www.maths.usyd.edu.au:8000/u/stevel/papers/psm.ps.gz Anyway, suppose T is a doctrine. Then we get a 2-category of T-algebras Cat^T, whose objects we should think of as categories equipped with extra structure of type T. The classic example would be categories with finite products. Just as Lawvere thought of these as algebraic theories, we can think of *any* T-algebra as a theory of type T, and define its category of models: given T-algebras C and D, the category of models of C in D is hom(C,D), where the hom is taken in Cat^T. Depending on what doctrine T we consider, we get many different forms of logic, and I'll just list a few to whet your appetite: Cat^T = categories with finite products = algebraic theories gives what one might call algebraic logic - purely equational reasoning about n-ary operations. The theory of groups, or abelian groups, or rings lives here. Cat^T = symmetric monoidal categories gives a sort of logic that allows for theories known as operads and PROPs - see week191 for more. This doctrine is weaker than the previous one, since we can only use equations where all the same variables appear on both sides, with no duplications or deletions. If we go further in this direction we obtain various sorts of quantum logic. Cat^T = categories with finite limits = essentially algebraic theories gives what one might call essentially algebraic logic. This doctrine is strong than that of algebraic theories, since it allows partially defined operations that are defined only when some equations hold. The theory of categories lives here, since composition of morphisms is an operation of this sort. Cat^T = regular categories gives regular logic. This doctrine is even stronger, since it allows for theories that involve relations as well as n-ary operations. Cat^T = cartesian closed categories gives the typed lambda-calculus. This allows for operations on operations on operations... etc. Cat^T = topoi gives topos logic. The typed lambda-calculus is very popular in theoretical computer science, and I recommend Crole's book cited above for more about how it's related to cartesian closed categories. A good introduction to topos logic is McLarty's book cited above. For an exhaustive study of many other sorts of logic that should be on this list but aren't, I recommend part D of this book: 24) Peter Johnstone, Sketches of an Elephant: a Topos Theory Compendium, Oxford U. Press, Oxford. Volume 1, comprising Part A: Toposes as Categories, and Part B: 2-categorical Aspects of Topos Theory, 720 pages, 2002. Volume 2, comprising Part C: Toposes as Spaces, and Part D: Toposes as Theories, 880 pages, 2002. We can do a lot of fun stuff with all these different forms of logic, and people have indeed done so... but I think I'll stop here. My point is merely that higher category theory and logic go hand-in-glove, and there is plenty of room for exploration here, especially if we keep categorifying - and also keep trying to craft our logic to real-world applications, both in quantum theory and computer science. I wish you all a Happy New Year, and good luck on all your adventures. ----------------------------------------------------------------------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === Subject: Re: Left-invariant metric Content-Length: 561 Originator: rusin@vesuvius > So how about weakening the hypothesis, to allow for anti-automorphisms? It is easy to see that the universal covering G of E(2), the group of oriented isometries of the affine euclidean plane, acts simply transitively on R^3 as isometries, and hence has a flat metric isometric to the 3-dimensional euclidean space. So, endowed with this metric, G has a three-dimensional compact group of isometries, whereas G has no nontrivial compact subgroups. Of course, G is not simple of higher rank, so that this does not contredict the result quoted by L. Kramer. === Subject: Re: Left-invariant metric Content-Length: 947 Originator: rusin@vesuvius > It is easy to see that the universal covering G of E(2), the group of > oriented isometries of the affine euclidean plane, acts simply > transitively on R^3 as isometries, and hence has a flat metric > isometric to the 3-dimensional euclidean space. > So, endowed with this metric, G has a three-dimensional compact group > of isometries, whereas [...] Here is the right argument, replacing [...]