Let f : S → S(^4) be an immersion of a Riemann surface in the 4-sphere. The thesis begins with a study of the adapted moving frame of / in order to produce conditions for certain naturally defined lifts to SO(5)/U(2) and S0(5)T(^2) to be conformal, harmonic and holomorphic with respect to two different but naturally occuring almost complex structures. This approach brings together the results of a number of authors regarding lifts of conformal, minimal immersions including the link with solutions of the Toda equations. Moreover it is shown that parallel mean curvature immersions have haj-monic lifts into S0(5)/U(2).A certain natural lift of / into CP(^3), the twistor space of S(^4), is studied more carefully via an explicit description and in the case of / being a conformal immersion this gives a beautiful and simple formula for the lift in terms of a stereographic co-ordinate associated to /. This involves establishing explicitly the two-to-one correspondence between elements of the matrix groups Sp(2) and SO(5) and working with quaternions. The formula enables properties of such lifts to be explored and in particular it is shown that the harmonic sequence of a harmonic lift is either finite or satisfies a certain symmetry property. Uniqueness properties of harmonic lifts are also proved. Finally, the ideas are extended to the hyperbolic space H(^4) and after an exposition of the twistor fibration for this case, a method for constructing superminimal immersions of surfaces into H'^ from those in S"' is given.

We consider the problem of characterising Besov-Lipshitz and Triebel-Lizorkin spaces using kernels with limited smoothness and decay. This extends the work of H.-Q. Bui et al in [4] and [5] from kernels in S to more general kernels, including the Poisson kernel. We overcome the difficulty of defining the convolution of a general kernel with a distribution by using the concept of a bounded distribution introduced by E. Stein [12]. The characterisations we obtain are valid for the full range of indices.

In this work we will study Hardy-Littlewood maximal function and maximal operator, basing on both classical and most up to date works. In the first chapter we will give definitions for different types of those objects and consider some of their most important properties. The second chapter is entirely devoted to an overview of the fundamental properties of Hardy-Littlewood maximal function, which are strong (p, p) and weak (1, 1) inequalities. Here we list the most actual results on this inequalities in correspondence to the way the maximal func-tion is defined. The third chapter presents the theorem on asymptotic behavior of the lower bound of the constant in the weak-type (1, 1) inequality for the maximal function associated with cubes of Rd, then the dimension d tends to infinity. In the last chapter a method forcomputing constant c, appearing in the main theorem of chapter 3, is given. / <p>QC 20140527</p>

The portion of energy converted by non-linear loads in the modern power system is increasing due to the energy-efficiency and sophistication possible with power electronics. Higher voltage and energy ratings are continuously forthcoming. These devices draw non-linear currents resulting in voltage waveform distortion at the Point of Common Coupling due to non-zero supply impedances between voltage source and the PCC. With the increase in waveform distortion comes the demand for better Quality of Supply management. The verification and quantification of the origin of waveform distortion in a power system is a continuous field of study and forms a critical part of the mitigation design. Methods utilizing single-point measurements, usually taken at the Point of Common Coupling, for the assessment of the harmonic distortion generated by loads are continuously being published. It’s been proven by means of computer simulations and laboratory experiments that in an interconnected network where multiple sources of distortion exist that loads have the ability to exchange harmonic active power between each other. This project investigates the latter statement by conducting practical experiments to conclude that loads have the ability to exchange harmonic active power and that multiple synchronized measurements should be taken to assess the harmonic distortion due to a load. Laboratory experiments are carried utilizing an acknowledged single point measurement method. The results are compared to the direction of harmonic active power obtained from utilizing multiple synchronized measurements. To further the information obtained from the laboratory experiments, practical experiments were conducted utilizing the same methods. The results obtained coincided with the results of previously conducted experiments of which the results were published. From the results obtained it was concluded that in an interconnected network where multiple sources of distortion exist that loads have the ability to exchange harmonic active power between each other. Furthermore it was proven that the single point measurement method investigated presented inconsistent results. Ultimately it was concluded that the reason for the inconsistency was due to the fact that loads have the ability to exchange harmonic active power and that the single point measurement failed to acknowledge this. / Thesis (M.Ing. (Electrical and Electronic Engineering))--North-West University, Potchefstroom Campus, 2012

We say that a restriction theorem holds for a curve (gamma) (t) in (//R)('n) if for all f(epsilon) ((//R)('n)) and for some p and q, there is a constant C(,p,q) such that / (VBAR)(VBAR) f (VBAR) (,(gamma)) (VBAR)(VBAR) (,L('q)(du)) (LESSTHEQ) C(,p,q) (VBAR)(VBAR) f (VBAR)(VBAR) (,L('P)((//R)('n))). / In Chapter 1, we prove restriction theorems for non-compact plane curves with non-negative affine curvature when 1 (LESSTHEQ) p < 4/3 and / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / There is an analogous result for space curves in the same chapter. / The Hilbert transform along the curve (gamma) is defined by / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / In Chapter 2, it is shown that when (gamma) has the rapidly decreasing positive affine curvature, H(,(gamma)) is a L('P)-bounded operator for / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 41-44). Also available in electronic format on the Internet.

Univ., Diss.--Sendai.

Thesis--University of Maryland. / Technical Note BN-87. AFOSR-TN-56-594, AD-115 021. "Supported in part by U.S. Air Force under contract No. AF 18 (600)-573."

Thesis--Copenhagen.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1962. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves [104]-108).