Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

On Paley-Wiener Theorems for the Heisenberg Group

In this paper we prove two Paley-Wiener-type theorems for the Heisenberg group. One is for the group Fourier transform which is the analogue of the classical Paley-Wiener theorem. The other one is for the spectral projections associated to the sub-Laplacian

Analogues of the Wiener-Tauberian and Schwartz theorems for radial functions on symmetric spaces

We prove a Wiener Tauberian theorem for the L-1 spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz-type theorem for complex groups. As a corollary we obtain a Wiener Tauberian type result for compactly supported distributions.

Holographic c-theorems in arbitrary dimensions

We re-examine holographic versions of the c-theorem and entanglement entropy in the context of higher curvature gravity and the AdS/CFT correspondence. We select the gravity theories by tuning the gravitational couplings to eliminate non-unitary operators in the boundary theory and demonstrate that all of these theories obey a holographic c-theorem. In cases where the dual CFT is even-dimensional, we show that the quantity that flow is the central charge associated with the A-type trace anomaly. Here, unlike in conventional holographic constructions with Einstein gravity, we are able to distinguish this quantity from other central charges or the leading coefficient in the entropy density of a thermal bath. In general, we are also able to identify this quantity with the coefficient of a universal contribution to the entanglement entropy in a particular construction. Our results suggest that these coefficients appearing in entanglement entropy play the role of central charges in odd-dimensional CFT's. We conjecture a new c-theorem on the space of odd-dimensional field theories, which extends Cardy's proposal for even dimensions. Beyond holography, we were able to show that for any even-dimensional CFT, the universal coefficient appearing the entanglement entropy which we calculate is precisely the A-type central charge.

Comparative studies on the effects of specific immunoneutralization of endogenous FSH or LH on testicular germ cell transformations in the adult bonnet monkey (Macaca radiata)

PROBLEM: It is yet to be determined clearly whether the two hormones FSH and T act synergistically in the same cell type-the Sertoli cells-to control overall spermatogenesis or influence independently the transformation of specific germ cell types during spermatogenesis in the adult mammal. METHOD: Adult male bonnet monkeys specifically deprived of either FSH or LH using immunoneutralization techniques were monitored for changes in testicular germ cell transformation by DNA flow cytometry. RESULTS: FSH deprivation caused a significant reduction (>40%; P < 0.05) in [H-3] thymidine incorporation into DNA of proliferating 2C (spermatogonial) cells, a marked inhibition (>50%) in the transformation of 2C to primary spermatocytes (4C) and a concomitant, belated reduction (50%) in the formation of round spermatids (1C). In contrast, specific LH/T deprivation led to an immediate arrest in the meiotic transformation of 4C to 1C/HC leading to an effective and significant block (<90%; P < 0.01) in sperm production. CONCLUSION: Thus, LH rather than FSH deprivation has a more pronounced and immediate effect as the former primarily blocks meiosis (4C --> 1C/HC) which controls production of spermatids. These data provide evidence for LH/T and FSH regulating spermatogenic process in the adult primate by primarily acting at specific germ cell transformation steps.

Modified Einstein's gravity as a possible missing link between sub- and super-Chandrasekhar type Ia supernovae

We explore the effect of modification to Einstein's gravity in white dwarfs for the first time in the literature, to the best of our knowledge. This leads to significantly sub- and super-Chandrasekhar limiting masses of white dwarfs, determined by a single model parameter. On the other hand, type Ia supernovae (SNeIa), a key to unravel the evolutionary history of the universe, are believed to be triggered in white dwarfs having mass close to the Chandrasekhar limit. However, observations of several peculiar, under- and over-luminous SNeIa argue for exploding masses widely different from this limit. We argue that explosions of the modified gravity induced sub- and super-Chandrasekhar limiting mass white dwarfs result in under- and over-luminous SNeIa respectively, thus unifying these two apparently disjoint sub-classes and, hence, serving as a missing link. Our discovery raises two fundamental questions. Is the Chandrasekhar limit unique? Is Einstein's gravity the ultimate theory for understanding astronomical phenomena? Both the answers appear to be no!

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.

On reconstruction theorems in noncommutative Riemannian geometry

We present a novel account of the theory of commutative spectral triples and their two closest noncommutative generalisations, almost-commutative spectral triples and toric noncommutative manifolds, with a focus on reconstruction theorems, viz, abstract, functional-analytic characterisations of global-analytically defined classes of spectral triples. We begin by reinterpreting Connes's reconstruction theorem for commutative spectral triples as a complete noncommutative-geometric characterisation of Dirac-type operators on compact oriented Riemannian manifolds, and in the process clarify folklore concerning stability of properties of spectral triples under suitable perturbation of the Dirac operator. Next, we apply this reinterpretation of the commutative reconstruction theorem to obtain a reconstruction theorem for almost-commutative spectral triples. In particular, we propose a revised, manifestly global-analytic definition of almost-commutative spectral triple, and, as an application of this global-analytic perspective, obtain a general result relating the spectral action on the total space of a finite normal compact oriented Riemannian cover to that on the base space. Throughout, we discuss the relevant refinements of these definitions and results to the case of real commutative and almost-commutative spectral triples. Finally, we outline progess towards a reconstruction theorem for toric noncommutative manifolds.

