Dissertationenhttps://kobra.uni-kassel.de:443/handle/123456789/20100617334332024-03-19T13:05:15Z2024-03-19T13:05:15ZGeneration Human Body Motion by the Centralized NetworksMorozov, Ivanhttps://kobra.uni-kassel.de:443/handle/123456789/152062023-11-22T16:20:04Z2023-01-01T00:00:00ZThe main goal of the thesis is to describe and study reduced models for efficient simulations of human body motions (HBM). To this end, we propose new coupled oscillator models, which are networks of dynamically coupled elements. These networks consist of a few of centers and many satellites. The centers evolve in time as periodical oscillators with different frequencies. The satellite states are defined via center states by a radial basis function (RBF) networks. To simulate different motions we adjust the parameter of the RBF networks and frequencies of centers. Moreover, our network includes a switching module that allows to turn from a motion to another one, say, from jumping to walking. It is shown that this model is capable to simulate complicated motions consisting of many different elements dynamical primitives (walking, running, jumping etc.).
2023-01-01T00:00:00ZMorozov, IvanThe main goal of the thesis is to describe and study reduced models for efficient simulations of human body motions (HBM). To this end, we propose new coupled oscillator models, which are networks of dynamically coupled elements. These networks consist of a few of centers and many satellites. The centers evolve in time as periodical oscillators with different frequencies. The satellite states are defined via center states by a radial basis function (RBF) networks. To simulate different motions we adjust the parameter of the RBF networks and frequencies of centers. Moreover, our network includes a switching module that allows to turn from a motion to another one, say, from jumping to walking. It is shown that this model is capable to simulate complicated motions consisting of many different elements dynamical primitives (walking, running, jumping etc.).Generalized Involutive Bases and Their Induced Free ResolutionsOrth, Matthiashttps://kobra.uni-kassel.de:443/handle/123456789/140412023-08-18T09:19:13Z2022-05-01T00:00:00ZIn this thesis, we generalize several types of involutive and marked bases for ideals in quotient rings of commutative polynomial rings. We apply these new types of bases to the analysis of infinite free resolutions and of Hilbert schemes defined over certain types of quotient rings. We are mostly concerned with Pommaret and Janet bases; the marked bases we consider are marked over monomial submodules that are quasi-stable, i.e., that possess finite Pommaret bases.
Involutive bases of the types we consider induce free resolutions of the ideals they generate and hence they yield estimates for homological invariants of these ideals, for example, for their Betti numbers. In the special case of monomial Pommaret bases, one even obtains an explicit formula for the differential of the resolution. However, the induced resolution is not necessarily minimal, because already the involutive bases themselves are in general not minimal generating systems. Moreover, the application of involutive and marked bases was up to now confined to ideals in ordinary polynomial rings.
The thesis addresses both of these problems. Its contributions are split into four parts. In the first part, we introduce involutive-like bases, which are types of Gröbner bases that preserve many of the algorithmic and combinatorial advantages of involutive bases, while needing in general much less generators. We show that Janet-like and Pommaret-like bases induce involutive-like bases of the same types for their syzygy modules, and thus induce free resolutions in the same way that involutive bases do. Moreover, we use involutive-like bases to design new efficient algorithms for the determination of complementary decompositions of monomial ideals and Hilbert functions.
Next, we generalize involutive and involutive-like bases to include also ideals in quotient rings. Our discussion is based on a comprehensive treatment of Gröbner bases for ideals in such rings, together with algorithms for their construction. We establish that Pommaret bases in quotient rings also induce Pommaret bases of their syzygy modules in a natural way.
The third part of contributions treats the application of these new types of bases to the computation and analysis of their induced infinite free resolutions. For the important special case of Clements-Lindström rings, we obtain closed formulas for the Betti numbers of the resolution. We identify several classes of quasi-stable monomial ideals in these quotient rings for which the induced resolution is minimal. Thus, we generalize several well-known resolution constructions, e.g. for stable and for square-free Borel ideals. We obtain explicit formulas for the differential which apply to some classes of quasi-stable ideals and their Pommaret-like bases.
In the final part, we introduce relative marked bases for ideals in quotient rings defined by ideals generated by Pommaret marked bases. We give an algorithm for the construction of relative marked families in case the quotient ring is defined by a monomial ideal. Lastly, we use these bases to obtain information about the lex-points and about quasi-stable open coverings of the Hilbert schemes defined on some quotient rings, e.g., Clements-Lindström rings.
