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International Congress of Mathematicians International Congress of Mathematicians Hyderabad, August 19–27, 2010 Abstracts Short Communications Posters Editor Rajendra Bhatia Co-Editors Arup Pal G. Rangarajan V. Srinivas M. Vanninathan Technical Editor Pablo Gastesi Contents Section 1: Logic and Foundations ....................... .............. 1 Section 2: Algebra................................... ................. 10 Section 3: Number Theory.............................. .............. 71 Section 4: Algebraic and Complex Geometry............... ........... 106 Section 5: Geometry.................................. ................ 116 Section 6: Topology.................................. ................. 150 Section 7: Lie Theory and Generalizations............... .............. 177 Section 8: Analysis.................................. .................. 188 Section 9: Functional Analysis and Applications......... .............. 254 Section 10: Dynamical Systems and Ordinary Differential Equations . 287 Section 11: Partial Differential Equations.............. ................. 338 Section 12: Mathematical Physics ...................... ................ 375 Section 13: Probability and Statistics................. .................. 431 Section 14: Combinatorics............................ .................. 460 Section 15: Mathematical Aspects of Computer Science..... ............ 509 Section 16: Numerical Analysis and Scientific Computing... ............ 533 Section 17: Control Theory and Optimization ............. ............. 556 Section 18: Mathematics in Science and Technology ........ ............ 589 Section 19: Mathematics Education and Popularization of Mathematics........................................ ....... 640 Section 20: History of Mathematics..................... ................ 657 Section 21: Mathematical Software..................... ................ 669 Supplement ................................................... ........ 673 Section 2: Algebra................................... ................. 675 Section 4: Algebraic and Complex Geometry............... ........... 678 Section 5: Geometry.................................. ................ 680 Section 6: Topology.................................. ................. 682 Section 7: Lie Theory and Generalizations............... .............. 684 Section 8: Analysis.................................. .................. 687 Section 9: Functional Analysis and Applications......... .............. 690 Section 10: Dynamical Systems and Ordinary Differential Equations . 693 Section 11: Partial Differential Equations.............. ................. 699 iv Contents Section 12: Mathematical Physics ...................... ................ 700 Section 13: Probability and Statistics................. .................. 703 Section 14: Combinatorics............................ .................. 707 Section 16: Numerical Analysis and Scientific Computing... ............ 709 Section 17: Control Theory and Optimization ............. ............. 737 Section 18: Mathematics in Science and Technology ........ ............ 741 Section 19: Mathematics Education and Popularization of Mathematics........................................ ....... 749 Section 20: History of Mathematics..................... ................ 751 Author Index........................................ ................... 755 Section 1 Logic and Foundations Axiom of Choice and Euclid Axiom 8 Sukhotin Alexander M. Tomsk Polytechnic University, Russia E-mail: [email protected] 2000 Mathematics Subject Classification. 01, 02, 01A55 1. The problem of infinity in mathematics is one of those “eternal problems” to which the science comes back on each coil of its development. Ancient greek mathematician Euclid (nearby 340–287 B.C.) introduced the Axiom 8 “And the Whole is more then its Part” [1, I, p. 15]. On the boundary of XIX and XX centuries founders of the sets theory rejected Euclid’s Axiom 8 by ignoring (or explisitly); an equivalency between any set and its own part has referred to as characteristic property of infinite sets. P. Cohen considered the existence of infinite collections as the most important issue, he recognized the neces- sity of inserting infinity axiom and emphasized, that “...by tradition the atti- tude to infinite sets was the criterion of discrepancy between mathematicians” [2, IV.13]. We think, at an axiomatization of the sets theory Euclid’s Axiom 8 was incorrectly transformed into “an axiom of infinity”. Any formulation of Axiom infinity [3, S. 11], as well as Peano’s Axiom of natural numbers set infinity [3, S. 12], is of potential character. Now, we formulate first statement. Hypothesis 1. Any infinite set is of potential nature. 