. The maximal compact groups of (anti)automorphisms of G are 1-dimensional. Indeed, Aut(G) is isomorphic to the group of (non)oriented similitaries of the plane, and its compact maximal subgroups are isometry stabilisators of points in the plane, so they are isomorphic to O(2). Therefore, many isometries of G are not (anti)automorphisms. I point out that it is really difficult to find an account about the geometry of Lie groups endowed with a left-invariant metric; the only === Subject: Re: Left-invariant metric Originator: baez@math-cl-n01.math.ucr.edu (John Baez) Content-Length: 2015 Originator: rusin@vesuvius >> Let G be a connected (unimodular) Lie group, endowed with a left-invariant >> metric. Let u be an isometry such that u(1)=1. Does it follows that u is a >> group automorphism? >No. >Consider the group SU(2) and see it as the set of quaternions >with norm 1. I will denote by ||q|| the norm of a quaternion >q (i.e. ||q|| = sqrt(q*q^*), wher q^* stands for the conjugate >quaternion). Consider in SU(2) the distance d defined by > d(q,r) = ||q - r||. >Then d is left-invariant (BTW, it is also right-invariant), >since the quaternionic norm preserves the product. >Now take u(q) = q^* = q^(-1). Then u is an isometry and u(1) = >= 1, but u is not a group homomorphism. Indeed, this counterexample works for every compact simple Lie group. Every such group has a metric which is both left- and right-invariant. In fact, the metric with these properties is unique up to multiplication by a positive constant. It's defined using the so-called Killing form on the Lie algebra. Every automorphism is an isometry with respect to this metric. But, so is the map sending each group element to its inverse ... and this map is never an automorphism, since taking inverses is an automorphism only for abelian groups, and I'm using the usual definition of compact simple Lie group, which explicitly excludes abelian groups. It is, however, true that if our compact simple Lie group is connected, and we use any metric that's both left- and right- invariant, every isometry mapping the identity to itself is *either* an automorphism *or* the composite of an automorphism and taking inverses. I should also warn Cornulier that we can easily find a compact Lie group with a metric on it that's both left- and right- invariant, and an automorphism of this group that's not an isometry. (Hint: use the torus.) So, the answer to the converse of his question is also No. === Subject: Re: Set Theoretic Statement Equivalent to the Existence of the Algebraic Closure Content-Length: 1397 Originator: rusin@vesuvius >> I've been told, that the existence of the algebraic closure of a >> field doesn't require full AC, but requires some. Is there a set >> theoretic formulation of the weaker version of AC, that is equivalent >> to the fact, that every field has an alegbraic closure? >You might want to look at the book The Axiom of Chioce by Thomas J. >Jech published in 1973 by North-Holland. It is out of print now, but >you may be >able to find a copy. >Many of the proofs are brief outlines. It discusses weaker versions of >the AC. >Hope this helps. Another place to look is the book by Paul Howard and Jean Rubin, _Consequences of the Axiom of Choice_. It seems to me that the Prime Ideal Theorem would be adequate, as the finite sets generated by the solutions of equations have the property that any finite number of them can be put together consistently, and thus the Tychonov Compactness Theorem for Hausdorff spaces would give a common solution. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Inequality with complex numbers Epigone-thread: jantorverm Originator: rusin@vesuvius let z(k)=exp(2*k*Pi*i/n); k=0,1, ... ,n-1 ,and f,g functions such that 1<=|f(0)|<=|f(1)|<= ... <=|f(n-1)|, and, 1<=g(0)<=g(1)<= ... <=g(n-1); A(n):=Sum[z(k)*f(k),{k,0,n-1}] B(n):=Sum[z(k)*f(k)*g(k),{k,0,n-1}] C(n):=Sum[z(k)*f(k)*(g(k)^2),{k,0,n-1}]. It seems to me that |B(n)-A(n)|<=|C(n)-A(n)| and |B(n)-C(n)|<=|C(n)-A(n)| for every n,f,g. === Subject: Re: Inequality with complex numbers Content-Length: 1132 Originator: rusin@vesuvius If I understand correctly... Let z(0)*f(0) = 1 z(1)*f(1) = (1 + i)/sqrt(2) z(2)*f(2) = (-2 - i)/sqrt(3) g(0) = sqrt(2) g(1) = sqrt(1 + sqrt(2)) g(2) = sqrt(1 + sqrt(3)) Then, (C(n) - A(n)) = 0, but not (B(n) - A(n)) or (B(n) - C(n)) unless I have calculated wrong. I have assumed that f is a complex sequence, whereas g is a real sequence. It seems to me that you can easily find a series of vectors of the form (z*f)*(g^2-1) that sums to zero by carefully choosing the magnitudes and phases. Pardon me if I have totally misunderstood the problem. Arnab > let z(k)=exp(2*k*Pi*i/n); k=0,1, ... ,n-1 ,and > f,g functions such that 1<=|f(0)|<=|f(1)|<= ... <=|f(n-1)|, > and, 1<=g(0)<=g(1)<= ... <=g(n-1); > A(n):=Sum[z(k)*f(k),{k,0,n-1}] > B(n):=Sum[z(k)*f(k)*g(k),{k,0,n-1}] > C(n):=Sum[z(k)*f(k)*(g(k)^2),{k,0,n-1}]. > It