Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.

Edge representations of operators on closed manifolds are known to induce large classes of operators that are elliptic on specific manifolds with edges, cf. [9]. We apply this idea to the case of boundary value problems. We establish a correspondence between standard ellipticity and ellipticity with respect to the principal symbolic hierarchy of the edge algebra of boundary value problems, where an embedded submanifold on the boundary plays the role of an edge. We first consider the case that the weight is equal to the smoothness and calculate the dimensions of kernels and cokernels of the associated principal edge symbols. Then we pass to elliptic edge operators for arbitrary weights and construct the additional edge conditions by applying relative index results for conormal symbols.

We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary.

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols.

A novel algorithm for the simplified tele-operation of mobile-manipulator systems is presented. The algorithm allows for unified, intuitive, and coordinated control of mobile manipulators, systems comprised of a robotic arm mounted on a mobile base. Unlike other approaches, the mobile-manipulator system is modeled and controlled as two separate entities rather than as a whole. The algorithm consists of thee states. In the rst state a 6-DOF (degree-of-freedom) joystick is used to freely control the manipulator's position and orientation. The second state occurs when the manipulator approaches a singular configuration, a con guration where the arm instantaneously loses a DOF of motion capability. This state causes the mobile base to proceed in such a way as to keep the end-effector moving in its last direction of motion. This is done through the use of a constrained optimization routine. The third state is triggered by the user: once the end-effector is in the desired position, the mobile base and manipulator both move with respect to one another keeping the end-effector stationary and placing the manipulator into an ideal configuration. The proposed algorithm avoids the problems of algorithmic singularities and simplifies the control approach. The algorithm has been implemented on the Jasper Mobile-Manipulator System. Test results show that the developed algorithm is effective at moving the system in an intuitive manner.

<p>In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal surface, if one exists. In physical terms, an equivalent statement is that the total mass of an asymptotically flat spacetime admitting a time-symmetric spacelike slice is at least the mass of any black holes that are present, assuming nonnegative energy density. The main goal of this thesis is to deduce geometric lower bounds for the ADM mass of manifolds to which neither the RPI nor the famous positive mass theorem (PMT) apply. This is the case, for instance, for manifolds that contain metric singularities or have boundary components that are not minimal surfaces.</p> <p>The fundamental technique is the use of conformal deformations of a given Riemannian metric to arrive at a new Riemannian manifold to which either the PMT or RPI applies. Along the way we are led to consider the geometry of certain types non-smooth metrics. We prove a result regarding the local structure of area-minimizing hypersurfaces with respect such metrics using geometric measure theory.</p> <p>One application is to the theory of ``zero area singularities,'' a type of singularity that generalizes the degenerate behavior of the Schwarzschild metric of negative mass. Another application deals with constructing and understanding some new invariants of the harmonic conformal class of an asymptotically flat metric.</p> / Dissertation

The dissertation consists of two parts.The first part is mainly to provide the algorithms and error estimates of the collocation Trefftz methods (CTMs) for seeking the solutions of partial differential equations. We consider several popular models of PDEs with singularities, including Poisson equations and the biharmonic equations. The second part is to present the collocation methods (CMs) and to give a unified framework of combinations of CMs with other numerical methods such as finite element method, etc. An interesting fact has been justified: The integration quadrature formulas only affect on the uniformly $V_h$-elliptic inequality, not on the solution accuracy. In CTMs and CMs, the Gaussian quadrature points will be chosen as the collocation points. Of course, the Newton-Cotes quadrature points can be applied as well. We need a suitable dense points to guarantee the uniformly $V_h$-elliptic inequality. In addition, the solution domain of problems may not be confined in polygons. We may also divide the domain into several small subdomains. For the smooth solutions of problems, the different degree polynomials can be chosen to approximate the solutions properly. However, different kinds of admissible functions may also be used in the methods given in this dissertation. Besides, a new unified framework of combinations of CMs with other methods will be analyzed. In this dissertation, the new analysis is more flexible towards the practical problems and is easy to fit into rather arbitrary domains. Thus is a great distinctive feature from that in the existing literatures of CTMs and CMs. Finally, a few numerical experiments for smooth and singularity problems are provided to display effectiveness of the methods proposed, and to support the analysis made.

In this thesis, we consider the Laplace quation on sector with various constant Dirichlet or Numann boundary conditions. Most of such problems have singularity in the solution. We first analyze the type of singularity on the corner and then survey some known methods to solve these problems. The boundary approximation method is used to compute some of their solutions with two singularities. Besides, a Laplace equation on a triangle with multiple solutions is solved by the method of separation of variables.

Thesis (Ph. D.)--University of Chicago, Dept. of Physics, 2002. / Includes bibliographical references. Also available on the Internet.

In this dissertation, we present some analysis of Ricci flow on complete noncompact manifolds. The first half of the dissertation concerns the formation of Type-II singularity in Ricci flow on [mathematical equation]. For each [mathematical equation] , we construct complete solutions to Ricci flow on [mathematical equation] which encounter global singularities at a finite time T such that the singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate [mathematical equation]. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on [mathematical equation] whose blow-ups near the origin converge uniformly to the Bryant soliton. In the second half of the dissertation, we fully analyze the structure of the Lichnerowicz Laplacian of a Bergman metric g[subscript B] on a complex hyperbolic space [mathematical equation] and establish the linear stability of the curvature-normalized Ricci flow at such a geometry in complex dimension [mathematical equation]. We then apply the maximal regularity theory for quasilinear parabolic systems to prove a dynamical stability result of Bergman metric on the complete noncompact CH[superscript m] under the curvature-normalized Ricci flow in complex dimension [mathematical equation]. We also prove a similar dynamical stability result on a smooth closed quotient manifold of [mathematical symbols]. In order to apply the maximal regularity theory, we define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties. / text