mm-3739 === Subject: Centers of Perfect Groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) In my thesis, group extensions of the form where H and P are given and P is perfect, have come up. (Just a quick review, a group is perfect iff its abelianization is trivial. The abelianization of group Q is Q/[Q,Q], where [Q,Q] is the commutator subgroup of Q. The commutator subgroup [Q,Q] of a group Q is the subgroup generated by all commutators ghg^(-1)h^(-1). The commutator subgroup is always a normal subgroup. See In reading _Cohomology of Groups_ by Brown, chapter 4, section 6 (pages 104-6), it is clear that knowing the center of P, Z(P), is important for classifying the solutions to the groups extension problem given by (*). Does anyone know what work has been done to classify centers Z(P) of perfect groups P? -- Jeffrey Rolland === Subject: Re: Centers of Perfect Groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For finite perfect groups, Z(P) is a quotient of M = H_2( P/Z(P), Z ), the second integral homology of P/Z(P). In fact for every finite perfect group P/Z(P) there is a unique perfect group D with M = Z(D) and D/Z(D) = P/Z(P). I don't have Brown handy, but this is just the idea of the Schur Multiplicator and Darstellunggruppe, and I think it is covered at least once there, if not multiple times (M has lots of different interpretations). I personally like the treatment in Aschbacher's Finite Groups textbook, but of course it may not be so useful for infinite groups. As far as which Z(P) can occur ignoring P other than requiring P/Z(P) to be finite, this is the expected answer of all finite abelian groups. SL(n,q) has center cyclic of order gcd(n,q-1), and so choosing n and q wisely one gets all cyclic groups, but then direct products of SL(n,q) will just have the direct products as centers. For a particular P/Z(P) there are a great many things to say on how large Z(P) can be, but perhaps the first and most surprising thing to notice is that Z(P) must be finite. Most of the best results along these lines are phrased for p-groups since the multiplicator of P/Z(P) is a quotient of the multiplicator of its Sylow subgroups (well take the direct product over primes dividing the order of the group). The book of Holt and Plesken on Perfect Groups contains a number of examples where the Schur multiplicators are calculated and you can see some variety in the types of centers that occur even for directly indecomposable and small perfect groups. === Subject: Re: Centers of Perfect Groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) | |For finite perfect groups, Z(P) is a quotient of M = H_2( P/Z(P), Z ), |the second integral homology of P/Z(P). In fact for every finite |perfect group P/Z(P) there is a unique perfect group D with M = Z(D) |and D/Z(D) = P/Z(P). I don't have Brown handy, but this is just the |idea of the Schur Multiplicator and Darstellunggruppe, and I think it |is covered at least once there, if not multiple times (M has lots of |different interpretations). I personally like the treatment in |Aschbacher's Finite Groups textbook, but of course it may not be so |useful for infinite groups. | |As far as which Z(P) can occur ignoring P other than requiring P/Z(P) |to be finite, this is the expected answer of all finite abelian |groups. SL(n,q) has center cyclic of order gcd(n,q-1), and so |choosing n and q wisely one gets all cyclic groups, but then direct |products of SL(n,q) will just have the direct products as centers. | |For a particular P/Z(P) there are a great many things to say on how |large Z(P) can be, but perhaps the first and most surprising thing to |notice is that Z(P) must be finite. Most of the best results along |these lines are phrased for p-groups since the multiplicator of P/Z(P) |is a quotient of the multiplicator of its Sylow subgroups (well take |the direct product over primes dividing the order of the group). | |The book of Holt and Plesken on Perfect Groups contains a number of |examples where the Schur multiplicators are calculated and you can see |some variety in the types of centers that occur even for directly |indecomposable and small perfect groups. I think I may have misstated my objective. I am more interested in calculating Z(P) given P, and in knowing what results are out there for accomplishing this calculation. The result for SL(n,q) is more along the lines of what I wanted. Knowing that Z(P) must be finite is most interesting. I have checked out the reference you and the other poster have -- Jeffrey Rolland === Subject: Re: Centers of Perfect Groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Jeffrey Rolland a .8ecrit : There should be a lot of informations on this in the book Perfect groups by D.Holt and W.Plesken (Oxford Mathematical Monographs). Serge. === Subject: Re: Centers of Perfect Groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) |Jeffrey Rolland a .8ecrit : |There should be a lot of informations on this in the book |Perfect groups by D.Holt and W.Plesken |(Oxford Mathematical Monographs). I have checked out the book, and will look into it for answers to my query. -- Jeffrey Rolland === Subject: Re: disproof of Riemann Hypothesis? Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) He explains a couple of pages back that A stands for some unspecified positive constant, which might not be the same constant each time it appears, even in consecutive lines of a series of inequalities. Sort of like big-O notation. -- 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: disproof of Riemann Hypothesis? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) But then the final contradiction of the kind f(t) < f(t) (for some f) poses no problem. If I am allowed to add some constant (not the same on both sides of the inequality), the contradiction vanishes. Stefan Wehmeier stefanw@math.upb.de === Subject: Re: disproof of Riemann Hypothesis? Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) True enough. But perhaps this issue could be fixed easily by insisting that A means A, and just changing A to A' at a few points in the paper. This wouldn't be any sloppier than many other papers I've seen. -- 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: disproof of Riemann Hypothesis? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Unfortunately, Louis de Branges has a proof of the Riemann Hypothesis on his home page.... Gary McGuire === Subject: Re: disproof of Riemann Hypothesis? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Maybe. === Subject: Re: Multilevel sequences in extended Galois fields Originator: @chiark.greenend.org.uk ([193.201.200.170]) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The order has to be a factor of 4^degree - 1, so you don't have to check all that many powers of x; and you can calculate x^2n by squaring x^n modulo the irreducible polynomial. What I'd recommend is using a fully-fledged computer algebra package; the one I know is Magma, you can evaluate things on-line at http://magma.maths.usyd.edu.au/calc/. Try the program deg:=10; for r in [1..1000] do tmp:=x^deg+P![Random(BaseRing(P)) : r in [1..deg]]; if (IsPrimitive(tmp)) then print tmp; end if; end for; which will produce you some primitive polynomials of degree 10; change the value of 'deg' to get primitive polynomials of a different degree. 