Periodic solutions of integro-differential equations which arise in population dynamics

The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.

The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.

Terahertz time domain spectroscopy of amino acids and sugars

A time-domain spectrometer for use in the terahertz (THz) spectral range was designed and constructed. Due to there being few existing methods of generating and detecting THz radiation, the spectrometer is expected to have vast applications to solid, liquid, and gas phase samples. In particular, knowledge of complex organic chemistry and chemical abundances in the interstellar medium (ISM) can be obtained when compared to astronomical data. The THz spectral region is of particular interest due to reduced line density when compared to the millimeter wave spectrum, the existence of high resolution observatories, and potentially strong transitions resulting from the lowest-lying vibrational modes of large molecules.

The heart of the THz time-domain spectrometer (THz-TDS) is the ultrafast laser. Due to the femtosecond duration of ultrafast laser pulses and an energy-time uncertainty relationship, the pulses typically have a several-THz bandwidth. By various means of optical rectification, the optical pulse carrier envelope shape, i.e. intensity-time profile, can be transferred to the phase of the resulting THz pulse. As a consequence, optical pump-THz probe spectroscopy is readily achieved, as was demonstrated in studies of dye-sensitized TiO₂, as discussed in chapter 4. Detection of the terahertz radiation is commonly based on electro-optic sampling and provides full phase information. This allows for accurate determination of both the real and imaginary index of refraction, the so-called optical constants, without additional analysis. A suite of amino acids and sugars, all of which have been found in meteorites, were studied in crystalline form embedded in a polyethylene matrix. As the temperature was varied between 10 and 310 K, various strong vibrational modes were found to shift in spectral intensity and frequency. Such modes can be attributed to intramolecular, intermolecular, or phonon modes, or to some combination of the three.

On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space.

The main results of the thesis are the following two theorems.

Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds.

Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense.

Besides this we also established the following results:

(1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds.

(2) MT holds for Ribet-type abelian varieties.

(3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds.

(4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I.

(5) For some abelian varieties either MT or the Hodge conjecture holds.

Propagation of the fast magnetosonic wave in a tokamak plasma

The propagation of the fast magnetosonic wave in a tokamak plasma has been investigated at low power, between 10 and 300 watts, as a prelude to future heating experiments.

The attention of the experiments has been focused on the understanding of the coupling between a loop antenna and a plasma-filled cavity. Special emphasis has been given to the measurement of the complex loading impedance of the plasma. The importance of this measurement is that once the complex loading impedance of the plasma is known, a matching network can be designed so that the r.f. generator impedance can be matched to one of the cavity modes, thus delivering maximum power to the plasma. For future heating experiments it will be essential to be able to match the generator impedance to a cavity mode in order to couple the r.f. energy efficiently to the plasma.

As a consequence of the complex impedance measurements, it was discovered that the designs of the transmitting antenna and the impedance matching network are both crucial. The losses in the antenna and the matching network must be kept below the plasma loading in order to be able to detect the complex plasma loading impedance. This is even more important in future heating experiments, because the fundamental basis for efficient heating before any other consideration is to deliver more energy into the plasma than is dissipated in the antenna system.

The characteristics of the magnetosonic cavity modes are confirmed by three different methods. First, the cavity modes are observed as voltage maxima at the output of a six-turn receiving probe. Second, they also appear as maxima in the input resistance of the transmitting antenna. Finally, when the real and imaginary parts of the measured complex input impedance of the antenna are plotted in the complex impedance plane, the resulting curves are approximately circles, indicating a resonance phenomenon.

The observed plasma loading resistances at the various cavity modes are as high as 3 to 4 times the basic antenna resistance (~ .4 Ω). The estimated cavity Q’s were between 400 and 700. This means that efficient energy coupling into the tokamak and low losses in the antenna system are possible.

A multiple scattering problem

The present work deals with the problem of the interaction of the electromagnetic radiation with a statistical distribution of nonmagnetic dielectric particles immersed in an infinite homogeneous isotropic, non-magnetic medium. The wavelength of the incident radiation can be less, equal or greater than the linear dimension of a particle. The distance between any two particles is several wavelengths. A single particle in the absence of the others is assumed to scatter like a Rayleigh-Gans particle, i.e. interaction between the volume elements (self-interaction) is neglected. The interaction of the particles is taken into account (multiple scattering) and conditions are set up for the case of a lossless medium which guarantee that the multiple scattering contribution is more important than the self-interaction one. These conditions relate the wavelength λ and the linear dimensions of a particle a and of the region occupied by the particles D. It is found that for constant λ/a, D is proportional to λ and that |Δχ|, where Δχ is the difference in the dielectric susceptibilities between particle and medium, has to lie within a certain range.

The total scattering field is obtained as a series the several terms of which represent the corresponding multiple scattering orders. The first term is a single scattering term. The ensemble average of the total scattering intensity is then obtained as a series which does not involve terms due to products between terms of different orders. Thus the waves corresponding to different orders are independent and their Stokes parameters add.