seems to me that |B(n)-A(n)|<=|C(n)-A(n)| and > |B(n)-C(n)|<=|C(n)-A(n)| for every n,f,g. === Subject: Re: Long division of large numbers Content-Length: 628 Originator: rusin@vesuvius > I've come up with a long division algorithm that is O(n^1.5) for large > intgers, as compared to O(n^2) for brute force. My Internet search > for similar algorithms has produced nothing. > A question: > Are there any known long division algorithms that perform better than > O(n^2), either published or proprietary? > Ed Schreiber > ed@estival.com > http://estival.com I haven't looked inside for myself but have you looked at the div method used in the bn (Big Number) code of OpenSSL? Just a thought. Good luck on your brute forcing. === Subject: Prediction of Profit/Loss Content-Length: 398 Originator: rusin@vesuvius Does anybody know if it is possible to calculate a distribution of the profit or loss from an option portfolio only knowing the greeks? e.g.: I have many different options for Vodafone and other stocks, I know all greeks from every option. Is it possible to calculate the probability of making a profit of, e.g. USD 1000, only from the greeks? I appreciate every help. dominik === Subject: Re: Prediction of Profit/Loss Content-Length: 1060 Originator: rusin@vesuvius > Does anybody know if it is possible to calculate a distribution of the > profit or loss from an option portfolio only knowing the greeks? Dominic - I'm assuming by the greeks, you mean the option's delta, lambda, gamma, theta, and kappa. Apart from these numbers you'll also need the purchase and current price for the option, the current underlying security price, the option exercise price, the time to expiration, forecast of the expected risk-free interest rate, cashflow data from the underlying security (for stocks, dividend schedule; for bonds, interest schedule), and a forecast for the underlying security volatility. You might be able to get away with estimating the implied risk-free rate from the composite data and the implied volatility using the options for the given security at any point in time, but when trying to then insert those implied parameters (both, actually distributions) back into the price formulae, I'd be worried about the predictions getting too mushy. faa === Subject: Re: Prediction of Profit/Loss Content-Length: 375 Originator: rusin@vesuvius Yes, this is possible and is called Delta-Gamma-Approximation. Its a simple approximation of the P&L function via second order Taylor Series and the resulting polynomial turns out to be a sum of delta normal and noncentral chi squared distributed independent random variables.« Try the key word Delta-Gamma-Apporximation here : http://www.gloriamundi.org/ Stefan Reitz === Subject: Re: Prediction of Profit/Loss Content-Length: 1437 Originator: rusin@vesuvius > Does anybody know if it is possible to calculate a distribution of the > profit or loss from an option portfolio only knowing the greeks? > e.g.: > I have many different options for Vodafone and other stocks, I know all > greeks from every option. Is it possible to calculate the probability of > making a profit of, e.g. USD 1000, only from the greeks? > I appreciate every help. > dominik Not quite clear how you want to make profit without stocks moving :-) I guess you want something like that. The bad answer: no. The better answer: only approximately. If you really 'know' the delta ~ change of option price per stock movement then for small changes you can do so for each - if elapsing time is not to large. Usually you do not have exact greeks without commercial environment (due to concurrent market prices = smile and dividends, rates, tax) and no model which catches dynamic changes (ok, http://www.ito33.com/ claim they do ...). Thus you can not write down expectation values for an arbitrary time. This was for 1 stock. If you want it for a portfolio there is no precise notion of greeks for it - there is no agreed 'distribution' for that (not with correlations or better: all the underlying components can not be reduced to one variable). So even a good approx for 1 stock gives you a poor answer. But predicting ... never. --- remove the no to email me === Subject: Is finding an optimal schedule NP-hard? Content-Length: 417 Originator: rusin@vesuvius Consider an M/*/1 queue where each arrival has a deadline, and the server knows the arrival time, service time and deadline of each arrival, permitting it to act like a clairvoyant scheduler. Then - finding the non-preemptive schedule that minimizes the number of missed