'g.1' and 'g.1^2' are the two elements of the GF(4) which aren't 0 and 1. Tom === Subject: Re: Multilevel sequences in extended Galois fields Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hello Tom is quite fast and not so difficult to use. But, it is quite unfortunate that there is no source code available for the functions implemented to have a look at the algorithm being used. Also, thank you for pointing out the factor of 4^degree - 1 part. ~Anup === Subject: Re: Multilevel sequences in extended Galois fields Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The algorithms being used will be an irreducibility test followed by simply trying all possible orders in the primitivity test; the irreducibility test is to compute the kernel of the matrix A+I where A is the matrix whose rows are the powers of (x^2). eg if we want to test whether x^7+x^2+1 is irreducible, we write down 1000000 because x^0 = 1 0010000 because x^2 = x^2 0000100 because x^4 = x^4 0000001 because x^6 = x^6 0101000 because x^8 = x^3+x 0001010 because x^10 = x^5+x^3 1010010 because x^12 = x^7+x^5 = x^5+x^2+1 adding the identity matrix we get 0000000 0110000 0010100 0001001 0101100 0001000 1010011 where the sum of rows 1, 2, 4 and 5 is 0000000 which tells you that x^7+x^2+1 is not irreducible, and in particular that it has a factor x^5+x^4+x^2+x^1+1. Factorising polynomials is much easier than integers. Tom === Subject: Re: Multilevel sequences in extended Galois fields Originator: bergv@math.uiuc.edu (Maarten Bergvelt) So essentially after establishing the irreducibility of the polynomials we should go and and check for each factor 'e' of q^n-1 whether the polynomial divides x^e-1. If this division leaves no remainder only when e = q^n-1 we have a primitive polynomial. Being on the topic, I was also wondering if there had been any attempts on deriving primitive polynomials in extended fields from the polynomials in original finite fields. A construction method rather than an exhaustive checking method. Tom, do you happen to of any results in this front ? -Anup === Subject: A coloring problem Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Given n^2 pairs (x1, x2). x1 is from 1 to n and x2 is also from 1 to n (It may be thought as an integer grid). We have several sets. Each (x1, x2) is in exactly m sets. Each set contains exactly k pairs and all the pairs in each set differ by at least one coordinate. This means (1,2) and (1,3) cannot be in the same set. I would like to put all the sets into a number of bins (which is equivalent to color the sets with different colors) such that in the end the set in each bin is a union of the sets that have been put into this bin. The question is how many bins are needed such that each bin does not contain both (x1, x2) and (x1, x2') or both (x1, x2) and (x1', x2) (in other words, all the pairs in each bin should differ by at least one coordinate, similarly (1,2) and (1,3) cannot be in the same bin). I first construct a regular graph, where each node represents a pair (x1,x2). There is an edge between two nodes if and only if the two pairs corresponding to the two nodes have one coordinate in common. Now the problem is a graph coloring problem. By using Lovasz Local lemma, we can show that O(km^2n) colors or bins are sufficient. But this bound is not tight as when k=1 we can readily find an example such that O(n) colors are enough (not O(m^2n)). Anyone has any idea to improve the bound or suggestions on other Best, Chris === Subject: Re: A coloring problem Originator: bergv@math.uiuc.edu (Maarten Bergvelt) There are several things that sound strange to me about your description of the problem, so I may be misunderstanding it - if so, apologies, perhaps you can clarify. When you say all the pairs differ by at least one coordinate, you seem to mean that all the pairs differ in both coordinates, since you say that (1,2) and (1,3) can't be in the same set. In your second paragraph you talk about constructing a regular graph, in which each node represents a pair (x1,x2) (a point in your n by n grid). Then you say that we have a graph coloring problem. But in the first paragraph we are trying to assign colors to _sets_, not pairs. So the graph coloring problem is not using the graph that you mention. So, if I understand correctly, we have the following situation: an n by n grid; each point in the grid belongs to m sets; each set contains k points, and any 2 points in the same set differ in both coordinates. We wish to assign a color (bin) to each set. If A and B are sets, and u is a point in A and v is a point in B, and u and v share at least one coordinate, then the color (bin) of A should be different from the color of B. Lower bound: the sum of the sizes of the sets is mn^2, since each of the n^2 points is in m sets. No bin can contain more than n points in total (since every point in the bin has to differ in both coordinates). So we need at least mn^2/n = mn bins. Upper bound: consider the graph in which each node represents a set. Put an edge between the nodes representing sets A and B if there exist u in A and v in B such that u and v share at least one coordinate. Now we need precisely a proper colouring of this graph. Let A be any set, and let u be a point in it. Then there are (2n-1) points which share at least one coordinate with u. Since each point is in m sets, there are at most (2n-1)m sets which contain a point sharing at least one coordinate with u, and one of these sets is A, so there are at most (2n-1)m - 1 other such sets. Finally, there are k possible choices for the point u in A, so there are at most k[(2n-1)m - 1] neighbours of A in the graph. That is, we have a graph of maximal degree no more than k[(2n-1)m -1]. Any graph with maximal degree d can be colored using d+1 colors. So we need at most k[(2n-1)m -1]+1 colors, which is < 2knm. So for any given k,m,n and any given sets satisfying the conditions, the number of colors needed is between mn and 2kmn. When you talk about O(.), you presumably mean that n goes to infinity. If so, you also need to say what happens to k and m. How do they depend on n? Also, you need to say whether you are interested in the best case, i.e. there exists an example of such sets which can be colored with so many colors, or worst case, i.e. _every_ such collection of sets can be colored with so many colors, or something else. Best wishes, Peter === Subject: IDA 2007 Final CFP Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Final Call for Papers IDA 2007 The 7th International Symposium on Intelligent Data Analysis Ljubljana, Slovenia September 6-8, 2007 http://www.ida2007.org IDA-2007 will take place in Ljubljana, Slovenia, September 6-8, 2007, under the auspices of the IDA council. The conference is organized by researchers and professors from the Universities of Ljubljana and Konstanz (Germany). It will consist of a stimulating program of invited talks by leading international experts in intelligent data analysis, contributed papers, poster sessions, and an exciting social program. Our aim for IDA-2007 is to bring together a wide variety of researchers - academic, industrial, and otherwise who are concerned with extracting knowledge from data, including researchers from statistics, machine learning, neural networks, computer science, pattern recognition, database