Maximal inequalities for fractional L\'evy and related processes

In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred L\'evy process, which covers the popular class of fractional L\'evy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding `convoluted martingale' is $p$-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale

PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues

Despite its common use in quantum theory, the mathematical requirement of Dirac Hermiticity of a Hamiltonian is sufficient to guarantee the reality of energy eigenvalues but not necessary. By establishing three theorems, this paper gives physical conditions that are both necessary and sufficient. First, it is shown that if the secular equation is real, the Hamiltonian is necessarily PT symmetric. Second, if a linear operator C that obeys the two equations [C,H]=0 and C^2=1 is introduced, then the energy eigenvalues of a PT-symmetric Hamiltonian that is diagonalizable are real only if this C operator commutes with PT. Third, the energy eigenvalues of PT-symmetric Hamiltonians having a nondiagonalizable, Jordan-block form are real. These theorems hold for matrix Hamiltonians of any dimensionality.Comment: 11 pages, no figure

Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart

In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian $H_{\rm PU}$ of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that $H_{\rm PU}$ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.Comment: 39 pages, 0 figure

Kinematic Structure of Merger Remnants

We use numerical simulations to study the kinematic structure of remnants formed from mergers of equal-mass disk galaxies. In particular, we show that remnants of dissipational mergers, which include the radiative cooling of gas, star formation, feedback from supernovae, and the growth of supermassive black holes, are smaller, rounder, have, on average, a larger central velocity dispersion, and show significant rotation compared to remnants of dissipationless mergers. The increased rotation speed of dissipational remnants owes its origin to star formation that occurs in the central regions during the galaxy merger. We have further quantified the anisotropy, three-dimensional shape, minor axis rotation, and isophotal shape of each merger remnant, finding that dissipational remnants are more isotropic, closer to oblate, have the majority of their rotation along their major axis, and are more disky than dissipationless remnants. Individual remnants display a wide variety of kinematic properties. A large fraction of the dissipational remnants are oblate isotropic rotators. Many dissipational, and all of the dissipationless, are slowly rotating and anisotropic. The remnants of gas-rich major mergers can well-reproduce the observed distribution of projected ellipticities, rotation parameter (V/\sigma)*, kinematic misalignments, Psi, and isophotal shapes. The dissipationless remnants are a poor match to this data. Our results support the merger hypothesis for the origin of low-luminosity elliptical galaxies provided that the progenitor disks are sufficiently gas-rich, however our remnants are a poor match to the bright ellipticals that are slowly rotating and uniformly boxy.Comment: 22 pages, 17 figures, accepted to Ap

In vitro evaluation of surface roughness, adhesion of periodontal ligament fibroblasts, and Streptococcus gordonii following root instrumentation with Gracey curettes and subsequent polishing with diamond-coated curettes

Objectives: The objective of the study was to evaluate the efficacy of an additional usage of a diamond-coated curette on surface roughness, adhesion of periodontal ligament (PDL) fibroblasts, and of Streptococcus gordonii in vitro. Materials and methods: Test specimens were prepared from extracted teeth and exposed to instrumentation with conventional Gracey curettes with or without additional use of diamond-coated curettes. Surface roughness (Ra and Rz) was measured before and following treatment. In addition, the adhesion of PDL fibroblasts for 72h and adhesion of S. gordonii ATCC 10558 for 2h have been determined. Results: Instrumentation with conventional Gracey curettes reduced surface roughness (median Ra before: 0.36μm/after: 0.25μm; p < 0.001; median Rz before: 2.34μm/after: 1.61μm; p < 0.001). The subsequent instrumentation with the diamond-coated curettes resulted in a median Ra of 0.31μm/Rz of 2.06μm (no significance in comparison to controls). The number of attached PDL fibroblasts did not change following scaling with Gracey curettes. The additional instrumentation with the diamond-coated curettes resulted in a two-fold increase in the number of attached PDL fibroblasts but not in the numbers of adhered bacteria. Conclusions: Treatment of root surfaces with conventional Gracey curettes followed by subsequent polishing with diamond-coated curettes may result in a root surface which provides favorable conditions for the attachment of PDL fibroblasts without enhancing microbial adhesion. Clinical relevance: The improved attachment of PDL fibroblasts and the limited microbial adhesion on root surfaces treated with scaling with conventional Gracey curettes followed by subsequent polishing with diamond-coated curettes may favor periodontal wound healin

Solution to the ghost problem in fourth order derivative theories

We present a solution to the ghost problem in fourth order derivative theories. In particular we study the Pais-Uhlenbeck fourth order oscillator model, a model which serves as a prototype for theories which are based on second plus fourth order derivative actions. Via a Dirac constraint method quantization we construct the appropriate quantum-mechanical Hamiltonian and Hilbert space for the system. We find that while the second-quantized Fock space of the general Pais-Uhlenbeck model does indeed contain the negative norm energy eigenstates which are characteristic of higher derivative theories, in the limit in which we switch off the second order action, such ghost states are found to move off shell, with the spectrum of asymptotic in and out S-matrix states of the pure fourth order theory which results being found to be completely devoid of states with either negative energy or negative norm. We confirm these results by quantizing the Pais-Uhlenbeck theory via path integration and by constructing the associated first-quantized wave mechanics, and show that the disappearance of the would-be ghosts from the energy eigenspectrum in the pure fourth order limit is required by a hidden symmetry that the pure fourth order theory is unexpectedly found to possess. The occurrence of on-shell ghosts is thus seen not to be a shortcoming of pure fourth order theories per se, but rather to be one which only arises when fourth and second order theories are coupled to each other.Comment: 36 pages, revtex. Prepared for the proceedings of the 2006 Biennial Meeting of the International Association for Relativistic Dynamics Version 2 contains an expanded discussion of the path integral quantization of the Pais-Uhlenbeck fourth order oscillator theor

Comprehensive Solution to the Cosmological Constant, Zero-Point Energy, and Quantum Gravity Problems

We present a solution to the cosmological constant, the zero-point energy, and the quantum gravity problems within a single comprehensive framework. We show that in quantum theories of gravity in which the zero-point energy density of the gravitational field is well-defined, the cosmological constant and zero-point energy problems solve each other by mutual cancellation between the cosmological constant and the matter and gravitational field zero-point energy densities. Because of this cancellation, regulation of the matter field zero-point energy density is not needed, and thus does not cause any trace anomaly to arise. We exhibit our results in two theories of gravity that are well-defined quantum-mechanically. Both of these theories are locally conformal invariant, quantum Einstein gravity in two dimensions and Weyl-tensor-based quantum conformal gravity in four dimensions (a fourth-order derivative quantum theory of the type that Bender and Mannheim have recently shown to be ghost-free and unitary). Central to our approach is the requirement that any and all departures of the geometry from Minkowski are to be brought about by quantum mechanics alone. Consequently, there have to be no fundamental classical fields, and all mass scales have to be generated by dynamical condensates. In such a situation the trace of the matter field energy-momentum tensor is zero, a constraint that obliges its cosmological constant and zero-point contributions to cancel each other identically, no matter how large they might be. Quantization of the gravitational field is caused by its coupling to quantized matter fields, with the gravitational field not needing any independent quantization of its own. With there being no a priori classical curvature, one does not have to make it compatible with quantization.Comment: Final version, to appear in General Relativity and Gravitation (the final publication is available at http://www.springerlink.com). 58 pages, revtex4, some additions to text and some added reference