2022-05-01T00:00:00ZOrth, MatthiasIn this thesis, we generalize several types of involutive and marked bases for ideals in quotient rings of commutative polynomial rings. We apply these new types of bases to the analysis of infinite free resolutions and of Hilbert schemes defined over certain types of quotient rings. We are mostly concerned with Pommaret and Janet bases; the marked bases we consider are marked over monomial submodules that are quasi-stable, i.e., that possess finite Pommaret bases.
Involutive bases of the types we consider induce free resolutions of the ideals they generate and hence they yield estimates for homological invariants of these ideals, for example, for their Betti numbers. In the special case of monomial Pommaret bases, one even obtains an explicit formula for the differential of the resolution. However, the induced resolution is not necessarily minimal, because already the involutive bases themselves are in general not minimal generating systems. Moreover, the application of involutive and marked bases was up to now confined to ideals in ordinary polynomial rings.
The thesis addresses both of these problems. Its contributions are split into four parts. In the first part, we introduce involutive-like bases, which are types of Gröbner bases that preserve many of the algorithmic and combinatorial advantages of involutive bases, while needing in general much less generators. We show that Janet-like and Pommaret-like bases induce involutive-like bases of the same types for their syzygy modules, and thus induce free resolutions in the same way that involutive bases do. Moreover, we use involutive-like bases to design new efficient algorithms for the determination of complementary decompositions of monomial ideals and Hilbert functions.
Next, we generalize involutive and involutive-like bases to include also ideals in quotient rings. Our discussion is based on a comprehensive treatment of Gröbner bases for ideals in such rings, together with algorithms for their construction. We establish that Pommaret bases in quotient rings also induce Pommaret bases of their syzygy modules in a natural way.
The third part of contributions treats the application of these new types of bases to the computation and analysis of their induced infinite free resolutions. For the important special case of Clements-Lindström rings, we obtain closed formulas for the Betti numbers of the resolution. We identify several classes of quasi-stable monomial ideals in these quotient rings for which the induced resolution is minimal. Thus, we generalize several well-known resolution constructions, e.g. for stable and for square-free Borel ideals. We obtain explicit formulas for the differential which apply to some classes of quasi-stable ideals and their Pommaret-like bases.
In the final part, we introduce relative marked bases for ideals in quotient rings defined by ideals generated by Pommaret marked bases. We give an algorithm for the construction of relative marked families in case the quotient ring is defined by a monomial ideal. Lastly, we use these bases to obtain information about the lex-points and about quasi-stable open coverings of the Hilbert schemes defined on some quotient rings, e.g., Clements-Lindström rings.Algorithmic Reduction of Biochemical Reaction NetworksLüders, Christophhttps://kobra.uni-kassel.de:443/handle/123456789/139822022-07-05T06:24:36Z2022-02-25T00:00:00ZThe dynamics of species concentrations of chemical reaction networks are given by autonomous first-order ordinary differential equations. Singular perturbation methods allow the computation of approximate reduced systems that make explicit several time scales with corresponding invariant manifolds. This thesis presents:
1. An algorithmic approach for the computation of such reductions on solid analytical grounds. Required scalings are derived using tropical geometry. The existence of invariant manifolds is subject to certain hyperbolicity conditions. These conditions are reduced to Hurwitz criteria and discrete combinatorial conditions on degrees, which are technically solved using SMT over nonlinear real and linear integer arithmetic, respectively. The approach is implemented in Python and applied to a large body of known biochemical models.
2. ODEbase, a repository of biochemical models that has been created to provide real-world input data in a form that is suitable for symbolic computation software. ODEbase has been populated with semi-automatic conversions of several hundred SBML models from the BioModels database and is available to everyone at https://odebase.org.
3. A calculus for model-driven computation of disjunctive normal forms of real constraints from conjunctions of such disjunctive normal forms, which is required for the tropicalization in 1. The calculus technically once more builds on SMT solving, here over linear real arithmetic. Compared to existing software like Redlog, its implementation generally shows significant speedups, and a number of otherwise infeasible computations finish within seconds.