deadlines - is it NP hard? PS: I don't rule out the possibility that the system is overloaded i.e. average arrival rate > average service rate. === Subject: Which part of Mathematics do I have to learn? Content-Length: 1504 Originator: rusin@vesuvius Hello I want to try to give a formerly posted question from me another formulation. I want to define a (volume)measure for convex sets in a space with norm where the norm comes from a field with an absolute value. The measur should be related to the norm as for example the Lebesgue measure to the euclidean norm in IR^n. and I do have no idea how I can define such a measure in a senseful way. BTW - the field could be restricted by a few conditions (for example the product formula should hold) but I think this is not important at a first glance. What is important is the fact, that I do not want the field to be a subfield of the complex numbers and so - I think - for example nothing known about Hilbert spaces could be used. Now my question: Which part of mathematics do I have to learn to get an idea for that? Is it topology or functional analysis or measure theory or algebraic number theory or somthing else? While scanning the internet I found something about so called mm-spaces but I'm not sure if this can help me. Also I heard about something called ergodic theory but I'm not quite sure what this is. A friend told me I should read Weil's Basic Number Theory first. I will have a look at it but I'm afraid this book only covers the case of global (and local) fields and this is not what I'm interested in. So can anyone give me a tip in which part of mathematics I do have to search. I'm sure this would be very helpful for me. Holger === Subject: Re: Which part of Mathematics do I have to learn? Content-Length: 1445 Originator: rusin@vesuvius >Hello >I want to try to give a formerly posted question from me another >formulation. >I want to define a (volume)measure for convex sets in a space with norm >where the norm comes from a field with an absolute value. The measur >should be related to the norm as for example the Lebesgue measure to the >euclidean norm in IR^n. and I do have no idea how I can define such a >measure in a senseful way. BTW - the field could be restricted by a few >conditions (for example the product formula should hold) but I think >this is not important at a first glance. What is important is the fact, >that I do not want the field to be a subfield of the complex numbers and >so - I think - for example nothing known about Hilbert spaces could be >used. The only complete Archimedean fields with an absolute value are the real numbers and the complex numbers. Non-Archimedean fields have |N| <= 1, where N is any integer. I do not see how there can be any reasonable notion of convexity not using the ordering of subsets of reals. Are the convex sets of any use without an Archimedean norm? -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Which part of Mathematics do I have to learn? Content-Length: 775 Originator: rusin@vesuvius [...] > I do not see how there can be any reasonable notion of convexity > not using the ordering of subsets of reals. Are the convex sets > of any use without an Archimedean norm? I called the sets convex because Mahler in his paper an analogue to Minkowski's geometry of numbers in a field of series did this. There he defines sets to be convex if they are of the type C := { X | F(X)<=t ) where F is what he calls a distance function in the finite dimensional space E over the field K which is complete with respect to the non-archimedean absolut value |.| A distance function is in his definition a function from E to the positive reals with F(X-Y)<= max(F(X),F(Y) and F(kX) = |k|*F(X) for all k in K. Holger Walliser === Subject: Re: Which part of Mathematics do I have to learn? Content-Length: 822 Originator: rusin@vesuvius > I do not see how there can be any reasonable notion of convexity > not using the ordering of subsets of reals. Are the convex sets > of any use without an Archimedean norm? > -- There exist non-Archimedean versions of the notion of convexity. See L. Narici, E. Beckenstein, and G. Bachman. Functional Analysis and Valuation Theory, Marcel Dekker, New York, 1971. W. H. Schikhof. Ultrametric Calculus, Cambridge Univ. Press, 1984. P. Schneider. Nonarchimedean Functional Analysis, Springer, Berlin, 2002. ----------------------------------------------- Anatoly N. Kochubei Institute of Mathematics, National Academy of Sciences of Ukraine. Tereshchenkivska 3, Kiev, 01601 Ukraine. E-mail: kochubei@i.com.ua === Subject: Re: Which part of Mathematics do