management, and other areas. The strategies adopted by people working in these areas are often different, and a synergy results if this is recognised. IDA-2007 is intended to stimulate interaction between these different areas, so that more powerful techniques and tools emerge for extracting knowledge from data and a better understanding is developed for the process of intelligent data analysis. During the IDA-2007 conference we plan for an interesting agenda of social events that include several tutorial tracks, open panel discussions, and keynote talks. ---------------- April 6, 2007 Deadline for submitting papers May 18, 2007 Notification of acceptance June 15, 2007 Deadline for submission of final papers Sept 6-8, 2007: IDA 2007 Conference Venue: ----------------- Ljubljana, the capital town of Slovenia, will be hosting IDA 2007 at the University of Ljubljana. Ljubljana maintains the friendliness of a small town and also has the flair and characteristics of a metropolis. The busy city centre is the old part of the town rich with medieval architecture. The river runs through the town, the river banks boasting numerous Italian style caf.8es and plazas for guests to enjoy. It is closer to more well-known cities than many people think. It is easy to reach with direct flights leaving from Vienna, Munich, Frankfurt, London and Paris. It is only a 2 hour drive from Vienna or Munich. Venice is also only 2 hours away and the Mediterranean coastline is within easy driving distance. Topics: ------- Paper submissions are sought in the following areas: Algorithms and Techniques: - Artificial neural networks - Bayesian networks - Heuristic methods - Optimization problems - Case-based reasoning - Computational models of human learning - Computational learning theory - Cooperative learning - Unsupervised learning - Decision and induction - Evolutionary computation - Grammatical inference - Incremental and on-line learning - Information retrieval and learning - Knowledge acquisition and learning - Data pre- and post-processing - Data visualisation - Statistical pattern recognition and analysis - Performance and optimization Theoretical Contributions: - Data Mining theories - Information retrieval restrictions - Legal data analysis restrictions - Innovative data analysis (models, information types, and objectives) - Theoretical IDA issues - New paradigms Application Fields: - Bio-informatics and bio-surveillance - Web analysis - Medical applications - Industrial data analysis - Commerce and finance information analysis - Government, legal analysis (socio-economic data, legal issues) This is the seventh Symposium on Intelligent Data Analysis after the (Lisboa), IDA-99 (Amsterdam), IDA-97 (London), and IDA-95 (Baden- Baden). The IDA conference series intends to provide an international forum for the discussion of the innovative outstanding research results in the field of intelligent data analysis, becoming one of the most significant conferences on this topic world-wide. Contributions in this field deal with either theoretical or applied real-world problems making special effort in introducing novel data analysis techniques. Publications: ------------- The proceedings will be published in the Lecture Notes in Computer Science Series of Springer. The proceedings of previous symposia appeared in this series as LNCS volumes 1280, 1642, 2189, 2810 and 3646. We also plan to have a special issue of the Intelligent Data Analysis journal with extended versions of a number of papers presented during the symposium. http://www.iospress.nl/html/1088467x.html Additional Information: ----------------------- Guidelines for submissions and information about the conference venue are available at the conference web site: http://www.ida2007.org === Subject: Ten papers published by Algebraic & Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Ten papers have been published by Algebraic & Geometric Topology (1) Algebraic & Geometric Topology 7 (2007) 135-156 A function on the homology of 3-manifolds by Vladimir Turaev URL: http://www.msp.warwick.ac.uk/agt/2007/07/p006.xhtml DOI: 10.2140/agt.2007.7.135 (2) Algebraic & Geometric Topology 7 (2007) 157-180 Almost periodic flows on 3-manifolds by Kelly Delp URL: http://www.msp.warwick.ac.uk/agt/2007/07/p007.xhtml DOI: 10.2140/agt.2007.7.157 (3) Algebraic & Geometric Topology 7 (2007) 181-196 Relationships between braid length and the number of braid strands by Cornelia A Van Cott URL: http://www.msp.warwick.ac.uk/agt/2007/07/p008.xhtml DOI: 10.2140/agt.2007.7.181 (4) Algebraic & Geometric Topology 7 (2007) 197-231 String bracket and flat connections by Hossein Abbaspour and Mahmoud Zeinalian URL: http://www.msp.warwick.ac.uk/agt/2007/07/p009.xhtml DOI: 10.2140/agt.2007.7.197 (5) Algebraic & Geometric Topology 7 (2007) 233-260 Infinity structure of Poincare duality spaces by Thomas Tradler and Mahmoud Zeinalian Appendix: Dennis Sullivan URL: http://www.msp.warwick.ac.uk/agt/2007/07/p010.xhtml DOI: 10.2140/agt.2007.7.233 (6) Algebraic & Geometric Topology 7 (2007) 261-284 Homological thickness and stability of torus knots by Marko Stošić URL: http://www.msp.warwick.ac.uk/agt/2007/07/p011.xhtml DOI: 10.2140/agt.2007.7.261 (7) Algebraic & Geometric Topology 7 (2007) 285-300 Growth series for vertex-regular CAT(0) cube complexes by Richard Scott URL: http://www.msp.warwick.ac.uk/agt/2007/07/p012.xhtml DOI: 10.2140/agt.2007.7.285 (8) Algebraic & Geometric Topology 7 (2007) 301-308 Super-exponential distortion of subgroups of CAT(-1) groups by Josh Barnard, Noel Brady and Pallavi Dani URL: http://www.msp.warwick.ac.uk/agt/2007/07/p013.xhtml DOI: 10.2140/agt.2007.7.301 (9) Algebraic & Geometric Topology 7 (2007) 309-325 String homology of spheres and projective spaces by Craig Westerland URL: http://www.msp.warwick.ac.uk/agt/2007/07/p014.xhtml DOI: 10.2140/agt.2007.7.309 (10) Algebraic & Geometric Topology 7 (2007) 327-338 Integrality of Homfly 1-tangle invariants by H R Morton URL: http://www.msp.warwick.ac.uk/agt/2007/07/p017.xhtml DOI: 10.2140/agt.2007.7.327 Abstracts follow (1) A function on the homology of 3-manifolds by Vladimir Turaev In analogy with the Thurston norm, we define for an orientable 3-manifold M a numerical function on H_2(M;Q/Z). This function measures the minimal complexity of folded surfaces representing a given homology class. A similar function is defined on the torsion subgroup of H_1(M;Z). These functions are estimated from below in terms of abelian torsions of M. (2) Almost periodic flows on 3-manifolds by Kelly Delp A 3-manifold which supports a periodic flow is a Seifert fibered space. We define a notion of almost periodic flow and give conditions under which a manifold supporting an almost periodic flow is Seifert fibered. It is well-known that R^3 does not support fixed point free periodic flows, and our results include that R^3 does not support certain almost periodic flows. (3) Relationships between braid length and the number of braid strands by Cornelia A Van Cott For a knot K, let l(K,n) be the minimum length of an n-stranded braid representative of K. Fixing a knot K, l(K,n) can be viewed as a function of n, which we denote by l_K(n). Examples of knots exist for which l_K(n) is a nonincreasing function. We investigate the behavior of l_K(n), developing bounds on the function in terms