Institut für Mathematik
https://kobra.uni-kassel.de:443/handle/123456789/2010061733331
Sat, 28 Nov 2020 17:47:27 GMT2020-11-28T17:47:27ZSkriptum zur Linearen Algebra und Analytischen Geometrie
https://kobra.uni-kassel.de:443/handle/123456789/11838
Vorlesungsskript zur Vorlesung „Lineare Algebra und Analytische Geometrie“ an der Universität Kassel im Sommersemester 2020. Es handelt sich um die Druckversion.
Wed, 01 Jul 2020 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/118382020-07-01T00:00:00ZKemm, FriedemannVorlesungsskript zur Vorlesung „Lineare Algebra und Analytische Geometrie“ an der Universität Kassel im Sommersemester 2020. Es handelt sich um die Druckversion.Explicit Description Of Isogeny And Isomorphism Classes Of Drinfeld Modules Of Higher Rank Over Finite Fields
https://kobra.uni-kassel.de:443/handle/123456789/11635
When jumping from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r ≥ 3. In this thesis we investigate the classes of those Drinfeld modules of higher rank r ≥ 3 defined over a finite field L. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules defined over a finite field L for any rank r ≥ 3, which generalizes what Yu already did for r = 2. We also provide a necessary and sufficient condition for an order O in the endomorphism algebra corresponding to some isogeny classes, to be the endomorphism ring of a Drinfeld module. To complete the classification, we define the notion of fine isomorphy invariants for any rank r Drinfeld module defined over a finite field L and we prove that the fine isomorphy invariants together with the J-invariants describe the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.
Wed, 01 Jan 2020 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/116352020-01-01T00:00:00ZNkotto Nkung Assong, SedricWhen jumping from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r ≥ 3. In this thesis we investigate the classes of those Drinfeld modules of higher rank r ≥ 3 defined over a finite field L. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules defined over a finite field L for any rank r ≥ 3, which generalizes what Yu already did for r = 2. We also provide a necessary and sufficient condition for an order O in the endomorphism algebra corresponding to some isogeny classes, to be the endomorphism ring of a Drinfeld module. To complete the classification, we define the notion of fine isomorphy invariants for any rank r Drinfeld module defined over a finite field L and we prove that the fine isomorphy invariants together with the J-invariants describe the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.Power Series Representations of Hypergeometric Type and Non-Holonomic Functions in Computer Algebra
https://kobra.uni-kassel.de:443/handle/123456789/11598
A Laurent-Puiseux series
$$ \sum\limits_{n = n_0}^{\infty }{a_n (z - z_0)^{n/k} (a_n \in K, k \in ℕ, n_0 \in ℤ ) } \quad (1) $$
where $ k $ denotes the corresponding Puiseux number and $ K $ an infinite computable field - mostly $ K= ℚ(α_1,\ldots,α_n) $ : a field of rational functions in several variables, is mainly characterized by the general coefficient. We consider the case where an is a term of an m-fold hypergeometric sequence.
That is $ a_{n+m} = r(n) a_n $ for all sufficiently large integers $ n, r(n) $ is a rational function over $ K $, and m is a positive integer. A Laurent-Puiseux series with an $ m $-fold hypergeometric sequence as general coefficient is said to be of hypergeometric type, with type $ m $. We call hypergeometric type function any expression (mostly meromorphic) that can be written as a hypergeometric type series.
To find the general coefficient in (1) of a given hypergeometric type function, three key steps are to be considered (see [1]). Given an expression $ f $ ,
1. find a holonomic differential equation (DE) satisfied by $ f $;
2. deduce a holonomic recurrence equation (RE) satisfied by the Taylor coefficients of $ f $;
3. find all m-fold hypergeometric term solutions of the obtained RE.
Last but not least, the series representation is handled by determining the linear combination of all the resulting hypergeometric type series provided some initial values using Taylor approximation of suitable order.
The understanding of these three steps is essential for our work. In [1], Koepf described the first two steps for getting holonomic recurrence equations of any given hypergeometric type function. But the third step was not complete as he considered three sub-families of hypergeometric type functions: exp-like functions, rational functions, and the functions whose recurrence equation obtained in step 2 is a two-term recurrence relation. In this thesis, we clearly solve the third step and develop a complete algorithm to compute power series of linear combinations of hypergeometric type functions by using a new algorithm which finds all m-fold hypergeometric term solutions of holonomic recurrence equations. Also, we investigate an algorithm to represent power series of non-holonomic and non-hypergeometric type functions like $ \tan (z), \frac{1 + \tan(z)}{1-\tan(z)} , \frac{z}{\exp(z)-1} , \frac{\tan^{-1}(z)}{z+1} , \exp(z²+z),\: etc.$
In addition, we confirm the asymptotically fast behavior of an algorithm based on holonomic recurrence equations to compute Taylor expansions of holonomic functions (see [2], Chapter 10), and present some interesting results for the automatic proof of certain identities that are generally difficult to prove (see [2], Chapter 9) like
$$ \frac{1+\tan(z)}{1-\tan(z)} = \exp \left(2 \tanh^{-1}\left(\frac{\sin(2z)}{cos(2z)+1}\right) \right) $$
by characterizing non-holonomic functions with non-linear recurrence equations and some initial values.