I have to learn? Content-Length: 2052 Originator: rusin@vesuvius > I want to define a (volume)measure for convex sets in a space with norm > where the norm comes from a field with an absolute value. The measur > should be related to the norm as for example the Lebesgue measure to the > euclidean norm in IR^n. and I do have no idea how I can define such a > measure in a senseful way. BTW - the field could be restricted by a few > conditions (for example the product formula should hold) but I think > this is not important at a first glance. What is important is the fact, > that I do not want the field to be a subfield of the complex numbers and > so - I think - for example nothing known about Hilbert spaces could be > used. In order to obtain a measure theory on vector spaces over some field, one has first to construct measures over the field itself. Measures with reasonable properties exist over locally compact fields, that is the fields of real/complex numbers and non-Archimedean local fields. For the latter case, there exists a well-developed finite-dimensional integration theory and even a theory of Gaussian (and some other) measures on infinite-dimensional spaces. See V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994. A.N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Marcel Dekker, New York, 2001. S. N. Evans, Local fields, Gaussian measures, and Brownian motions. In: Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 28, 11-50 (2001). K. Yasuda, Extension of measures to infinite dimensional spaces over p-adic field. Osaka J. Math. 37, No.4, 967-985 (2000). ---------------------------------------------------------- Anatoly N. Kochubei Institute of Mathematics, National Academy of Sciences of Ukraine. Tereshchenkivska 3, Kiev, 01601 Ukraine. E-mail: kochubei@i.com.ua === Subject: Re: Which part of Mathematics do I have to learn? Content-Length: 1806 Originator: rusin@vesuvius First of all thank you for your quick answer! > Measures with > reasonable properties exist over locally compact fields, that is the fields > of real/complex numbers and non-Archimedean local fields. This is the case I know. In this case one can work over the adelic ring and define a haar measure. But I am interested in much more general spaces as I posted in my original posting. I want the field to be not local or global! An easy example would be the field K(X) of rational functions over an _arbitrary_ field K with the absolute value v(f):=e^d, wher d is the difference of the degrees of the numerator and the denominator. Then - in general - the copmletion is _not_ locally compact and so - as far as I understood - one cannot define a haar measure in the usual manner for this field. This is one of the cases I am interested in. > For the latter > case, there exists a well-developed finite-dimensional integration theory > and even a theory of Gaussian (and some other) measures on > infinite-dimensional spaces. See > V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adic Analysis and > Mathematical Physics, World Scientific, Singapore, 1994. > A.N. Kochubei, Pseudo-Differential Equations and Stochastics over > Non-Archimedean Fields, Marcel Dekker, New York, 2001. > S. N. Evans, Local fields, Gaussian measures, and Brownian motions. In: > Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. > Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 28, > 11-50 (2001). > K. Yasuda, Extension of measures to infinite dimensional spaces over p-adic > field. Osaka J. Math. 37, No.4, 967-985 (2000). This references would be interesting in any case - so thank you again. Holger Walliser === Subject: Re: Which part of Mathematics do I have to learn? Content-Length: 2544 Originator: rusin@vesuvius >First of all thank you for your quick answer! >> Measures with >> reasonable properties exist over locally compact fields, that is the fields >> of real/complex numbers and non-Archimedean local fields. >This is the case I know. In this case one can work over the adelic ring and >define a haar measure. But I am interested in much more general spaces as I >posted in my original posting. I want the field to be not local or global! An >easy example would be the field K(X) of rational functions over an _arbitrary_ >field K with the absolute value v(f):=e^d, wher d is the difference of the >degrees of the numerator and the denominator. Then - in general - the >copmletion is _not_ locally compact and so - as far as I understood - one >cannot define a haar measure in the usual manner for this field. This is one of >the cases I am interested in. One cannot define a countably additive