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1、<p><p>&lt;p&gt;&amp;lt;p&amp;gt;&amp;amp;lt;p&amp;amp;gt;&amp;amp;amp;lt;p&amp;amp;amp;gt;&amp;amp;amp;amp;lt;p&amp;amp;amp;amp;gt;Formal Groups, Integrable Systems and Number Theory Universidad Compl
2、utense, Madrid, Spain and Scuola Normale Superiore, Pisa, Italy Piergiulio Tempesta Gallipoli, June 18, 2008 Outline: the main characters ? Finite operator theory ? Number Theory Exact (and Quasi Exact) Solvability Formal solutions of linear difference equations Application: Superintegrability Integ
3、rals of motion Generalized Riemann zeta functions New Appell polynomias of Bernoulli type ? Formal groups: a brief introduction Symmetry preserving Discretization of PDEs Finite Operator Calculus Formal group laws Algebraic Topology Riemann-type zeta functions Bernoulli-type polynomials Delta operat
4、ors Symmetry preserving discretizations nP. Tempesta., C. Rend. Acad. Sci. Paris, 345, 2007 nP. Tempesta., J. Math. Anal. Appl. 2008 nS. Marmi, P. Tempesta, generalized Lipschitz summation formulae and hyperfunctions 2008, submitted nP. Tempesta, L - series and Hurwitz zeta functions associated with
5、 formal groups and universal Bernoulli polynomials (2008) nP. Tempesta, A. Turbiner, P. Winternitz, J. Math. Phys, 2002 nD. Levi, P. Tempesta. P. Winternitz, J. Math. Phys., 2004 nD. Levi, P. Tempesta. P. Winternitz, Phys Rev D, 2005 Formal group laws Let R be a commutative ring with identity be the
6、 ring of formal power series with coefficients in R Def 1. A one-dimensional formal group law over R is a formal power series s.t. s.t. When the formal group is said to be commutative. a unique formal series such that Def 2. An n-dimensional formal group law over R is a collection of n formal power
7、series in 2n variables, such that 1) The additive formal group law 2) The multiplicative formal group law 3) The hyperbolic one ( addition of velocities in special relativity) 4) The formal group for elliptic integrals (Euler) Indeed: Examples Connection with Lie groups and algebras Let us write: An
8、y n- dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part : ? More generally, we can construct a formal group law of dimension n from any algebraic group or Lie group of the same dimension n, by taking coordinates at the identity and
9、 writing down the formal power series expansion of the product map. An important special case of this is the formal group law of an elliptic curve (or abelian variety) ? Viceversa, given a formal group law we can construct a Lie algebra. Algebraic groups Formal group laws Lie algebras ? Bochner, 194
10、6 ? Serre, 1970 - ? Novikov, Bukhstaber, 1965 - The associated formal group exponential is defined by so that Def 4. The formal group defined by the Lazard Universal Formal Group The Lazard Ring is the subring of generated by the coefficients of the power series Bukhstaber, Mischenko and Novikov : A
11、ll fundamental facts of the theory of unitary cobordisms, both modern and classical, can be expressed by means of Lazards formal group. ? Algebraic topology: cobordism theory ? Analytic number theory ? Combinatorics is called Def. 3. Let be indeterminates over The formal group logarithm is Given a f
12、unction G(t), there is always a delta difference operator with specific properties whose representative is G(t) Main idea nThe theory of formal groups is naturally connected with finite operator theory. nIt provides an elegant approach to discretize continuous systems, in particular superintegrable
13、systems, in a symmetry preserving way nSuch discretizations correspond with a class of interesting number theoretical structures (Appell polynomials of Bernoulli type, zeta functions), related to the theory of formal groups. IntroductionIntroduction toto finite operator finite operator theorytheory
14、? G. C. Rota and coll., M.I.T., 1965- Umbral Calculus ? Silvester, Cayley, Boole, Heaviside, Bell, F Ft, P Pt F ? Di Bucchianico, Loeb (Electr.J. Comb., 2001, survey) algebra of f.p.s. algebra isomorphic to P* subalgebra of L (P) F F : subalgebra of shift-invariant operators Def 5. Q F is a delta op
15、erator if Q x = c 0. : polynomial in x of degree n. Def 6.is a sequence of basic polynomials for Q if Q F Def 7. An umbral operator R is an operator mapping basic sequences into basic sequences: ?Finite operator theory and Algebraic Topology . E: complex orientable spectrum AppellAppell polynomialsp
16、olynomials Additional structure in F : Heisenberg-WeylHeisenberg-Weyl algebra F , Q, x = 1Q: delta operator, D. Levi, P. T. and P. Winternitz, J. Math. Phys. 2004, D. Levi, P. T. and P. Winternitz, Phys. Rev. D, 2004 Lemma a) = , b) : basic sequence of operators for Q R: L(P) L(P) R R R R DeltaDelta
17、 operators, formal groupsoperators, formal groups and basic and basic sequencessequences Theorem 1: The sequence of polynomials satisfies: : generalized Stirling numbers of first kind : generalized Stirling numbers of second kind (Appell property)? Simplest example: = 1 Discrete derivatives: (Formal group exponentials) Finite operator Finite operator t&amp;amp;amp;amp;lt;/p&amp;amp;amp;amp;gt;&amp;amp;amp;lt;/p&amp;amp;amp;gt;&amp;amp;lt;/p&amp;amp;gt;&amp;lt;/p&amp;gt;&lt;/p&gt;</p></p>
