Linear Fractional PDE, Uniqueness of Global Solutions

Mathematics Subject Classification: 26A33, 47A60, 30C15.

Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation

Mathematics Subject Classification: 45G10, 45M99, 47H09

Reducible von Neumann models and uniqueness

This paper can be regarded as a result of basic research on the technological characteristics of the von Neumann models and their consequences. It introduces a new taxonomy of reducible technologies, explores their key distinguishing features, and specifies which ones ensure the uniqueness of von Neumann equilibrium. A comprehensive comparison is also given between the familiar (in)decomposability ideas and the reducibility concepts suggested here. All these are carried out with a modern approach. Simultaneously, the reader may also acquire a complete picture of and guidance on the fundamental von Neumann models here.

The existence and source of stock return predictability : Evidence from dividend, output and consumption ratios

Date of Acceptance: 13/03/2015

The existence and source of stock return predictability : Evidence from dividend, output and consumption ratios

Date of Acceptance: 13/03/2015

Existence and wandering of bumps in a spiking neural network model

We study spatially localized states of a spiking neuronal network populated by a pulse coupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to those of the standard Amari model. For non-slow synaptic connections we are able to go beyond the standard firing rate analysis of localized solutions allowing us to explicitly construct a family of co-existing one-bump solutions, and then track bump width and firing pattern as a function of system parameters. We also present an analysis of the model on a discrete lattice. We show that multiple width bump states can co-exist and uncover a mechanism for bump wandering linked to the speed of synaptic processing. Moreover, beyond a wandering transition point we show that the bump undergoes an effective random walk with a diffusion coefficient that scales exponentially with the rate of synaptic processing and linearly with the lattice spacing.

Kinematics and Local Motion Planning for Quasi-static Whole-body Mobile Manipulation

This thesis studies mobile robotic manipulators, where one or more robot manipulator arms are integrated with a mobile robotic base. The base could be a wheeled or tracked vehicle, or it might be a multi-limbed locomotor. As robots are increasingly deployed in complex and unstructured environments, the need for mobile manipulation increases. Mobile robotic assistants have the potential to revolutionize human lives in a large variety of settings including home, industrial and outdoor environments.

Mobile Manipulation is the use or study of such mobile robots as they interact with physical objects in their environment. As compared to fixed base manipulators, mobile manipulators can take advantage of the base mechanism’s added degrees of freedom in the task planning and execution process. But their use also poses new problems in the analysis and control of base system stability, and the planning of coordinated base and arm motions. For mobile manipulators to be successfully and efficiently used, a thorough understanding of their kinematics, stability, and capabilities is required. Moreover, because mobile manipulators typically possess a large number of actuators, new and efficient methods to coordinate their large numbers of degrees of freedom are needed to make them practically deployable. This thesis develops new kinematic and stability analyses of mobile manipulation, and new algorithms to efficiently plan their motions.

I first develop detailed and novel descriptions of the kinematics governing the operation of multi- limbed legged robots working in the presence of gravity, and whose limbs may also be simultaneously used for manipulation. The fundamental stance constraint that arises from simple assumptions about friction and the ground contact and feasible motions is derived. Thereafter, a local relationship between joint motions and motions of the robot abdomen and reaching limbs is developed. Baseeon these relationships, one can define and analyze local kinematic qualities including limberness, wrench resistance and local dexterity. While previous researchers have noted the similarity between multi- fingered grasping and quasi-static manipulation, this thesis makes explicit connections between these two problems.

The kinematic expressions form the basis for a local motion planning problem that that determines the joint motions to achieve several simultaneous objectives while maintaining stance stability in the presence of gravity. This problem is translated into a convex quadratic program entitled the balanced priority solution, whose existence and uniqueness properties are developed. This problem is related in spirit to the classical redundancy resoxlution and task-priority approaches. With some simple modifications, this local planning and optimization problem can be extended to handle a large variety of goals and constraints that arise in mobile-manipulation. This local planning problem applies readily to other mobile bases including wheeled and articulated bases. This thesis describes the use of the local planning techniques to generate global plans, as well as for use within a feedback loop. The work in this thesis is motivated in part by many practical tasks involving the Surrogate and RoboSimian robots at NASA/JPL, and a large number of examples involving the two robots, both real and simulated, are provided.

Finally, this thesis provides an analysis of simultaneous force and motion control for multi- limbed legged robots. Starting with a classical linear stiffness relationship, an analysis of this problem for multiple point contacts is described. The local velocity planning problem is extended to include generation of forces, as well as to maintain stability using force-feedback. This thesis also provides a concise, novel definition of static stability, and proves some conditions under which it is satisfied.

Systems of coupled clamped beams equations with full nonlinear terms: Existence and location results

This work gives sufficient conditions for the solvability of the fourth order coupled system┊ u⁽⁴⁾(t)=f(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) v⁽⁴⁾(t)=h(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) with f,h: [0,1]×ℝ⁸→ℝ some L¹- Carathéodory functions, and the boundary conditions {┊ u(0)=u′(0)=u′′(0)=u′′(1)=0 v(0)=v′(0)=v′′(0)=v′′(1)=0. To the best of our knowledge, it is the first time in the literature where two beam equations are considered with full nonlinearities, that is, with dependence on all derivatives of u and v. An application to the study of the bending of two elastic coupled campled beams is considered.

Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Stochastic Filtering

The stochastic filtering has been in general an estimation of indirectly observed states given observed data. This means that one is discussing conditional expected values as being one of the most accurate estimation, given the observations in the context of probability space. In my thesis, I have presented the theory of filtering using two different kind of observation process: the first one is a diffusion process which is discussed in the first chapter, while the third chapter introduces the latter which is a counting process. The majority of the fundamental results of the stochastic filtering is stated in form of interesting equations, such the unnormalized Zakai equation that leads to the Kushner-Stratonovich equation. The latter one which is known also by the normalized Zakai equation or equally by Fujisaki-Kallianpur-Kunita (FKK) equation, shows the divergence between the estimate using a diffusion process and a counting process. I have also introduced an example for the linear gaussian case, which is mainly the concept to build the so-called Kalman-Bucy filter. As the unnormalized and the normalized Zakai equations are in terms of the conditional distribution, a density of these distributions will be developed through these equations and stated by Kushner Theorem. However, Kushner Theorem has a form of a stochastic partial differential equation that needs to be verify in the sense of the existence and uniqueness of its solution, which is covered in the second chapter.

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space X. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of global solution. The existence and convergence of the Faedo–Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator A. Finally we give an example to illustrate the applications of the abstract results.

Thermomechanical behaviour of pin joints subjected to sheet loads

An important problem regarding pin joints in a thermal environment is addressed. The motivation emerges from structural safety requirements in nuclear and aerospace engineering. A two-dimensional model of a smooth, rigid misfit pin in a large isotropic sheet is considered as an abstraction. The sheet is subjected to a biaxial stress system and far-field unidirectional heat flow. The thermoelastic analysis is complex due to non-linear load-dependent contact and separation conditions at the pin-hole interface and the absence of existence and uniqueness theorems for the class of frictionless thermoelastic contact problems. Identification of relevant parameters and appropriate synthesis of thermal and mechanical variables enables the thermomechanical generalization of pin-joint behaviour. This paper then proceeds to explore the possibility of multiple solutions in such problems, especially interface contact configuration.

On kuhnels 9-vertex complex projective plane

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C P2. The original proof of existence, due to Kuhnel, as well as the original proof of uniqueness, due to Kuhnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kuhnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.