2. Remaining within the framework of Halmosh sets naive theory [3], we proved Euclid’s Axiom 8 in the form of following theorem. Theorem 1. B ⊂ A ⇒ (∀ϕ : A → B∃(a,q),a,q ∈ A : a 6= q & ϕ(a)= ϕ(q)). Theorem 1 has the following canonically short form: B ⊂ A ⇒ ¬(A ∼ B), in particular, q∈ / A ⇒ ¬(A ∼ (A ∪ {q})). This theorem demonstration contains, 2 Logic and Foundations in particular, references to Axiom of Choice [3, S. 15], the concept of exact rearrangement of set A and concept of the maximal element of chains family formed from set A subsets and others too. Remaining within Theorem 1, now we formulate the following two sugges- tions: Hypothesis 2. Euclid Axiom 8 is unprovable without Axiom of Choice. Statement 1. The power sets theory and continuum hypothesis ([2]–[3]) has new way of development with Euclid’s Axiom 8. Work is executed at financial support ADTP, the project N o4207 References [1] Euclid’s, “Elements”,: transl. from Greek.–M.–L.: OGIZ.–Books I–VI, 1948. (In Russian). [2] P.J. Cohen, Set Theory and the Continuum Hypothesis,–New York: W.A. 1964. [3] Paul.R. Halmosh, Naive Set Theory, Princeton: D. Van Nostrand Company, Inc., 1960. ❖ ❖ ❖ Effective Eilenberg Machines Benoˆıt Razet STCS, Tata Institute of Fundamental Research, Homi Bhabha road 400005 Mumbai, India E-mail: [email protected] 2000 Mathematics Subject Classification. 68Q10, 68Q45 We revisit the idea of Eilenberg [1] to use automata labelled with binary re- lations as a general computational model. The Eilenberg machines model has interesting properties. First, It is general because it unifies most of the devices having a finite-state control (including Turing machines). Second, it is expres- sive thanks to the following modularity property: an Eilenberg machine defines a characteristic relation that shall be used in another Eilenberg machine. We study the effective fragment of Eilenberg machines for which we propose a sim- ulation engine [2, 3]. The correction of the simulation engine is proved using a proof assistant. This approach of effective Eilenberg machines may be used to solve properly problems which use multiple stages of traditional finite-state machines. Logic and Foundations 3 References [1] S. Eilenberg, Automata, Languages and Machines, Volume A, Academic Press, New York, USA, 1974. [2] G. Huet and B. Razet, Computing with Relational Machines, Tutorial at ICON2008, Pune, India, 2008. [3] B. Razet, Machines d’Eilenberg Effectives, Ph.D. Thesis, University of Paris- Diderot, France, 2009. ❖ ❖ ❖ Another Arithmetization: and Godels Second Incompleteness Theorem Khanindra Chandra Chowdhury∗ Department of Mathematics, Gauhati University, Guwahati-781014, INDIA E-mail: [email protected] Maitrayee Chowdhury 2000 Mathematics Subject Classification. 03B10, 03B25, 03B53 Here we present a discussion to arithmetize a consistency statement for any formal system with a recursive set of axioms, a finite number of finitary and recursive rules of inference including modus ponens. In informal language, the consistency of S shall be expressed by the sentence: there are no meaningful formulae y and z where z is the negation of y such that both are theorems of S. we then show that the formalized consistency statement for the arbitrary system S cannot be proved in a certain sub-theory T of Peanos arithmetic. This arithmetic theory T with + and ., is much stronger than Q. It will follow that the consistency of Q formalized in the manner we do, cannot be proved in T, let alone in Q itself, which is a sub-theory of T. When formalizing the notion of theorem-hood in any arithmetic system with + and ., we have to refer to sequences in the formal language, since a proof is a sequence of formulae. In the construction of the proof-predicate the usual method is to allot (recursively) a single Godel number (g.n.) to each formula and then a single g.n. to a sequence of formulae (forming a proof, say of the last formula ). This is what we have done. The novelty in our present construction is that instead of codifying sequences by single natural numbers, we use a number triple. This device enables us, as we shall see, to do away with the need to find a defining formula for the recursive function xy. With a Godel triple to codify a sequence, our proof-predicate is a formula ProvS (c,d,l,y) with 4 free variables, 3 of which are thought of as always appearing in a bunch. ProvS (c,d,l,y) says that the triple < c,d,l > yields a sequence of expressions (i.e. their g.n.’s) constituting a proof in S of the formulae with g.n.y. Thus the triple < c,d,l > 4 Logic and Foundations merely indicates a sequence of expressions without mentioning anything about it being finite or infinite. We shall exploit this to our advantage, using an infinite sequence of expressions to constitute a proof of its last member. ❖ ❖ ❖ Analogy of Hilbert’s Tenth Problem in Observers Mathematics
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