The second and third order intensity terms are explicitly computed. The method used suggests a general approach for computing any order. It is found that in general the first order scattering intensity pattern (or phase function) peaks in the forward direction Θ = 0. The second order tends to smooth out the pattern giving a maximum in the Θ = π/2 direction and minima in the Θ = 0 , Θ = π directions. This ceases to be true if ka (where k = 2π/λ) becomes large (> 20). For large ka the forward direction is further enhanced. Similar features are expected from the higher orders even though the critical value of ka may increase with the order.

The first order polarization of the scattered wave is determined. The ensemble average of the Stokes parameters of the scattered wave is explicitly computed for the second order. A similar method can be applied for any order. It is found that the polarization of the scattered wave depends on the polarization of the incident wave. If the latter is elliptically polarized then the first order scattered wave is elliptically polarized, but in the Θ = π/2 direction is linearly polarized. If the incident wave is circularly polarized the first order scattered wave is elliptically polarized except for the directions Θ = π/2 (linearly polarized) and Θ = 0, π (circularly polarized). The handedness of the Θ = 0 wave is the same as that of the incident whereas the handedness of the Θ = π wave is opposite. If the incident wave is linearly polarized the first order scattered wave is also linearly polarized. The second order makes the total scattered wave to be elliptically polarized for any Θ no matter what the incident wave is. However, the handedness of the total scattered wave is not altered by the second order. Higher orders have similar effects as the second order.

If the medium is lossy the general approach employed for the lossless case is still valid. Only the algebra increases in complexity. It is found that the results of the lossless case are insensitive in the first order of k_imD where k_im = imaginary part of the wave vector k and D a linear characteristic dimension of the region occupied by the particles. Thus moderately extended regions and small losses make (k_imD)² ≪ 1 and the lossy character of the medium does not alter the results of the lossless case. In general the presence of the losses tends to reduce the forward scattering.

Theoretical investigations of experimental gravitation

This thesis has two basic themes: the investigation of new experiments which can be used to test relativistic gravity, and the investigation of new technologies and new experimental techniques which can be applied to make gravitational wave astronomy a reality.

Advancing technology will soon make possible a new class of gravitation experiments: pure laboratory experiments with laboratory sources of non-Newtonian gravity and laboratory detectors. The key advance in techno1ogy is the development of resonant sensing systems with very low levels of dissipation. Chapter 1 considers three such systems (torque balances, dielectric monocrystals, and superconducting microwave resonators), and it proposes eight laboratory experiments which use these systems as detectors. For each experiment it describes the dominant sources of noise and the technology required.

The coupled electro-mechanical system consisting of a microwave cavity and its walls can serve as a gravitational radiation detector. A gravitational wave interacts with the walls, and the resulting motion induces transitions from a highly excited cavity mode to a nearly unexcited mode. Chapter 2 describes briefly a formalism for analyzing such a detector, and it proposes a particular design.

The monitoring of a quantum mechanical harmonic oscillator on which a classical force acts is important in a variety of high-precision experiments, such as the attempt to detect gravitational radiation. Chapter 3 reviews the standard techniques for monitoring the oscillator; and it introduces a new technique which, in principle, can determine the details of the force with arbitrary accuracy, despite the quantum properties of the oscillator.

The standard method for monitoring the oscillator is the "amplitude- and-phase" method (position or momentum transducer with output fed through a linear amplifier). The accuracy obtainable by this method is limited by the uncertainty principle. To do better requires a measurement of the type which Braginsky has called "quantum nondemolition." A well-known quantum nondemolition technique is "quantum counting," which can detect an arbitrarily weak force, but which cannot provide good accuracy in determining its precise time-dependence. Chapter 3 considers extensively a new type of quantum nondemolition measurement - a "back-action-evading" measurement of the real part X₁ (or the imaginary part X₂) of the oscillator's complex amplitude. In principle X₁ can be measured arbitrarily quickly and arbitrarily accurately, and a sequence of such measurements can lead to an arbitrarily accurate monitoring of the classical force.

Chapter 3 describes explicit gedanken experiments which demonstrate that X₁ can be measured arbitrarily quickly and arbitrarily accurately, it considers approximate back-action-evading measurements, and it develops a theory of quantum nondemolition measurement for arbitrary quantum mechanical systems.

In Rosen's "bimetric" theory of gravity the (local) speed of gravitational radiation v_g is determined by the combined effects of cosmological boundary values and nearby concentrations of matter. It is possible for v_g to be less than the speed of light. Chapter 4 shows that emission of gravitational radiation prevents particles of nonzero rest mass from exceeding the speed of gravitational radiation. Observations of relativistic particles place limits on v_g and the cosmological boundary values today, and observations of synchrotron radiation from compact radio sources place limits on the cosmological boundary values in the past.