This paper presents the results of the observations of the detectors participating in the International Gravitational Event Collaboration (IGEC) from 1997 to 2000 and reviews the data analysis methods. The analysis is designed to search for coincident excitations in multiple detectors. The data set analyzed in this article covers a longer period and is more complete than that given in previous reports. The current analysis is more accurate for determining the false dismissal probability for a time coincidence search and it optimizes the search with respect to a target amplitude and direction of the signal. The statistics of the accidental coincidences agrees with the model used for drawing the results. The observations of this IGEC search are consistent with no detection of gravitational wave burst events. A new conservative upper limit has been set on the rate of gravitational wave bursts with a Fourier component $H>\mathrm{}2\mathrm{}\times {10}^{-21}{\mathrm{Hz}}^{-1},$ both for searches with and without a filter for the galactic center direction. This study confirms that the false alarm rate of the observation can be negligible when at least three detectors are operating simultaneously.

We obtain exact results for fractional equations of Fokker-Planck type using the evolution operator method. We employ exact forms of one-sided L\'evy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.

In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with noninteger exponents. We apply the method to equations resembling generalizations of the heat equations and discuss the possibility of extending the procedure to the relativistic Schr\"odinger and Dirac equations.

We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique is used to solve the associated Liouville equations, yielding the phase-space evolution of an ensemble of relativistic particles, subject to a ``harmonic'' potential. The nonharmonic distortion of the spatial and momentum distributions due to the intrinsic nonlinear nature of the relativistic contributions is discussed. We analyze the relativistic dynamics induced by two types of Hamiltonian, which can be ascribed to those of harmonic oscillator type. Finally, we briefly discuss the quantum aspects of the problem by considering possible strategies for the solution of the associated Salpeter equation.

There is no reason why the energy spectra of the relic gravitons amplified by the pumping action of the background geometry should not increase at high frequencies. A typical example of this behavior is quintessential inflation where the slopes of the energy spectra can be either blue or mildly violet. In comparing the predictions of scenarios leading to blue and violet graviton spectra we face the problem of correctly deriving the sensitivities of the interferometric detectors. Indeed the expression of the signal-to-noise ratio not only depends upon the noise power spectra of the detectors but also upon the spectral form of the signal and, therefore, one can reasonably expect that models with different spectral behaviors will produce different signal-to-noise ratios. By assuming monotonic (blue) spectra of relic gravitons we will give general expressions for the signal-to-noise ratio in this class of models. As an example we study the case of quintessential gravitons. The minimum achievable sensitivity to $h_0^2{\Omega }_{\mathrm{GW}}$ of different pairs of detectors is computed, and compared with theoretical expectations.