On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

A mathematical and experimental investigation of microcycle spectral estimates of semiconductor flicker noise

The experimental portion of this thesis tries to estimate the density of the power spectrum of very low frequency semiconductor noise, from 10^-6.3 cps to 1. cps with a greater accuracy than that achieved in previous similar attempts: it is concluded that the spectrum is 1/f^α with α approximately 1.3 over most of the frequency range, but appearing to have a value of about 1 in the lowest decade. The noise sources are, among others, the first stage circuits of a grounded input silicon epitaxial operational amplifier. This thesis also investigates a peculiar form of stationarity which seems to distinguish flicker noise from other semiconductor noise.

In order to decrease by an order of magnitude the pernicious effects of temperature drifts, semiconductor "aging", and possible mechanical failures associated with prolonged periods of data taking, 10 independent noise sources were time-multiplexed and their spectral estimates were subsequently averaged. If the sources have similar spectra, it is demonstrated that this reduces the necessary data-taking time by a factor of 10 for a given accuracy.

In view of the measured high temperature sensitivity of the noise sources, it was necessary to combine the passive attenuation of a special-material container with active control. The noise sources were placed in a copper-epoxy container of high heat capacity and medium heat conductivity, and that container was immersed in a temperature controlled circulating ethylene-glycol bath.

Other spectra of interest, estimated from data taken concurrently with the semiconductor noise data were the spectra of the bath's controlled temperature, the semiconductor surface temperature, and the power supply voltage amplitude fluctuations. A brief description of the equipment constructed to obtain the aforementioned data is included.

The analytical portion of this work is concerned with the following questions: what is the best final spectral density estimate given 10 statistically independent ones of varying quality and magnitude? How can the Blackman and Tukey algorithm which is used for spectral estimation in this work be improved upon? How can non-equidistant sampling reduce data processing cost? Should one try to remove common trands shared by supposedly statistically independent noise sources and, if so, what are the mathematical difficulties involved? What is a physically plausible mathematical model that can account for flicker noise and what are the mathematical implications on its statistical properties? Finally, the variance of the spectral estimate obtained through the Blackman/Tukey algorithm is analyzed in greater detail; the variance is shown to diverge for α ≥ 1 in an assumed power spectrum of k/|f|^α, unless the assumed spectrum is "truncated".

Spectral shaping of femtosecond pulses with a prism-waveguide coupler

We report a novel technique for spectral shaping of femtosecond pulses employing a prism-waveguide coupler (PWC). It is demonstrated that the PWC is capable of producing a frequency-dependent loss with greater attenuation at the peak of the spectrum profile of femtosecond pulses than in the wings, which is especially useful for compensation for gain narrowing in most chirped-pulse amplification laser systems.

Improved uniformity of target illumination by combining a lens array and the technique of spectral dispersion

A scheme using a lens array and the technique of spectral dispersion is presented to improve target illumination uniformity in laser produced plasmas. Detailed two-dimensional simulation shows that a quasi-near-field target pattern, of steeper edges and without side lobes, is achieved with a lens array, while interference stripes inside the pattern are smoothed out by the use of the spectral dispersion technique. Moving the target slightly from the exact focal plane of the principal focusing lens can eliminate middle-scale-length intensity fluctuation further. Numerical results indicate that a well-irradiated laser spot with small nonuniformity and great energy efficiency can be obtained in this scheme. (c) 2007 American Institute of Physics.

Beam smoothing by lens array with spectral dispersion

A scheme of combining technology of lens array (LA) and smoothing by spectral dispersion (SSD) is introduced to improve the irradiation uniformity in laser fusion based on the earlier works on LA. The feasibility of the scheme is also analyzed by numerical simulation. It shows that a focal pattern with flat-top and sharp-edge profile could be obtained, and the irradiation nonuniformity can fall down from 14% with only LA to 3% with both SSD and LA. And this smoothing scheme is depended less on the incidence comparing to other smoothing methods. The preliminary experiment has demonstrated its effectiveness. (c) 2006 Elsevier B.V. All rights reserved.

Smoothing the side lobes of a focused pattern by spectral dispersion in the broadband laser

Starting from the Huygens-Fresnel diffraction integral, the propagation equations of a broadband laser passing through a dispersive lens and a dispersive wedge are derived. Smoothing effect on the side lobes of the focused pattern is achieved as the broadband laser passes through the lens because of the spectral dispersion of the lens. By inserting a dispersive wedge behind the lens, better smoothing effect is realized because a relative position shift between focused patterns of different frequency components is generated due to the spectral dispersion of the wedge. Smoothing effect on the side lobe is obtained even with small bandwidth of the broadband laser as the wedge is used. (c) 2006 Elsevier GmbH. All rights reserved.

Two-dimensional performance of uniform irradiation with the use of an edge-softened lens array and spectral dispersion

A lens array composed of edge-softened elements is used to improve on-target irradiation uniformity in the Shenguang II Laser Facility, with which a Fresnel pattern of suppressed diffraction peaks is obtained. Additional uniformity can be reached by reducing short-wavelength interference speckles inside the pattern when the technique of smoothing by spectral dispersion is also used. Two-dimensional performance of irradiation is simulated and the results indicate that a pattern of steeper edges and a flat top can be achieved with this joint technique. (c) 2007 Optical Society of America.

Synthesis, spectral, and thermal characterizations of Ni(II) and Cu(II) beta-diketone complexes with thenoyltrifluoroacetone ligand

Two kinds of nickel(II) and copper(II) P-diketone complexes derived from thenoyltrifluoroacetone ligand with blue-violet light absorption were synthesized by reacting free ligand and different metal(II) ions in sodium methoxide solution. Their structures were postulated based on elemental analysis, ESI-MS, FT-IR spectra and UV-vis electronic absorption spectra. Smooth films on K9 glass substrates were prepared using the spin-coating method. Their solubility in organic solvents, absorption properties of thin film and thermal stability of these complexes were evaluated. (c) 2006 Elsevier B.V. All rights reserved.

Nickel(II) and copper(II) complexes containing 2-(2-(5-substitued isoxazol-3-yl)hydrazono)-5,5-dimethylcyclohexane-1,3-dione ligands: Synthesis, spectral and thermal characterizations

Two new hydrazone chelating ligands, 2-(2-(5-methylisoxazol-3-yl)hydrazono)-5,5-dimethylcyclohexane-1,3-dione (HL1) and 2-(2-(5-tert-butylisoxazol-3-yl)hydrazono)-5,5-dimethylcyclohexane- 1,3-dione (HL2), and their nickel(II) and copper(II) complexes were synthesized using the procedure of diazotization, coupling and metallization. Their structures were postulated based on elemental analysis, H-1 NMR, ESI-MS, FT-IR spectra and UV-vis electronic absorption spectra. Smooth films of these complexes on K9 glass substrates were prepared using spin-coating and their absorption properties were evaluated. The thermal properties of the metal(II) complexes were investigated by thermogravimetry (TG) and differential thermogravimetry (DTG). Different thermodynamic and kinetic parameters namely activation energy (E