Our implementations are done in the computer algebra system (CAS) Maxima 5.37.2, and regrouped in our package FPS. The CAS Maple is also used for comparison in order to show the improvement given by our algorithms and their implementations.
References
[1] Koepf, W.: Power series in computer algebra. Journal of Symbolic Computation 13, 1992, 581-603
[2] Koepf, W.: Computeralgebra. Eine algorithmisch orientierte Einführung. Springer, Berlin-Heidelberg, 2006, ISBN 3-540-29894-0
Wed, 10 Jun 2020 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/115982020-06-10T00:00:00ZTeguia Tabuguia, BertrandA Laurent-Puiseux series
$$ \sum\limits_{n = n_0}^{\infty }{a_n (z - z_0)^{n/k} (a_n \in K, k \in ℕ, n_0 \in ℤ ) } \quad (1) $$
where $ k $ denotes the corresponding Puiseux number and $ K $ an infinite computable field - mostly $ K= ℚ(α_1,\ldots,α_n) $ : a field of rational functions in several variables, is mainly characterized by the general coefficient. We consider the case where an is a term of an m-fold hypergeometric sequence.
That is $ a_{n+m} = r(n) a_n $ for all sufficiently large integers $ n, r(n) $ is a rational function over $ K $, and m is a positive integer. A Laurent-Puiseux series with an $ m $-fold hypergeometric sequence as general coefficient is said to be of hypergeometric type, with type $ m $. We call hypergeometric type function any expression (mostly meromorphic) that can be written as a hypergeometric type series.
To find the general coefficient in (1) of a given hypergeometric type function, three key steps are to be considered (see [1]). Given an expression $ f $ ,
1. find a holonomic differential equation (DE) satisfied by $ f $;
2. deduce a holonomic recurrence equation (RE) satisfied by the Taylor coefficients of $ f $;
3. find all m-fold hypergeometric term solutions of the obtained RE.
Last but not least, the series representation is handled by determining the linear combination of all the resulting hypergeometric type series provided some initial values using Taylor approximation of suitable order.
The understanding of these three steps is essential for our work. In [1], Koepf described the first two steps for getting holonomic recurrence equations of any given hypergeometric type function. But the third step was not complete as he considered three sub-families of hypergeometric type functions: exp-like functions, rational functions, and the functions whose recurrence equation obtained in step 2 is a two-term recurrence relation. In this thesis, we clearly solve the third step and develop a complete algorithm to compute power series of linear combinations of hypergeometric type functions by using a new algorithm which finds all m-fold hypergeometric term solutions of holonomic recurrence equations. Also, we investigate an algorithm to represent power series of non-holonomic and non-hypergeometric type functions like $ \tan (z), \frac{1 + \tan(z)}{1-\tan(z)} , \frac{z}{\exp(z)-1} , \frac{\tan^{-1}(z)}{z+1} , \exp(z²+z),\: etc.$
In addition, we confirm the asymptotically fast behavior of an algorithm based on holonomic recurrence equations to compute Taylor expansions of holonomic functions (see [2], Chapter 10), and present some interesting results for the automatic proof of certain identities that are generally difficult to prove (see [2], Chapter 9) like
$$ \frac{1+\tan(z)}{1-\tan(z)} = \exp \left(2 \tanh^{-1}\left(\frac{\sin(2z)}{cos(2z)+1}\right) \right) $$
by characterizing non-holonomic functions with non-linear recurrence equations and some initial values.
Our implementations are done in the computer algebra system (CAS) Maxima 5.37.2, and regrouped in our package FPS. The CAS Maple is also used for comparison in order to show the improvement given by our algorithms and their implementations.
References
[1] Koepf, W.: Power series in computer algebra. Journal of Symbolic Computation 13, 1992, 581-603
[2] Koepf, W.: Computeralgebra. Eine algorithmisch orientierte Einführung. Springer, Berlin-Heidelberg, 2006, ISBN 3-540-29894-0No Chaos in Dixon's System
https://kobra.uni-kassel.de:443/handle/123456789/11470
The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.
Wed, 01 Jan 2020 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/114702020-01-01T00:00:00ZSeiler, Werner M.Seiß, MatthiasThe so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.