of the genus of K. The bounds lead to the conclusion that for any knot K the function l_K(n) is eventually stable. We study the stable behavior of l_K(n), with stronger results for homogeneous knots. For knots of nine or fewer crossings, we show that l_K(n) is stable on all of its domain and determine the function completely. (4) String bracket and flat connections by Hossein Abbaspour and Mahmoud Zeinalian manifold M of dimension m=2d. We construct a map of Lie algebras of the equivariant homology of LM, the free loop space of M, and MC is the Maurer-Cartan moduli space of the graded differential Lie algebra Omega*(M, ad P), the differential forms with values in the associated adjoint bundle of P. For a 2-dimensional manifold M, our Lie algebra map reduces to that constructed by Goldman [Invent. Math. 85 (1986) 263-302]. We treat different Lie algebra structures on H_{2*}(LM) depending on the choice of the linear reductive Lie group G in our discussion. This paper provides a mathematician-friendly formulation and proof of the main result of Cattaneo, Frohlich and Pedrini together with its natural generalization to other reductive Lie groups. (5) Infinity structure of Poincare duality spaces by Thomas Tradler and Mahmoud Zeinalian Appendix: Dennis Sullivan We show that the complex C_* X of rational simplicial chains on a compact and triangulated Poincare duality space X of dimension d is an A_infinity coalgebra with infinity duality. This is the structure required for an A_infinity version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology HH^{*+d} (C^* X, C_* X) of the cochain algebra C^*X with values in C_* X has a BV structure. This implies, if X is moreover simply connected, that the shifted homology H_{*+d}LX of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of infinity structures. (6) Homological thickness and stability of torus knots by Marko Stosic In this paper we show that the nonalternating torus knots are homologically thick, ie that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing a certain part of its homology, and consequently, there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in [Experiment. Math. 15 (2006) 129-159]. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov--Rozansky $sl(n)$ homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured by Dunfield, Gukov and Rasmussen. (7) Growth series for vertex-regular CAT(0) cube complexes by Richard Scott We show that the known formula for the growth series of a right-angled Coxeter group holds more generally for any CAT(0) cube complex whose vertex links all have the same f-polynomial. (8) Super-exponential distortion of subgroups of CAT(-1) groups by Josh Barnard, Noel Brady and Pallavi Dani We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential. (9) String homology of spheres and projective spaces by Craig Westerland We study a spectral sequence that computes the S^1-equivariant homology of the free loop space LM of a manifold M (the emphstring homology of M). Using it and knowledge of the BV operations on HH^*(H^*(M), H^*(M)), we compute the (mod 2) string homology of M when M is a sphere or a projective space. (10) Integrality of Homfly 1-tangle invariants by Hugh R Morton Given an invariant J(K) of a knot K, the corresponding 1-tangle invariant J'(K)=J(K)/J(U) is defined as the quotient of J(K) by its value J(U) on the unknot U. We prove here that when J is the Homfly satellite invariant determined by decorating K with any eigenvector of the meridian map in the Homfly skein of the annulus then J' is always an integer 2-variable Laurent polynomial. Specialisation of the 2-variable polynomials for suitable choices of eigenvector shows that the 1-tangle irreducible quantum sl(N) invariants of K are integer 1-variable Laurent polynomials. === Subject: This Week's Finds in Mathematical Physics (Week 248) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Also available as http://math.ucr.edu/home/baez/week248.html March 28, 2007 This Week's Finds in Mathematical Physics (Week 248) John Baez This week I'll continue the Tale of Groupoidification, but first - relativity on the world-wide web, and some new views of the Sun! Chris Hillman has always been one of the most erudite and enigmatic explainers of mathematical physics on the internet, from the early days of sci.physics, to sci.physics.research, to the rise of Wikipedia. I know him fairly well, but I've never actually met him. Feared by crackpots worldwide, some claim he is a software agent - an artificial intelligence run amuck. He has never denied this. In fact, I'm beginning to believe it's true. Anyway, he has just updated his wonderful guide to relativity: 1) Chris Hillman, Relativity on the World-Wide Web, http://math.ucr.edu/home/baez/RelWWW/ Regardless of where you stand on the road to knowledge - whether you just want to see animations of black holes, or need software for doing tensor calculations, or want to learn more about advanced astrophysics - this has something for you! Speaking of astrophysics - here's a cool movie of the Moon passing in front of the Sun, as viewed from the STEREO B spacecraft: 2) Astronomy Picture of the Day, March 3 2007, Lunar transit from STEREO, http://antwrp.gsfc.nasa.gov/apod/ap070303.html As the name hints, there's a pair of STEREO satellites in orbit around the Sun. One is leading the Earth a little, the other lagging behind a bit, to provide a stereoscopic view of coronal mass ejections. What's a coronal mass ejection? It's an event where the Sun shoots off a blob of ionized gas - billions of tons of it - at speeds around 1000 kilometers per second! That sounds cataclysmic... but it happens between once a day and 5-6 times a day, depending on where we are in the 11-year solar cycle, also known as the sunspot cycle. Right now we're near the minimum of this cycle. Near the maximum, coronal mass ejections can really screw up communication systems here on Earth. For example, in 1998 a big one seems to have knocked out a communication satellite called Galaxy 4, causing 45 million people in the US to lose their telephone pager service: 3) Gordon Holman and Sarah Benedict, Solar Flare Theory: Coronal mass ejections, solar flares, and the Earth-Sun connection, So, it's not only fun but also practical to understand coronal mass ejections. Here's a movie of one taken by the Solar and Heliospheric observatory (SOHO): 4) NASA, Cannibal coronal mass ejections, http://science.nasa.gov/headlines/y2001/ast27mar_1.htm As I mentioned in week150, SOHO is a satellite orbiting the Sun right in front of the Earth, at an unstable equilibrium - a Lagrange point - called L1. SOHO is bristling with detectors and telescopes of all sorts, and this movie was taken by a coronagraph, which is a telescope specially designed to block out the Sun's disk and see the fainter corona. If a coronal mass ejection hits the Earth, it does something like this: 5) NASA, What is a CME?, http://www.nasa.gov/mpg/111836main_what_is_a_cme_NASA%20WebV_1.mpg In this artist's depiction you can see the plasma shoot off from the Sun, hit the Earth's magnetic field - this actually takes one to five days - and squash it, pushing field lines around to the back side of the Earth. When the magnetic field lines reconnect in back, trillions of watts of power come cascading down through