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result(s) for
"Hyperbolic structures."
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Shock Capturing and High-Order Methods for Hyperbolic Conservation Laws
Long description:
This thesis is concerned with the numerical treatment of hyperbolic conservation laws. These play an important role in describing many natural phenomena. Challenges in their theoretical as well as numerical study stem from the fact that spontaneous shock discontinuities can arise in their solutions, even in finite time and smooth initial states.
Moreover, the numerical treatment of hyperbolic conservations laws involves many different fields from mathematics, physics, and computer science. As a consequence, this thesis also provides contributions to several different fields of research -- which are still connected by numerical conservation laws, however. These contributions include, but are not limited to, the construction of stable high order quadrature rules for experimental data, the development of new stable numerical methods for conservation laws, and the investigation and design of shock capturing procedures as a means to stabilize high order numerical methods in the presence of (shock) discontinuities.
eBook
Gluing equations for real projective structures on 3-manifolds
Given an orientable ideally triangulated 3-manifold M, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on M. These equations represent a unifying framework for the classical Thurston gluing equations in hyperbolic geometry and their more recent counterparts in Anti-de Sitter and half-pipe geometry. Moreover, these equations can be used to detect properly convex structures on M. The paper also includes explicit examples where the equations are used to construct properly convex structures.
Journal Article
A Primer on Mapping Class Groups (PMS-49)
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.
eBook
Hyperbolic and Euclidean Structures on Cone-Manifolds over Trefoil Knot with a Bridge
We study cone-manifolds whose singular set is the trefoil knot with a bridge and whose underlying space is the 3-dimensional sphere. We also establish necessary and sufficient conditions for the existence of such manifolds in both Euclidean and hyperbolic geometries, and derive explicit volume formulas in each case.
Journal Article
Hyperbolic structures : Shukhov's lattice towers - forerunners of modern lightweight construction
Hyperbolic structures analyses the interactions of form with the structural behaviour of hyperbolic lattice towers, and the effects of the various influencing factors were determined with the help of parametric studies and load capacity analyses. This evaluation of Shukhov's historical calculations and the reconstruction of the design and development process of his water towers shows why the Russian engineer is considered not only a pathfinder for lightweight structures but also a pioneer of parametrised design processes.
eBook
Symmetries of hyperbolic spatial graphs and realization of graph symmetries
In a closed connected orientable 3-manifold associated with an orientation-preserving smooth finite group action, we construct setwise invariant hyperbolic spatial graphs with given singularity. As an application, we provide a condition under which symmetries of abstract graphs are realizable by symmetries of the 3-sphere through hyperbolic spatial embeddings.
Journal Article
Octahedral developing of knot complement I: Pseudo-hyperbolic structure
It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also of cusp shape. We demonstrate explicit solutions for T(2, N) torus knots, J(N, M) knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspects of the knot diagram.
Journal Article
Global Conjugacy of Vector Fields via Tame Hyperbolic Structure
In this work, we first show that the closed subspace
E
of smooth vector fields on
R
n
which are rapidly decreasing together with all derivatives admits a tame hyperbolic structure first for a hyperbolic linear vector field
X
0
then with respect to a perturbation
X
0
+
Y
of
X
0
where
Y
is an infinitely flat vector field at the origin 0 with small support. As a consequence of the tameness property of
E
, we show that these fields are globally conjugate to their perturbations by any vector field from a small neighborhood of 0 in
E
.
Journal Article
Analytical construction and visualization of nonlinear waves in the (2+1) dimensional Kadomtsev-Petviashvili-Sawada-Kotera-Ramani equation with stability analysis
In this study, we investigate the (2+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani (KPSKR) equation, a physically significant model describing nonlinear wave phenomena in higher-dimensional spaces. Utilizing the improved modified extended tanh-function method, we derive a diverse spectrum of exact analytical solutions. These include bright solitons, singular solitons, singular periodic waves, and hyperbolic function solutions. The physical characteristics and dynamical behaviors of the obtained solutions are further elucidated through comprehensive two-dimensional and three-dimensional graphical visualizations, offering insight into the complex wave structures governed by the KPSKR equation. The results highlight the versatility of the proposed method and the rich nonlinear dynamics inherent in the model.
Journal Article