SIGMA THEORY AND TWISTED CONJUGACY CLASSES

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut(G) such that for phi is an element of H the Reidemeister number R(phi) is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP(infinity); groups G/G `` of finite Prufer rank; groups G of type FP(2) without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group. Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups. In some cases, such as with the generalized Thompson`s groups F(n,0) and their finite direct products, H = Aut(G).

Probabilistic Recursion Theory and Implicit Computational Complexity

In this thesis we provide a characterization of probabilistic computation in itself, from a recursion-theoretical perspective, without reducing it to deterministic computation. More specifically, we show that probabilistic computable functions, i.e., those functions which are computed by Probabilistic Turing Machines (PTM), can be characterized by a natural generalization of Kleene's partial recursive functions which includes, among initial functions, one that returns identity or successor with probability 1/2. We then prove the equi-expressivity of the obtained algebra and the class of functions computed by PTMs. In the the second part of the thesis we investigate the relations existing between our recursion-theoretical framework and sub-recursive classes, in the spirit of Implicit Computational Complexity. More precisely, endowing predicative recurrence with a random base function is proved to lead to a characterization of polynomial-time computable probabilistic functions.

Planck stars: theory and phenomenology

General Relativity (GR) is one of the greatest scientific achievements of the 20th century along with quantum theory. Despite the elegance and the accordance with experimental tests, these two theories appear to be utterly incompatible at fundamental level. Black holes provide a perfect stage to point out these difficulties. Indeed, classical GR fails to describe Nature at small radii, because nothing prevents quantum mechanics from affecting the high curvature zone, and because classical GR becomes ill-defined at r = 0 anyway. Rovelli and Haggard have recently proposed a scenario where a negative quantum pressure at the Planck scales stops and reverts the gravitational collapse, leading to an effective “bounce” and explosion, thus resolving the central singularity. This scenario, called Black Hole Fireworks, has been proposed in a semiclassical framework. The purpose of this thesis is twofold: - Compute the bouncing time by means of a pure quantum computation based on Loop Quantum Gravity; - Extend the known theory to a more realistic scenario, in which the rotation is taken into account by means of the Newman-Janis Algorithm.

Equivariant inverse spectral theory and toric orbifolds

Let O-2n be a symplectic toric orbifold with a fixed T-n-action and with a tonic Kahler metric g. In [10] we explored whether, when O is a manifold, the equivariant spectrum of the Laplace Delta(g) operator on C-infinity(O) determines O up to symplectomorphism. In the setting of tonic orbifolds we shmilicantly improve upon our previous results and show that a generic tone orbifold is determined by its equivariant spectrum, up to two possibilities. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature. (C) 2012 Elsevier Inc. All rights reserved.

Dynamics of settling charged particles in turbulence : theory and experiments

It has been proposed that inertial clustering may lead to an increased collision rate of water droplets in clouds. Atmospheric clouds and electrosprays contain electrically charged particles embedded in turbulent flows, often under the influence of an externally imposed, approximately uniform gravitational or electric force. In this thesis, we present the investigation of charged inertial particles embedded in turbulence. We have developed a theoretical description for the dynamics of such systems of charged, sedimenting particles in turbulence, allowing radial distribution functions to be predicted for both monodisperse and bidisperse particle size distributions. The governing parameters are the particle Stokes number (particle inertial time scale relative to turbulence dissipation time scale), the Coulomb-turbulence parameter (ratio of Coulomb ’terminalar speed to turbulence dissipation velocity scale), and the settling parameter (the ratio of the gravitational terminal speed to turbulence dissipation velocity scale). For the monodispersion particles, The peak in the radial distribution function is well predicted by the balance between the particle terminal velocity under Coulomb repulsion and a time-averaged ’drift’ velocity obtained from the nonuniform sampling of fluid strain and rotation due to finite particle inertia. The theory is compared to measured radial distribution functions for water particles in homogeneous, isotropic air turbulence. The radial distribution functions are obtained from particle positions measured in three dimensions using digital holography. The measurements support the general theoretical expression, consisting of a power law increase in particle clustering due to particle response to dissipative turbulent eddies, modulated by an exponential electrostatic interaction term. Both terms are modified as a result of the gravitational diffusion-like term, and the role of ’gravity’ is explored by imposing a macroscopic uniform electric field to create an enhanced, effective gravity. The relation between the radial distribution functions and inward mean radial relative velocity is established for charged particles.

Relative Predicativity and dependent recursion in second-order set theory and higher-orders theories

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

Molecular theories and mathematics.

"An address delivered at the inauguration of the Rice institute, by Emile Borel ... Translated from the French by Professor Albert Léon Guérard of the Rice institute."

Set theory and metric spaces. Math. 261, autumn, 1950.

Cover title.

Transducer diffraction theory and development of an ultrasonic tomogram

The work described in this thesis is the development of an ultrasonic tomogram to provide outlines of cross-sections of the ulna in vivo. This instrument, used in conjunction with X-ray densitometry previously developed in this department, would provide actual bone mineral density to a high resolution. It was hoped that the accuracy of the plot obtained from the tomogram would exceed that of existing ultrasonic techniques by about five times. Repeat measurements with these instruments to follow bone mineral changes would involve very low X-ray doses. A theoretical study has been made of acoustic diffraction, using a geometrical transform applicable to the integration of three different Green's functions, for axisymmetric systems. This has involved the derivation of one of these in a form amenable to computation. It is considered that this function fits the boundary conditions occurring in medical ultrasonography more closely than those used previously. A three dimensional plot of the pressure field using this function has been made for a ring transducer, in addition to that for disc transducers using all three functions. It has been shown how the theory may be extended to investigate the nature and magnitude of the particle velocity, at any point in the field, for the three functions mentioned. From this study. a concept of diffraction fronts has been developed, which has made it possible to determine energy flow also in a diffracting system. Intensity has been displayed in a manner similar to that used for pressure. Plots have been made of diffraction fronts and energy flow direction lines.

Physics and mathematics of dispersion-managed optical solitons

We review the main physical and mathematical properties of dispersion-managed (DM) optical solitons. Theory of DM solitons can be presented at two levels of accuracy: first, simple, but nevertheless, quantitative models based on ordinary differential equations governing evolution of the soliton width and phase parameter (the so-called chirp); and second, a comprehensive path-average theory that is capable of describing in detail both the fine structure of DM soliton form and its evolution along the fiber line. An analogy between DM soliton and a macroscopic nonlinear quantum oscillator model is also discussed. © 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Deformation Lemma, Ljusternik-Schnirellmann Theory and Mountain Pass Theorem on C1-Finsler Manifolds

∗Partially supported by Grant MM409/94 Of the Ministy of Science and Education, Bulgaria. ∗∗Partially supported by Grant MM442/94 of the Ministy of Science and Education, Bulgaria.

Interior Boundaries for Degenerate Elliptic Equations of Second Order Some Theory and Numerical Observations

2010 Mathematics Subject Classification: Primary 35J70; Secondary 35J15, 35D05.

Probabilistic Models in Cryptography, Coding Theory and Tests for PRNG

2000 Mathematics Subject Classification: 94A29, 94B70

Viability Theory and Economic Modeling

A brief introduction into the theory of differential inclusions, viability theory and selections of set valued mappings is presented. As an application the implicit scheme of the Leontief dynamic input-output model is considered.