the upper atmosphere, producing auroras. Here's a nice movie of what *those* can look like: 6) YouTube, Aurora (Northern Lights), http://www.youtube.com/watch?v=qIXs6Sh0DKs I wish I understood this magnetic field line trickery better! Magnetohydrodynamics - the interactions between electromagnetic fields and plasma - is a branch of physics that always gave me the shivers. The Navier-Stokes equations describing fluid flow are bad enough - if you can prove they have solutions, you'll win $1,000,000 from the Clay Mathematics Institute. Throw in Maxwell's equations and you get a real witches' brew of strange phenomena. In fact, this subject is puzzling even to experts. For example, why is the Sun's upper atmosphere - the corona - so hot? Here's a picture of the Sun in X-rays taken by another satellite: 7) Transition Region and Coronal Explorer (TRACE), Images of the sun, http://trace.lmsal.com/POD/TRACEpodarchive26.html This lets you see plasma in the corona with temperatures between 1 million kelvin (shown as blue) and 2 million kelvin (red). By comparison, the visible surface of the Sun is a mere 5800 kelvin! Where does the energy come from to heat the corona? There are lots of competing theories. It could even be due to magnetic field reconnection, the same topological phenomenon that triggers auroras when coronal mass ejections smash into the Earth's magnetic field, as in that movie above. For more, try this: 8) Andrew L. Haynes, Clare E. Parnell, Klaus Galsgaard and Eric R. Priest, Magnetohydrodynamic evolution of magnetic skeletons, Proc. Roy. Soc. Lond. A 463 (2007) 1097-1115. Also available as astro-ph/0702604. A new satellite called Hinode is getting a good look at what's going on, and it seems the magnetic field on the Sun's surface is much more dynamic than before thought: 9) NASA, Hinode: investigating the Sun's magnetic field, http://www.nasa.gov/mission_pages/solar-b/ In fact, weather on the Sun may be more complex than on the Earth. There's rain when plasma from the corona cools and falls back down to the Sun's surface... and sometimes there are even tornados! You think tornados on Earth are scary? Check out this movie made during an 8-hour period in August 2000, near the height of the solar cycle: 10) TRACE, Tornados and fountains in a filament on 2 Aug. 2000, movie 13, http://trace.lmsal.com/POD/ Besides the tornados, near the end you can see glowing filaments of plasma following magnetic field lines! Now for something simpler: the Tale of Groupoidification. I don't want this to be accessible only to experts, since a bunch of it is so wonderfully elementary. So, I'm going to proceed rather slowly. This may make the experts impatient, so near the end I'll zip ahead and sketch out a bit of the big picture. Last time I introduced spans of sets. A span of sets is just a set S equipped with functions to X and Y: S / / F/ G / v v X Y Simple! But the important thing is to understand this thing as a witnessed relation. Have you heard how computer scientists use the term witness? They say the number 17 is a witness to the fact that the number 221 isn't prime, since 17 evenly divides 221. That's the idea here. Given a span S as above, we can say an element x of X and an element y of Y are related if there's an element s of S with F(s) = x and G(s) = y. The element s is a witness to the relation. Last week, I gave an example where a Frenchman x and an Englishwoman y were related if they were both the favorites of some Russian s. Note: there's more information in the span than the relation it determines. The relation either holds or fails to hold. The span does more: it provides a set of witnesses. The relation holds if this set of witnesses is nonempty, and fails to hold if it's empty. At least, that's how mathematicians think. When I got married last month, we discovered the state of California demands TWO witnesses attend the ceremony and sign the application for a marriage license. Here the relation is being married, and the witnesses attest to that relation - but for the state, one witness is not enough to prove that the relation holds! They're using a more cautious form of logic. To get the really interesting math to show up, we need to look at other examples of witnessed relations - not involving Russians or marriages, but geometry and symmetry. For example, suppose we're doing 3-dimensional geometry. There's a relation the point x and the line y lie on a plane, but it's pretty dull, since it's always true. More interesting is the witnessed relation the point x and the line y lie on the plane z. The reason is that sometimes there will be just one plane containing a point and a line, but when the point lies on the line, there will be lots. To think of this witnessed relation as a span S / / F/ G / v v X Y we can take X to be the set of points and Y to be the set of lines. Can we take S to be the set of planes? No! Then there would be no way to define the functions f and g, because the same plane contains lots of different points and lines. So, we should take S to be the set of triples (x,y,z) where x is a point, y is a line, and z is a plane containing x and y. Then we can take F(x,y,z) = x and G(x,y,z) = y. A witness to the fact that x and y lie on a plane is not just a plane containing them, but the entire triple. (If you're really paying attention, you'll have noticed that we need to play the same trick in the example of witnesses to a marriage.) Spans like this play a big role in incidence geometry. There are lots of flavors of incidence geometry, with projective geometry being the most famous. But, a common feature is that we always have various kinds of figures - like points, lines, planes, and so on. And, we have various kinds of incidence relations involving these figures. But to really understand incidence geometry, we need to go beyond relations and use spans of sets. Actually, we need to go beyond spans of sets and use spans of groupoids! The reason is that incidence geometries usually have interesting symmetries, and a groupoid is like a set with symmetries. For example, consider lines in 3-dimensional space. These form a set, but there are also symmetries of 3-dimensional space mapping one line to another. To take these into account we need a richer structure: a groupoid! Here's the formal definition: a groupoid consists of a set of objects, and for any objects x and y, a set of morphisms which we think of as symmetries taking x to y. We can compose a morphism think of fg as the result of doing first f and then g. So, we demand the associative law (fg)h = f(gh) whenever either side is well-defined. We also demand that every object x has an identity morphism We think of this as the symmetry that doesn't do anything to x. f 1_y = f = 1_x f. So far this is the definition of a category. What makes it a with the property that f f^{-1} = 1_x and f^{-1} f = 1_y. In other words, we can undo any symmetry. So, in our spans from incidence geometry: S / / F/ G / v v X Y X, Y and S will be groupoids, while F and G will be maps between groupoids: that is, functors! What's a functor? Given groupoids A and B, clearly a functor should send any object x in A to an object F(x) in B. But also, it should send any morphism in A: to a morphism in B: And, it should preserve all the structure that a groupoid has, namely