Studies in nuclear reactor dynamics : I. The accuracy of point kinetics. II. The effect of delayed neutrons on the spectrum of the group-diffusion operator

This thesis is a theoretical work on the space-time dynamic behavior of a nuclear reactor without feedback. Diffusion theory with G-energy groups is used.

In the first part the accuracy of the point kinetics (lumped-parameter description) model is examined. The fundamental approximation of this model is the splitting of the neutron density into a product of a known function of space and an unknown function of time; then the properties of the system can be averaged in space through the use of appropriate weighting functions; as a result a set of ordinary differential equations is obtained for the description of time behavior. It is clear that changes of the shape of the neutron-density distribution due to space-dependent perturbations are neglected. This results to an error in the eigenvalues and it is to this error that bounds are derived. This is done by using the method of weighted residuals to reduce the original eigenvalue problem to that of a real asymmetric matrix. Then Gershgorin-type theorems .are used to find discs in the complex plane in which the eigenvalues are contained. The radii of the discs depend on the perturbation in a simple manner.

In the second part the effect of delayed neutrons on the eigenvalues of the group-diffusion operator is examined. The delayed neutrons cause a shifting of the prompt-neutron eigenvalue s and the appearance of the delayed eigenvalues. Using a simple perturbation method this shifting is calculated and the delayed eigenvalues are predicted with good accuracy.

Some theorems in classical elastodynamic

This investigation is concerned with various fundamental aspects of the linearized dynamical theory for mechanically homogeneous and isotropic elastic solids. First, the uniqueness and reciprocal theorems of dynamic elasticity are extended to unbounded domains with the aid of a generalized energy identity and a lemma on the prolonged quiescence of the far field, which are established for this purpose. Next, the basic singular solutions of elastodynamics are studied and used to generate systematically Love's integral identity for the displacement field, as well as an associated identity for the field of stress. These results, in conjunction with suitably defined Green's functions, are applied to the construction of integral representations for the solution of the first and second boundary-initial value problem. Finally, a uniqueness theorem for dynamic concentrated-load problems is obtained.

Pata contractions and coupled type fixed points

A new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered complete metric spaces is established. It is shown that the coupled fixed point can be unique under some extra suitable conditions involving mid point lower or upper bound properties. Also the corresponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point.

A New Type of Coincidence and Common Fixed Point Theorem with Applications

Coincidence and common fixed point theorems for a class of 'Ciric-Suzuki hybrid contractions involving a multivalued and two single-valued maps in a metric space are obtained. Some applications including the existence of a common solution for certain class of functional equations arising in a dynamic programming are also discussed..

First-principles study of p-type transparent conductive oxides CuXO2 (X=Y, Sc, and Al)

Using first-principles methods we have calculated electronic structures, optical properties, and hole conductivities of CuXO2 (X=Y, Sc, and Al). We show that the direct optical band gaps of CuYO2 and CuScO2 are approximately equal to their fundamental band gaps and the conduction bands of them are localized. The direct optical band gaps of CuXO2 (X=Y, Sc, and Al) are 3.3, 3.6, and 3.2 eV, respectively, which are consistent with experimental values of 3.5, 3.7, and 3.5 eV. We find that the hole mobility along long lattice c is higher than that along other directions through calculating effective masses of the three oxides. By analyzing band offset we find that CuScO2 has the highest valence band maximum (VBM) among CuXO2 (X=Y, Sc, and Al). In addition, the approximate transitivity of band offset suggests that CuScO2 has a higher VBM than CuGaO2 and CuInO2 [Phys. Rev. Lett. 88, 066405 (2002)]. We conclude that CuScO2 has a higher p-type doping ability in terms of the doping limit rule. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2991157]

RADIATIVE RECOMBINATION IN N-TYPE AND P-TYPE GAAS COMPENSATED WITH LI

We report fundamental changes of the radiative recombination in a wide range of n-type and p-type GaAs after diffusion with the group-I element Li. These optical properties are found to be a bulk property and closely related to the electrical conductivity of the samples. In the Li-doped samples the radiative recombination is characterized by emissions with excitation-dependent peak positions which shift to lower energies with increasing degree of compensation and concentration of Li. These properties are shown to be in qualitative agreement with fluctuations of the electrostatic potential in strongly compensated systems. For Li-diffusion temperatures above 700-800-degrees-C semi-insulating conditions with electrical resistivity exceeding 10(7) OMEGA cm are obtained for all conducting starting materials. In this heavy Li-doping regime, the simple model of fluctuating potentials is shown to be inadequate for explaining the. experimental observations unless the number of charged impurities is reduced through complexing with Li. For samples doped with low concentrations of Li, on the other hand, the photoluminescence properties are found to be characteristic of impurity-related emissions.

Chromatid-type aberrations following irradiation in G0 lymphocytes with heavy ions

We describe a low level of chromatid-type aberrations which included the relatively rare isochromatid/chromatid triradial in peripheral blood lymphocytes that were irradiated, ostensibly in GO, with accelerated heavy C-12 ions. These were produced only at the energies of 69 MeV/n (34.6 keV/ mu m), almost absent at the energy of either 58.6 McV/n (46.07 keV/mu m) or 19.3 MeV/n (97 keV/mu m), nor were they found after low-LET X-rays. Mechanisms potentially responsible for their formation are discussed.