Regular measures of noncompactness and Ascoli-Arzela type compactness criteria in spaces of vector-valued functions

In this paper we estimate the Kuratowski and the Hausdorff measures of noncompactness of bounded subsets of spaces of vector-valued bounded functions and of vector-valued bounded differentiable functions. To this end, we use a quantitative characteristic modeled on a new equicontinuity-type concept and classical quantitative characteristics related to pointwise relative compactness. We obtain new regular measures of noncompactness in the spaces taken into consideration. The established inequalities reduce to precise formulas in some classes of subsets. We derive Ascoli-Arzela type compactness criteria

Rotor-vortex interaction noise

A theoretical and experimental study was conducted to develop a validated first principles analysis for predicting noise generated by helicopter main-rotor shed vortices interacting with the tail rotor. The generalized prediction procedure requires a knowledge of the incident vortex velocity field, rotor geometry, and rotor operating conditions. The analysis includes compressibility effects, chordwise and spanwise noncompactness, and treats oblique intersections with the blade planform. Assessment of the theory involved conducting a model rotor experiment which isolated the blade-vortex interaction noise from other rotor noise mechanisms. An isolated tip vortex, generated by an upstream semispan airfoil, was convected into the model tail rotor. Acoustic spectra, pressure signatures, and directivity were measured. Since assessment of the acoustic prediction required a knowledge of the vortex properties, blade-vortes intersection angle, intersection station, vortex stength, and vortex core radius were documented. Ingestion of the vortex by the rotor was experimentally observed to generate harmonic noise and impulsive waveforms

Harmonic mappings between Riemannian manifolds

These notes originated from a series of lectures I delivered at the Centre for Mathematical Analysis at Canberra. The purpose of the lectures was to introduce mathematicians familiar with the basic notions and results of linear elliptic partial differential equations and Riemannian geometry to the subject of harmonic mappings. I selected some topics to the presentation of which I felt I could contribute something, while on the other hand it was possible to provide complete and detailed proofs of them during these lectures. Thus, these notes are not meant to cover all that is known about harmonic maps, but nevertheless I believe that they give a good account of many of the interesting aspects of the subject and a fair idea of the variety of techniques used in the field

Fixed point indices and existence theorems for semilinear equations in cones

The purpose of this thesis is to develop fixed point indices for A-proper semilinear operators defined on cones in Banach spaces and use the results to obtain existence theorems to semilinear equations. We consider semilinear equations of the form Lx = Nx where L is a linear Fredholm operation of index zero, N a nonlinear operator such that L - N is A-proper at zero relative to a projection scheme L. Chapter 1 is an introduction to basic concepts used throughout the thesis, including; Banach spaces, linear operators, A-proper maps, Fredholm operators of index zero, and the definition and properties of the generalised degree for A-proper maps. In Chapter 2, we define a fixed point index for A-proper maps on cones in terms of the generalised degree and derive the basic properties of this index. We then extend the definition to include unbounded sets. A more general fixed point index than that of Chapter 2 is developed in Chapter 3 for A-proper maps based on limits of a finite dimensionally defined index. Properties of the index are given and a definition for unbounded sets is provided. Chapter 4 extends the Lan-Webb fixed point index for weakly inward A-proper at 0 maps to semilinear operators. This index is also extended to include unbounded sets. Existence theorems of positive and non-negative solutions to semilinear equations on cones are established in Chapter 5 using the fixed point indices of Chapters 2, 3, and 4. Finally, in Chapter 6, we apply some of the existence theorems of Chapter 5 to several differential and integral equations. We prove the existence of: a positive solution to a Picard boundary value problem; a non-negative solution to a periodic boundary value problem; and, a non-negative solution to a Volterra integral equation

Some results in weak KPZ universality

Stochastic partial differential equations (SPDEs) are a central object of study in the field of stochastic analysis. Their study involves a number of different tools coming from probability theory, functional analysis, harmonic analysis, statistical mechanics, and dynamical systems. Conversely SPDEs are an extremely useful paradigm to study scaling limit phenomena encountered throughout many other areas of mathematics and physics. The present thesis is concerned mainly with one particular SPDE called the Kardar-Parisi-Zhang (KPZ) equation, which appears universally as a fluctuation limit of height profiles of microscopic models such as interacting particle systems, directed polymers, and corner growth models. Such limit results are deemed instances of ``weak KPZ universality," a field born from the seminal paper of Bertini and Giacomin. We extend results on weak KPZ universality in a number of different directions. In one direction, we prove a version of Bertini-Giacomin's result in a half-space by adapting their methods to this setting, thus extending a result of Corwin and Shen and completing the final step towards the proof of a conjecture about fluctuation behavior of half-space KPZ. In another direction, we also prove a result for the free energy for directed polymers in an octant converging to the KPZ equation in a half-space with a nontrivial normalization at the boundary. In a third direction, we return to the whole-space regime and extend the Bertini-Giacomin result to the case of several different initial data coupled together, proving joint convergence of ASEP with its basic coupling to KPZ driven by the same realization of its noise. Finally we prove a ``nonlinear" version of the law of the iterated logarithm for the KPZ equation in a weak-noise but strong-nonlinearity regime. Beyond their intrinsic purpose, one application of all these extensions and generalizations is to take limits of known results and identities for discrete systems and pass them to the limit to obtain nontrivial information about the KPZ equation itself, which is a well-known methodology launched by I. Corwin and coauthors

Differentiable positive definite kernels on two-point homogeneous spaces

In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5