composition: F(fg) = F(f) F(g) and identities: F(1_x) = 1_{F(x)}. It then automatically preserves inverses too: F(f^{-1}) = F(f)^{-1} Given this, what's the meaning of a span of groupoids? You could say it's a invariant witnessed relation - that is, a relation with witnesses that's *preserved* by the symmetries at hand. These are the very essence of incidence geometry. For example, if we have a point and a line lying on a plane, we can rotate the whole picture and get a new point and a new line lying on a new plane. Indeed, a symmetry in incidence geometry is precisely something that preserves all such incidence relations. For those of you not comfy with groupoids, let's see how this actually works. Suppose we have a span of groupoids: S / / F/ G / v v X Y and the object s is a witness to the fact that x and y are related: F(s) = x and G(s) = y. Also suppose we have a symmetry sending s to some other object of S: This gives morphisms in X and in Y. And if we define F(s') = x' and G(s') = y', we see that s' is a witness to the fact that x' and y' are related. Let me summarize the Tale so far: Spans of groupoids describe invariant witnessed relations. Invariant witnesses relations are the essence of incidence geometry. There's a way to turn spans of groupoids into matrices of numbers, so that multiplying matrices corresponds to some nice way of composing spans of groupoids (which I haven't really explained yet). of incidence geometry, we should be able to get a bunch of matrices. Facts about incidence geometry will give facts about linear algebra! Groupoidification is an attempt to reverse-engineer this process. We will discover that lots of famous facts about linear algebra are secretly facts about incidence geometry! To prepare for what's to come, the maniacally diligent reader might like to review week178, week180, week181, week186 and week187, where I explained how any Dynkin diagram gives rise to a flavor of incidence geometry. For example, the simplest-looking Dynkin diagrams, the A_n series, like this for n = 3: o------o------o points lines planes give rise to n-dimensional projective geometry. I may have to review this stuff, but first I'll probably say a bit about the theory of group representations and Hecke algebras. (There will also be other ways to get spans of groupoids, that don't quite fit into what's customarily called incidence geometry, but still fit very nicely into our Tale. For example, Dynkin diagrams become quivers when we give each edge a direction, and the groupoid of representations of a quiver gives rise to linear-algebraic structures related to a quantum group. In fact, I already mentioned this in item E of week230. Eventually this will let us groupoidify the whole theory of quantum groups! But, I don't want to rush into that, since it makes more sense when put in the right context.) By the way, some of you have already pointed out how unfortunate it is that *last* Week was devoted to E8, instead of *this* one. Sorry. ----------------------------------------------------------------------- Addendum: I thank logopetria for catching typos. For more discussion, go to the n-Category Cafe: http://golem.ph.utexas.edu/category/2007/03/this_weeks_finds_in_mathematic_9 .html ----------------------------------------------------------------------- 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/twfcontents.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: Physical Applications of Tetration Originator: bergv@math.uiuc.edu (Maarten Bergvelt) have any of you come across any references which link tetration to any physical processes? applications he cites in biology are sort of lukewarm and are only indirectly related to tetration. I tried to look into nuclear fission but it looks as though such processes are mainly growing as simple exponentials. Tetration seems to be too fast (or too powerful if you wish) to describe anything physical in this universe. I was thinking that perhaps this was a good reason why the hierarchy of operators {+,*,^,^^,...} has a discrepancy at ^^. Maybe tetration is too powerful a process to describe anything physical. If any of you have come across any actual physical applications, (such as something growing tetrationally, for example), I would appreciate the reference. -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/ === Subject: Re: Physical Applications of Tetration Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) All of physics reduces to optimizing the action. O.K., maybe that's an overstatement, but it seems hard to construct an action-minimization problem whose solution is tetration. Is that a good enough reason? -- 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: Physical Applications of Tetration Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I don't have an answer, but would like to point out that the issue is deeper than just nothing physical grows fast enough -- it is that there is no known satisfactory way of making tetration continuous (that is, no canonical way that doesn't depend on arbitrary choices). If there were some physical process which involved a generalized functional square root then that would already be enough to explain tetration (that is, if you could find f(x) so that f(f(x)) was an exponential, and g(x) so that g(g(x))=f(x), etc., then you already can interpolate to get a set of functions indexed by the reals, which can be regarded as a 2-variable function, which when the other variable is fixed specializes to tetration). There is not even a good example in physics that I know of for a half- exponential -- a function which arises in physics whose rate of growth is the functional square root of an exponential -- this is much weaker than having a generalized functional square root, so it would not solve the problem of interpreting tetration physically, but it is a prerequisite that may be easier. -- Joe Shipman === Subject: Re: Physical Applications of Tetration Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Why is this the issue? Many exponential models in physics and biology simply manifest only with discrete values. Continuation is not necessary for the understanding of the model. Consider cancer for example: The cell division pattern follows the sequence 1,2,4,8,16,32,...,2^n,... The previous process does not have to exhibit a continuous behavior to be understood. Cells are distinct entities, and their division is a discrete process. In particular, the function f(x) = 2^x = exp(ln(2)*x) is far from necessary to understand cancer. Something similar holds for the number of neutrons produced in nuclear fission and fusion. The total numbers of nucleons produced, are in all cases simple exponential sequences of time. The above processes are completely understood (barring the cause in the case of cancer of course) without resorting to a continuous function. In a similar spirit, I cannot see why some phenomenon in nature cannot grow as 1,2,4,16,65536,....,2^^n,... or even as [f(n)], where f(f(x))=exp(x), and where [] is the integer part function, particularly since in this case f(x) < exp(x). -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/ === Subject: Re: Physical Applications of Tetration Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hm, at least for a similar topic exists a unique continuation. We know that for those higher operations the bracketing is somewhat arbitrary. Tetration is *right-bracketed* x^(x^(x^....))), while the *left- bracketing* (..