Haar measure in any other manner, either. As the field is a commutative ring under addition, there always is a finitely additive probability invariant probability measure on it. >> For the latter >> case, there exists a well-developed finite-dimensional integration theory >> and even a theory of Gaussian (and some other) measures on >> infinite-dimensional spaces. See >> V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adic Analysis and >> Mathematical Physics, World Scientific, Singapore, 1994. >> A.N. Kochubei, Pseudo-Differential Equations and Stochastics over >> Non-Archimedean Fields, Marcel Dekker, New York, 2001. >> S. N. Evans, Local fields, Gaussian measures, and Brownian motions. In: >> Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. >> Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 28, >> 11-50 (2001). >> K. Yasuda, Extension of measures to infinite dimensional spaces over p-adic >> field. Osaka J. Math. 37, No.4, 967-985 (2000). >This references would be interesting in any case - so thank you again. Finitely additive measures have some nasty properties, including no reasonable Radon-Nikodym derivatives. There is a Fubini theorem, but it is not what one would expect. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: A problem of quadrilaterals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i037YKw25161 ABCD is a convex quadrilateral (a normal one, where the sides don't intersect and none of the interior angles exceeds 180 degrees). You are given that angles A and B sum up to greater than 180 degrees. Prove that: AC + BD > AD + BC i.e. sum of diagonals exceeds sum of sides AD and BC. I think I have a proof, but it is not a simple one. I believe that something simple and elegant should be able to do it. Arnab === Subject: Re: A problem of quadrilaterals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i05E03w15387 > ABCD is a convex quadrilateral (a normal one, where the sides don't > intersect and none of the interior angles exceeds 180 degrees). You > are given that angles A and B sum up to greater than 180 degrees. > Prove that: > AC + BD > AD + BC > i.e. sum of diagonals exceeds sum of sides AD and BC. > I think I have a proof, but it is not a simple one. I believe that > something simple and elegant should be able to do it. > Arnab The simplest way I can think of (and I might be wrong as well): Since ABCD is convex, the diagonals meet. Call the meeting point O. Then we have, using the triangle inequality: AO + OD > AD BO + OC > BC We know: AC = AO + OC BD = BO + OD And we get (adding the first two, replacing using the last two): AC + BD > AD + BC (It's > and not >= for obvious reasons). === Subject: Re: A problem of quadrilaterals Content-Length: 981 Originator: rusin@vesuvius [Arnab] | ABCD is a convex quadrilateral (a normal one, where the sides don't | intersect and none of the interior angles exceeds 180 degrees). You | are given that angles A and B sum up to greater than 180 degrees. | Prove that: | | AC + BD > AD + BC | | i.e. sum of diagonals exceeds sum of sides AD and BC. | | I think I have a proof, but it is not a simple one. I believe that | something simple and elegant should be able to do it. Let E be the point of intersection of the diagonals. Then by the triangle inequality AE + ED > AD BE + EC > BC Add to get AE + EC + BE + ED > AD + BC. By convexity, E is inside, so AE + EC = AC and BE + ED = BD. The condition on the angles A and B is irrelevant. SA -- Stein Arild Str¿mme +47 55584825, +47 95801887 Universitetet i Bergen Fax: +47 55589672 Matematisk institutt www.mi.uib.no/stromme/ Johs Brunsg 12, N-5008 BERGEN stromme@mi.uib.no === Subject: automorphisms of C commuting with exp Content-Length: 415 Originator: rusin@vesuvius Do there exist automorphisms h of C (the field of complex numbers) such that h(exp(z))=exp(h(z)) for all z, other than the obvious h(z)=z and h(z)=zbar? Of course I'm not requiring h to be continuous, and I allow use of the axiom of choice. I'm afraid I don't know enough model theory to settle this by myself. -- David A. Madore (david.madore@ens.fr, http://www.eleves.ens.fr:8080/home/madore/ ) === Subject: Re: automorphisms of C commuting with exp Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) Content-Length: 1300 Originator: rusin@vesuvius >Do there exist automorphisms h of C (the field of complex numbers) >such that h(exp(z))=exp(h(z)) for all z, other than the obvious h(z)=z >and h(z)=zbar? Of course I'm not requiring h to be continuous, and I >allow use of the axiom of choice. I'm not sure about this, but I suspect that the existence of such a thing might depend on Schanuel's conjecture. http://mathworld.wolfram.com/SchanuelsConjecture.html The natural way to try to construct h is to start with the subfield E of elementary numbers of C, and set h(z) = z for all z in E. Then pick any element x not in E and define h(x) more or less arbitrarily, and continue by transfinite induction. The catch is that you have to make sure you don't accidentally assign inconsistent values to h(x), and this might require assuming Schanuel's conjecture (so that no unexpected identities show up). I'd be interested in hearing if you get a more satisfactory answer. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: computability with random Oracles Content-Length: 3493 Originator: rusin@vesuvius My question is probably very na.95ve: I apologize in advance. In short (a longer, more precise statement follows): is it true that any (deterministic) function that is (almost surely) computable by a Turing machine with a random Oracle is, in fact, computable? Here is a rigorously formulated question. If A is a subset of N (the set of naturals), a Turing machine with A for Oracle is a Turing machine which is given the power, at any stage of its computation, to ask the question does this number belong to A?. We can then consider the set of subsets of N which are recursive relative to the Oracle A (meaning that their characteristic function is computable with a Turing machine having A for Oracle); certainly A is recursive relative to itself, and if A is recursive (in the usual sense) then the subsets of N recursive relative to A are exactly the recursive subsets of N in the usual sense. Now I can take A randomly (in 2^N endowed with the product measure): I call a subset of N [almost surely] recursive relative to a random Oracle when the probability that it be recursive relative to A, when A is randomly chosen, is 1. Obviously any recursive subset of N is recursive relative to a random Oracle; my question is then: is it true that, conversely, any subset that is recursive relative to a random Oracle recursive in the usual sense? Intuitively I tend to believe that the answer is yes, because I really can't see what computing power could be brought by a random Oracle when the final result must be deterministic. Also, it seems that the answer no would somehow endager (a moderately strong form of) the Church-Turing thesis (since the notion of being computable relative to a random Oracle is a not-too-far-fetched notion of physical computability). But I can't imagine any way of proving that the answer would be yes. As a matter of fact, I'm not even sure how I would proceed to prove that there exists *some* subset of N which is not recursive relative to a random Oracle (the most straightforward attempt with Fubini seems to fail). I am also aware that random Oracles have been introduced in complexity theory. For example, there are such landmark results as almost surely relative to a random Oracle A, P is not equal to NP. (This is why I imagine my question about computability must be extremely na.95ve, if anything is known about complexity.) This doesn't tell me, however, anything about functions which are almost surely P relative to a random Oracle, or about functions which are almost surely NP relative to a random Oracle, or even about these two sets being different (there is a difference in the order of the quantifiers). In case I am wrong in believing that computable relative to a random Oracle means the same as computable, I can also ask about *uniformly* computable relative to a random Oracle, which is a stronger statement, namely that there should exist a _fixed_ Turing machine which, when given a random Oracle A, will almost surely compute the given function. Lastly, I can ask about generic Oracles instead of random ones: a function is computable relative to a generic Oracle when the set of Oracles relative to which it is not computable is meager in the Baire sense (meaning that is contained in a countable union of nowhere dense closed sets of 2^N). I'll be grateful for any light on this problem. -- David A. Madore (david.madore@ens.fr, http://www.eleves.ens.fr:8080/home/madore/ ) === Subject: Re: computability with random Oracles Content-Length: 987 Originator: rusin@vesuvius > My question is probably very na.95ve: I apologize in advance. > In short (a longer, more precise statement follows): is it true that > any (deterministic) function that is (almost surely) computable by a > Turing machine with a random Oracle is, in fact, computable? Let me first check that I understand your question correctly. You assume that you have an