2022-02-25T00:00:00ZLüders, ChristophThe dynamics of species concentrations of chemical reaction networks are given by autonomous first-order ordinary differential equations. Singular perturbation methods allow the computation of approximate reduced systems that make explicit several time scales with corresponding invariant manifolds. This thesis presents:
1. An algorithmic approach for the computation of such reductions on solid analytical grounds. Required scalings are derived using tropical geometry. The existence of invariant manifolds is subject to certain hyperbolicity conditions. These conditions are reduced to Hurwitz criteria and discrete combinatorial conditions on degrees, which are technically solved using SMT over nonlinear real and linear integer arithmetic, respectively. The approach is implemented in Python and applied to a large body of known biochemical models.
2. ODEbase, a repository of biochemical models that has been created to provide real-world input data in a form that is suitable for symbolic computation software. ODEbase has been populated with semi-automatic conversions of several hundred SBML models from the BioModels database and is available to everyone at https://odebase.org.
3. A calculus for model-driven computation of disjunctive normal forms of real constraints from conjunctions of such disjunctive normal forms, which is required for the tropicalization in 1. The calculus technically once more builds on SMT solving, here over linear real arithmetic. Compared to existing software like Redlog, its implementation generally shows significant speedups, and a number of otherwise infeasible computations finish within seconds.Symmetrien von Differentialgleichungen via Vessiot-TheorieUrich, Maximhttps://kobra.uni-kassel.de:443/handle/123456789/135822023-09-29T07:07:10Z2021-04-01T00:00:00ZDie übliche Definition des Symmetriebegriffs einer Differentialgleichung lautet wie folgt: Symmetrien sind Transformationen, die Lösungen wieder in Lösungen überführen. Modelliert man Differentialgleichungen als Untermannigfaltigkeiten eines Jetbündels, so lassen sich zwei Arten von Symmetrien unterscheiden: innere und äußere. Der erste Fall entspricht einer Transformation, die ausschließlich auf der Differentialgleichung definiert ist. Im zweiten Fall ist die betrachtete Transformation auf dem gesamten umgebenden Jetbündel erklärt. Dabei stellt sich die naheliegende Frage, wann es mehr innere als äußere Symmetrien gibt. Es handelt sich um ein äußerst schwieriges Fortsetzungsproblem, da die Lösungen der betrachteten Differentialgleichung zudem praktisch unbekannt sind.
Zur Klärung dieser Frage stellen wir in der vorliegenden Arbeit eine neue geometrische Formulierung der Symmetrietheorie mittels Distributionen auf, die zu einem tieferen Verständnis der Zusammenhänge verschiedener Symmetriearten beiträgt. Zu diesem Zweck führen wir den Begriff der abgeleiteten Distribution ein. Diese entsteht aus einer gegebenen Distribution durch Hinzunahme von Lie-Klammern der beteiligten Vektorfeldern. Wir zeigen, dass die mehrfach abgeleiteten Distributionen invariant unter geometrischen Symmetrien sind. Mehr noch, es stellt sich heraus, dass die so konstruierten abgeleiteten Distributionen spezielle invariante Subdistributionen enthalten, welche ausschließlich aus Cauchy-Cartan-Charakteristiken bestehen.
Anschließend wenden wir die beschriebene geometrische Symmetrietheorie auf die Kontaktdistribution bzw. die Vessiot-Distribution an. Dabei finden wir mehrere unterschiedliche Beweise des klassischen Bäcklund-Theorems für Kontakttransformationen, welche die dahinterstehende geometrische Struktur besonders deutlich aufzeigen. Im Speziellen beweisen wir eine verschärfte globale Version des Bäcklund-Theorems. Die für innere Symmetrien wichtige Erkenntnis ist, dass diese im Lokalen genau dann äußere Punktsymmetrien sind, wenn die Vertikalbündel invariant sind. Durch die Einführung der abgeleiteten Vessiot-Distributionen können wir dafür hinreichende und leicht verifizierbare Kriterien angeben. Die Situation vereinfacht sich enorm, wenn wir ausschließlich infinitesimale innere Symmetrien betrachten. Wir beschreiben ausführlich, wie die infinitesimalen Lie-Kontaktsymmetrien innerhalb der Lie-Algebra der infinitesimalen inneren Symmetrien geometrisch charakterisiert werden können. In diesem Fall genügt es, die Dimensionen der zugehörigen involutiven bestimmenden Systeme zu vergleichen. Damit sind wir in der Lage algorithmisch zu entscheiden, wann eine gegebene Differentialgleichung mehr infinitesimale innere als infinitesimale äußere Symmetrien besitzt, und zwar ohne die bestimmenden Systeme vorher zu lösen.