(x^x)^..)^x has already a standard continuation, i.e. x^(x^(r-1)). But I want to point on something I would call *middle-bracketing*. We want an operation # such that x # 2 = x^x x # 4 = (x^x)^(x^x) = (x#2)^(x#2) x # 8 = (x#4)^(x#4) .... x # 2^(n+1) = (x # 2^n )^(x # 2^n) For doing this let $ be the operation defined inductively by x $ 0 = x x $ (n+1) = (x$n)^(x$n) Then we can easily see that x $ n = x # 2^n. The interesting thing however is that we can regard x$n as the n-times application of the function f(x)=x^x, i.e. x $ n = f^n(x) := fo...of(x). Now there is a unique analytic iteration group (though to put out all details would take too much room here) for f, especially f^(1/n) has has asymptotically at (the fixed point) 1 the unique formal powerseries f^(1/n). Hence we can define x$r in a unique way as f^r(x). And then we go back, defining x#r = x $ (log_2 r). x#r is then also a quite rapidly increasing operation (though probably slower than x^^r but faster than any x^(x^(...x^r)) for a fixed number of x's). Of course nobody can tell physics applications for *this* function, if it was unknown yet. There is a bunch of similar operations #, by letting the above f for example be f(x)=x^(x^x) or f(x)=x^(x^2) and then taking the log_3 instead of log_2, or doing so with other bracketings of n x's and taking log_n instead. === Subject: Re: Physical Applications of Tetration Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Am 29.03.2007 21:30 schrieb Ioannis: I found a text called WexZal , which deals with the x^^2 term. Don't know about the relevance regarding your question. It was some years ago, so I don't know, whether this document was continued, or whether it is still online at all. cite from Preface of WexZal: HTH - Gottfried Helms === Subject: commutative algebra nomenclature Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let k be a field; in the cases I care about, k has characteristic p. Let A be the ring A = k[x_n: n in Z] / ((x_n)^p - x_{n+1} : n in Z). Think of A as being a modification of a polynomial algebra on a single variable x, where x_n corresponds to x^{p^n}: in A, you have all (p^n)th roots of x. So for example, A contains as a subring which is isomorphic to k[x]. Is there a standard name for A, as obtained from k[x]? (When k has characteristic p, I'm hoping for some name like the Frobenius localization, but maybe it's called a root algebra?) name for this, in relation to k[x]. (The Frobenius dual??) -- J. H. Palmieri Associate Professor of Mathematics University of Washington Box 354350, Seattle, WA 98195-4350 palmieri@math.washington.edu http://www.math.washington.edu/~palmieri/ === Subject: Statistical Challenges and Advances in Brain Science (Statistica Sinica) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) This is a gentle reminder that the deadline for paper submissions to the Brain Science theme is approaching -- it's April 15, 2007. Please upload your submissions via our on-line system at http://www.stat.sinica.edu.tw/statistica/submission/ . Please contact me if you have any problems with uploading your files or have questions regarding the special issue. With all the best, Karen Li---on behalf of the Guest Editors Editorial Assistant Statistica Sinica --------------------------------------------------------------- Guest Editors: John Aston, Academia Sinica Emery Brown, MIT/Massachusetts General Hospital Keith Worsley, McGill University Yingnian Wu, UCLA Summary: EMP spam === === Subject: Re: Refreshing basics for postgraduate study Rather than forgetting any rule, you may just have been misled by bad notation. In algebraic formulae people sometimes insert a dot to indicate multiplication, for example a.b instead of ab . But IMHO it's a bad practice to carry over that convention to arithmetical formulae, as the multiplication dot is easily confused with a decimal point. The traditional multiplication sign is clearer, in your example (3^2 x 2^2)^2 . (That last dot was just a fullstop. :-) Ken Pledger. === Subject: Re: Refreshing basics for postgraduate study Well ,yes but someone is bound to think 'x' is a variable and get 1296x^2 .Whether you use '.' or 'x' somebody has to tell you what the symbol denotes ,or you have to have the gift of guess.Hmm,I was just being humerous but === Subject: My old clock Good morning to all, As the Beatles sang: with a little help from my friends............. My previous puzzle, about the paper hat, got a surprising turn. The solution I had in mind possibly can be surpassed. But in order to be able to establish its correctness, I kindly call for your help. Please refer to Oplossing 171 on my website, underneath puzzle 172. My new puzzle, 172, is about a clock I found earlier this week, while cleaning the barn. It is not running anymore, but it does show some peculiar properties. Have fun with the puzzle ! Peter http://home.planet.nl/~p.j.hendriks/ppvdw.htm (click on the Union Jack. Please answer by email, and NOT in this newsgroup. === Subject: Calculating_Primes Mail-To-News-Contact: abuse@dizum.com Rogue scientist Jim McCanney has published a book in which he shows how to calculate prime numbers. He claims anyone with a grade three education can now do it. Hear him laugh at the megalomaniac vermin who have stockpiled warehouses full of chips to be injected into the masses - They are useless now, because with a prime number calculator the encryption can be broken. http://www.jmccanneyscience.com/WeeklyRadioShowArchivesSubPage.HTM March 22 Summary: EMP spam === === Subject: basic differentiation i have an equation s = (t+1)/(t+3). I want to differentiate it to get ds/dt ? whats the rule for this. === Subject: Re: basic differentiation Here's a way to approach all questions like that. Suppose you were to put in a numerical value for t, for example t = 2. First you'd find t + 1 = 2 + 1 = 3, and t + 3 = 2 + 3 = 5, and finally you'd divide 3 by 5 to get 3/5. The _last_ step in such a calculation tells you the _first_ step in differentiating. In this example the last step was a division, so begin the differentiation by using the quotient rule. HTH Ken Pledger. === Subject: simple Problem... I have a very simple Problem. I have a matrix C (6*6). And i would like to get the conditions about the elements of this matrix if my matrix is invariant of rotation. I mean: Rot(C)=C But I dont 'know how i can do it. Subject: Binary subtraction hiya I'd like to learn how to subtract binary numbers ( here I don't mean subtracting by finding two's complement, but instead actually subtracting the number ), but I realised I don't even know how to subtract decimal numbers using only paper and a pen ( of course I can subtract decimals mentaly )! Can you show me how to subtract decimals ? Can you also show me how to subtract binary numbers, and more importantly explain why the method works? thank you === Subject: Re: Binary subtraction -- Paul Sperry Columbia, SC (USA) === Subject: Re: Binary subtraction Didn't you learn this in elementary school? Or haven't got around to this. -- Geo. Michael Henry And one of the hot topics for me is the number of Christian atheists who are fully committed to living according to the teachings of