| 1. |
- Lindahl, Karl-Olof
(författare)
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On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider the problem of local linearization of power series defined over complete valued fields. The complex field case has been studied since the end of the nineteenth century, and renders a delicate number theoretical problem of small divisors related to diophantine approximation. Since a work of Herman and Yoccoz in 1981, there has been an increasing interest in generalizations to other valued fields like p-adic fields and various function fields. We present some new results in this domain of research. In particular, for fields of prime characteristic, the problem leads to a combinatorial problem of seemingly great complexity, albeit of another nature than in the complex field case. In cases for which linearization is possible, we estimate the size of linearization discs and prove existence of periodic points on the boundary. We also prove that transitivity and ergodicity is preserved under the linearization. In particular, transitivity and ergodicity on a sphere inside a non-Archimedean linearization disc is possible only for fields of p-adic numbers.
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| 2. |
- de Gosson de Varennes, Serge
(författare)
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Multi-oriented Symplectic Geometry and the Extension of Path Intersection Indices
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Symplectic geometry can be traced back to Lagrange and his work on celestial mechanics and has since then been a very active field in mathematics, partly because of the applications it offers but also because of the beauty of the objects it deals with.I this thesis we begin by the simplest fact of symplectic geometry. We give the definition of a symplectic space and of the symplectic group, Sp(n). A symplectic space is the data of an even-dimensional space and of a form which satisfies a number of properties. Having done this we give a definition of the Lagrangian Grassmannian Lag(n) which consists of all n-dimensional subspaces of the symplectic space on which the symplectic form vanishes. We carefully study the topology of these spaces and their universal coverings.It is of great interest to know how the elements of the Lagrangian Grassmannian intersect each other. A lot of efforts have therefore been made to construct intersection indices for elements of Lag(n). They have gone under many names but have had a sole purpose, namely to give us a way to determine how these elements intersect. We show how these elements are constructed and extend the definition to paths of elements of Lag(n) and Sp(n). We end this thesis by extending the definition of an index defined by Conley and Zehnder bu using the properties of the Leray index. Their index plays a significant role in the theory of periodic Hamiltonian orbit.
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| 3. |
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| 4. |
- Nyqvist, Robert
(författare)
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Algebraic Dynamical Systems, Analytical Results and Numerical Simulations
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we study discrete dynamical system, given by a polynomial, over both finite extension of the fields of p-adic numbers and over finite fields. Especially in the p-adic case, we study fixed points of dynamical systems, and which elements that are attracted to them. We show with different examples how complex these dynamics are.For certain polynomial dynamical systems over finite fields we prove that the normalized average of the numbers of linear factors modulo prime numbers exists. We also show how to calculate the average, by using Chebotarev's Density Theorem. The non-normalized version of the average of the number of linear factors of linearized polynomials modulo prime numbers, tends to infinity, so in that case we find an asymptotic formula instead.We have also used a computer to study different behaviors, such as iterations of polynomials over the p-adic fields and the asymptotic relation mention above. In the last chapter we present the computer programs used in different part of the thesis.
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| 5. |
- Petersson, Henrik, 1973-
(författare)
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Infinite dimensional holomorphy in the ring of formal power series : partial differential operators
- 2001
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study holomorphy in the ring of formal power series in an infinite number of variables. Thus we restrict our study to (infinite dimensional) holomorphy on sequence spaces and we show that we obtain a rich theory without requiring any topological structure on the domain space. We make a comprehensive PDO-study for the spaces under consideration.As a basis we establish the Martineau duality, described by the Fourier-Borel transform, between spaces of entire and of exponential type functions in an infinite number of variables. In order to study PDO:s and PDE:s in spaces in the ring of formal power series, such an established duality is a useful tool for the transpose of a differential operator become the operator of multiplication by the corresponding symbol. The second part of the thesis is devoted to applications of the Martineau duality for various PDO-related problems. The following topics are considered: Existence theorems, Approximation theorems, Fischer decompositions, Cauchy problems, PDE-preserving projectors (Kergin projector) and Pseudo-differential operators.Some of the main results are the following. We prove Malgrange type existence theorems for infinite dimensional differential operators on A, Exp and F and we show that homogenous solutions can be approximated by such solutions consisting of exponential (finitely supported) polynomials. Here A and Exp are the spaces of entire respectively exponential type functions and F is the Fischer-Fock Hilbert space (in an infinite number of variables). We extend the notions of Fischer decomposition and Fischer pair, studied by H. Shapiro, J. Aniansson etc. A Fischer pair for a space is a pair of maps (here differential operators) whose kernel respective transpose image decompose the space into a direct sum. We establish some necessary and sufficient conditions for that a given pair of maps will make up a Fischer pair. By generalizing a known result for finite dimensional domain spaces, we show the existence of non-trivial Fischer pairs for A and Exp.Moreover, we prove some infinite dimensional generalizations of results obtained by H. Shapiro. In particular we show that densely defined differential operators together with their adjoints constitute Fischer pairs for F. We show that Cauchy-and dual Cauchy problems, w.r.t differential operators, in Exp respective in Exp'=A are well-posed. We prove that the infinite dimensional Kergin operator has interpolating and PDE-preserving properties and that it is uniquely determined by these properties. A PDE-preserving projector is a projector that preserves homogeneous solutions to differential equations.
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