Im abschließenden Kapitel gehen wir auf die sogenannten verallgemeinerten Symmetrien ein. Es handelt sich um eine weitreichende Verallgemeinerung der infinitesimalen inneren Symmetrien. Wir zeigen, dass jede infinitesimale innere Symmetrie eine verallgemeinerte Symmetrie festlegt. Weiter beschreiben wir genau, unter welchen intrinsischen Bedingungen verallgemeinerte Symmetrien von infinitesimalen inneren Symmetrien herkommen.
2021-04-01T00:00:00ZUrich, MaximDie übliche Definition des Symmetriebegriffs einer Differentialgleichung lautet wie folgt: Symmetrien sind Transformationen, die Lösungen wieder in Lösungen überführen. Modelliert man Differentialgleichungen als Untermannigfaltigkeiten eines Jetbündels, so lassen sich zwei Arten von Symmetrien unterscheiden: innere und äußere. Der erste Fall entspricht einer Transformation, die ausschließlich auf der Differentialgleichung definiert ist. Im zweiten Fall ist die betrachtete Transformation auf dem gesamten umgebenden Jetbündel erklärt. Dabei stellt sich die naheliegende Frage, wann es mehr innere als äußere Symmetrien gibt. Es handelt sich um ein äußerst schwieriges Fortsetzungsproblem, da die Lösungen der betrachteten Differentialgleichung zudem praktisch unbekannt sind.
Zur Klärung dieser Frage stellen wir in der vorliegenden Arbeit eine neue geometrische Formulierung der Symmetrietheorie mittels Distributionen auf, die zu einem tieferen Verständnis der Zusammenhänge verschiedener Symmetriearten beiträgt. Zu diesem Zweck führen wir den Begriff der abgeleiteten Distribution ein. Diese entsteht aus einer gegebenen Distribution durch Hinzunahme von Lie-Klammern der beteiligten Vektorfeldern. Wir zeigen, dass die mehrfach abgeleiteten Distributionen invariant unter geometrischen Symmetrien sind. Mehr noch, es stellt sich heraus, dass die so konstruierten abgeleiteten Distributionen spezielle invariante Subdistributionen enthalten, welche ausschließlich aus Cauchy-Cartan-Charakteristiken bestehen.
Anschließend wenden wir die beschriebene geometrische Symmetrietheorie auf die Kontaktdistribution bzw. die Vessiot-Distribution an. Dabei finden wir mehrere unterschiedliche Beweise des klassischen Bäcklund-Theorems für Kontakttransformationen, welche die dahinterstehende geometrische Struktur besonders deutlich aufzeigen. Im Speziellen beweisen wir eine verschärfte globale Version des Bäcklund-Theorems. Die für innere Symmetrien wichtige Erkenntnis ist, dass diese im Lokalen genau dann äußere Punktsymmetrien sind, wenn die Vertikalbündel invariant sind. Durch die Einführung der abgeleiteten Vessiot-Distributionen können wir dafür hinreichende und leicht verifizierbare Kriterien angeben. Die Situation vereinfacht sich enorm, wenn wir ausschließlich infinitesimale innere Symmetrien betrachten. Wir beschreiben ausführlich, wie die infinitesimalen Lie-Kontaktsymmetrien innerhalb der Lie-Algebra der infinitesimalen inneren Symmetrien geometrisch charakterisiert werden können. In diesem Fall genügt es, die Dimensionen der zugehörigen involutiven bestimmenden Systeme zu vergleichen. Damit sind wir in der Lage algorithmisch zu entscheiden, wann eine gegebene Differentialgleichung mehr infinitesimale innere als infinitesimale äußere Symmetrien besitzt, und zwar ohne die bestimmenden Systeme vorher zu lösen.
Im abschließenden Kapitel gehen wir auf die sogenannten verallgemeinerten Symmetrien ein. Es handelt sich um eine weitreichende Verallgemeinerung der infinitesimalen inneren Symmetrien. Wir zeigen, dass jede infinitesimale innere Symmetrie eine verallgemeinerte Symmetrie festlegt. Weiter beschreiben wir genau, unter welchen intrinsischen Bedingungen verallgemeinerte Symmetrien von infinitesimalen inneren Symmetrien herkommen.