Jesus, but unwilling to accept the idea of God. === Subject: Re: Metric space open ball problem I have an idea for a proof, but I'm not sure it's very rigorous. Please critique it. center of all the open balls B(x;r) (i.e. x is a member of all the open balls), therefore x is a member of the intersection over all the Now, if we assume another member of the intersection over the open construct a new open ball B(x;r_1) in which y is not contained. No 0 = say r_0, and we can always construct an open ball with radius r_0 contained in at least one open ball centered at x. This is because pick, there are open balls which do not contain y and therefore this therefore this y cannot be contained in the intersection over all open The only case where we cannot construct an open ball B(x;r_0) which does not contain y is when d(x,y) = 0, since there are no open balls B(x;r) with d(x,y) < 0. And also, by definition of a metric, when d(x,y) = 0, y = x. Therefore (I think), the theorem is proven. Please let me know how I can improve my proof, and thank you very much everybody for your wonderful help. === Subject: Re: Metric space open ball problem OK, but would be a good idea to mention the property of a metric that you are using ( d(x, y) = 0 if x = y). [ Incidentally, / (back slash - forward slash) is better for intersection than ^ which is usually used for exponentiation.] This is what I didn't care for previously. You show that the statement y is not in the intersection or y = x when what you really want is only y = x. Here is what you had previously which is OK: [...] I ordinarily wouldn't do this, but you have obviously thought about the problem and maybe an example would help. I don't claim that what follows is a paragon of exposition - you might want to flesh it out a little; it is more or less what you have previously said but rewritten a bit. { x } is a subset of I. is not in I. It follows that if y is in I then y = x so I is a subset of { x }. Since I is a subset of { x } ond { x } is a subset of I, I = { x }.// Beginning student to teacher, it wouldn't fly. Now, if you want, try it for _closed_ balls. -- Paul Sperry Columbia, SC (USA) === Subject: Re: Metric space open ball problem I think this is the part I was missing in my understanding about how to go about a formal proof. I missed the connection between what I see this now. === Subject: Re: Metric space open ball problem Note that Paul is citing the definition of the metric to Then from this statement he concludes x in I. Each statement has support from something previously defined, stated, proven, etc. Note how clear Paul's statements are. There's nothing like Well if we kinda did this...then something like this would happen. - it's very cut and dry. It's good to be able to explain a proof (and the idea behind it), but when you write a proof you do not usually want it to sound like a casual conversation. It should sound like a chain of reasoning with as little excess words as possible. This is another good point - the formality depends on the audience. One of my professors this semester was really frustrated because students were not including their hypotheses in the beginning of proofs, nor were they justifying obvious statements. His advice to the class, Pretend I'm stupid and have no idea what I'm talking about... ..because I usually don't! ^_^ === Subject: mechanical vibration with external force I have a differential equation: y''+ay'+y = coswt where a is damping coefficient and w is forcing frequency. I solved for y(t), and here's my question. Let R be the amplitude of y(t), which is periodic motion with some damping and external force in it, and R is a function of a and w. R(a,w) = 1 / sqrt((w^2-1)^2 + a^2w^2)) I got lim R(w,a) = 1 and lim R(w,a) = 0 I was having trouble interpreting the bottom limit in terms of the mechanical oscillation. If the forcing frequency goes to infinity, the amplitude goes to zero. What's really happening physically? Joseph === Subject: Re: mechanical vibration with external force If the forcing frequency goes to infinity, how much TIME does the oscillator have to go up before it is being forced back down again? === Subject: Re: trig question I don't want to start any new convention, I just made an observation. === Subject: Why Google Is Failing as a Online Advertising Company -Eric. Webcon 2007 baby lets do texas. We want deep pockets we want deep pockets. Leave Deep Voicemail! 206-666-4400 Ext 666 This guys a right off. I pretend to work for them, they pretend to pay me === Subject: Graphs make math more interesting Graphs make math more interesting, do you think so? John ------------------------------------------------------- http://www.graphnow.com === Subject: Re: Graphs make math more interesting there are packages freely available to download that will do all that for nothing === Subject: Re: Graphs make math more interesting 1. I don't know what you mean by graphs. The word has several different meanings in mathematics. 2. I don't know what you consider interesting. Different people will consider different thing interesting. 3. You may learn visually. Some people learn better by seeing objects, some learn better by reading explanations. It's a matter of style. === Subject: Integer programing I have a scheduling problem to do and I'm not sure where to start. were given 10 employee's who can work either five eight hour shifts or four ten hour shifts. If an employee works five eight hour shifts the shifts must be consecutive, if an employee works four ten hour shifts there must be a break of atleast one day after the second day is worked. All shifts start at the same time. Last each employee works 40 hours a week. So if Bill starts work on Monday at 6 am he then works tuesday, wednesday,thursday, and friday from 6 am till 2 pm. I'm an optimal schedule that I need to minimize the sum of the absolute values of the deviations from the optimal schedule. Any help would be appreciated. ME === Subject: Re: Integer programing well you've defined the problem, next thing you need is to formulate a function, write down your constraints, and use the simplex method to minimise it according to the constraints === Subject: topology compact Hi there, consider an infinite dimensional Banach space X and denote by V a finite dimensional subspace of X. Let C(B(R)) denote the closed ball of radius R centered at the origin of X and consider the intersection of C(B(R)) and V. Denote such an intersection with J. Then J is a closed bounded subset of V, so (by Heine-Borel) J is a compact subset of V, V viewed as a topological subspace of X. Is J also a compact subspace of X? === Subject: GMAT question - can you help? Can you help me with this GMAT question? In a rectangular coordinate system, there are three points which form a triangle, point A(2,3) , B(4,4) and C.What are the coordinates of point C? (1) The distance between A and C are 2 feet (2) The distance between B and C are 2 feet Ron Summary: EMP spam === === Subject: Trying to find 2 functions I have two rows of *random* data and for each one of them, I am trying to find a corresponding function that matches to this data, if such exist. The two rows are something like (replacing the values with abstract letters): Av Ev Av-1 Ev-1 . . . . . . A1 E1 A0 E0 What I consider as axioms: f(Av)< f(Av-1) < ...