oracle Turing machine A, such that for every x, and a uniform randomly chosen oracle O, A^O stops, given x and answers correctly with probability 1. Then, you wonder if there is a Turing machine B which decides the same language, but without an oracle. Now, this should certainly hold: first note that if A stops for any oracle O on input x, then the answer has to be correct. This is because A can only make finitely long queries to the oracle, so the probability of such an oracle cannot be 0. On the other hand, it seems easy to find an oracle for which A stops. Thomas === Subject: radius of zeroness Content-Length: 1156 Originator: rusin@vesuvius Define Sigma to be the set of all univalent functions of the form: f(z) = 1/z + a(0) + a(1)*z + a(2)*z^2 + a(3)*z^3 + ... That is, f is one-to-one and complex analytic in the disk |z|<1 except for a simple pole at the origin. If N is the supremum of all r such that an arbitrary f in Sigma never vanishes on the disk |z|<=r, then Goluzin [1] proved that 0.86 < N <= sqrt(3)/2 = 0.8660254037... Has anyone seen a sharpening of this theorem anywhere? Goodman [2] referred to N as the radius of zeroness (although he didn't mention Goluzin's particular problem). Help or references would be gratefully received. For more background, please visit http://pauillac.inria.fr/algo/bsolve/ and open the file entitled Radii in geometric function theory. Steve Finch 1. G. M. Goluzin, Zur Theorie der schlichten konformen Abbildungen (in Russian), Mat. Sbornik 42 (1935) 169--190. 2. A. W. Goodman, Univalent Functions, v. 2, Mariner, 1983, pp. 85 & 110. _________________________________________________________________ Tired of slow downloads? Compare online deals from your local high-speed providers now. https://broadband.msn.com === Subject: Special functions Epigone-thread: drangzeeyay Content-Length: 1900 Originator: rusin@vesuvius Hello! I have a question for experts in special functions. I have been studying a function given by [ F(z)=prod_{k=0}^{infty} frac{1+overline{q^{2k}z}}{1+q^{2k}z} ] for $z$ in (a special subset of) ${mathbb C}$ and $q$ a complex parameter of absolute value strictly less than 1 (also chosen in a particuar way). Obviously this function is related to well known special functions (theta?, quotient of q-exponential functions?). My first question is: to what known special functions is it related? The special choice of $q$ is the following: [ q=exp{i r^{-1}} ] where $r$ is a complex number of strictly negative real part and imaginary part equal to $frac{2pi}{N}$ where $N$ is a non zero even integer. Now I restrict the domain of $F$ to the closure (in the complex plane) of the multiplicative group generated by $q$ and ${q^{it}: tinmathbb R}$. This is a colection of $N$ logarhitmic spirals plus ${0}$. On this collection of spirals I have a function [ alpha(q^nq^{it})= exp{iIm frac{(n+it)^2}{2 r}} ] ($r$ is the inverse of the logarithm of $q$ - remember?). O.K. It looks strange, but I alo have the following formula [ F(z)F(q^2 z^{-1})=Calpha(q^{-1} z) ] where $C$ is a constant expresible with some infinite product. The point is that as $z$ tends to infinity $F(q^2 z^{-1})$ goes to 1, and so my formula gives an assymptotic behaviour of $F$. My second question is: are such formulae known for the good old special functions that have been around for ages? The function $F$ is very special for the folowing reason: [ F(R+S)=F(R)F(S) ] for pairs of normal Hilbert space operators $(R,S)$ with spectra contained in the domain of definition of $F$, satisfying [ SR=q^2RS, SR^*=R^*S. ] I am not an expert in special functions and I do not know any references for these topics. I would apperciate any help with my two questions above. Piotr === Subject: Re: Polyhedra Content-Length: 957 Originator: rusin@vesuvius bossavit schrieb im Newsbeitrag > While studying mesh-refinement schemes in the context of so-called > multigrid methods, I stumbled upon the following issue. Can anyone > give me hints? > Start from a simplex S0 in n-dimensional space. Take the mid-edges, > take their convex hull, S1 say. Take the barycenters of the What did you mean by the mid-edges? The midpoints of edges? If yes, then S1 is an octahedron, S2 is a parallelepiped. How do you want to proceed then? If by taking the facet barycenters of S2, then S3 is an octahedron homothetic to S1. > 2-dimensional faces, take their convex hull, S2. And so on. It happens > that each polyhedron Si is a tessellation of pieces homothetic to Si > and other Sj's, with the reduction factor 2. Have these polyhedra been > studied in other contexts? What is known about them?