Single Function Of Time Research Articles

Quantum quench in non-relativistic fermionic field theory: harmonic traps and 2d string theory

We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the c = 1 Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asymptote to constant masses and frequencies at early times, and cross or approach a gapless potential. In a right side up harmonic oscillator potential we determine the scaling behavior of the one point function and the entanglement entropy of a subregion by obtaining analytic approximations to the exact answers. The results are consistent with Kibble-Zurek scaling for slow quenches and with perturbation calculations for fast quenches. For cis-critical quench protocols the entanglement entropy oscillates at late times around its initial value. For end-critical protocols the entanglement entropy monotonically goes to zero inversely with time, reflecting the spread of fermions over the entire line. For the inverted harmonic oscillator potential, the dual collective field description is a scalar field in a time dependent metric and dilaton background.

Time average estimation in the fraction-of-time probability framework

Abstract The problem of time average estimation is addressed in the fraction-of-time probability framework. In this approach, the observed signal is modeled as a single function of time rather than as a sample path of a stochastic process. Under mild regularity assumptions on temporal cumulants of the signal, a central limit theorem (CLT) is proved for the normalized error of the time average estimate, where the normalizing factor is the square root of the observation interval length. This rate of convergence is the same as that obtained in the classical stochastic approach, but is derived here without resorting to mixing assumptions. That is, no ergodicity hypothesis is needed. Examples of functions of interest in communications are provided whose time average estimate satisfies the CLT. For the class of the almost-periodic functions, a non normal limit distribution for the normalized error of the time average estimate can be obtained when the normalizing factor equals the observation interval length. The limit distribution, nevertheless, depends on the sequence adopted to perform the limit, a result that cannot be obtained in the classical stochastic approach. Numerical examples illustrate the theoretical results and an application to test the presence of a nonzero-mean cyclostationary signal is presented.

Central Limit Theorem in the Functional Approach

The central limit theorem is proved within the framework of the functional approach for signal analysis. In this framework, a signal is modeled as a single function of time rather than a stochastic process. Distribution function, expectation, and all the familiar probabilistic parameters are built starting from this single function of time by resorting to the concept of relative measure. Furthermore, the concept of independence among functions of time can be introduced. In the paper it is shown that if a sequence of independent signals fulfills some mild regularity assumptions, then the asymptotic distribution of the appropriately scaled average of such signals has a limiting normal distribution. The approach is shown to be useful when only one realization of a signal is available and no ensemble of realizations is observed or exists. The obtained results also allow one to rigorously justify stochastic models for signals and channels that up to now have been derived starting from a deterministic description of phenomena and for which the inferred stochastic model is built invoking a not proved ergodicity property. An application to the statistical characterization of the output signal of a multipath Doppler channel is presented.

Repeatable light paths in the shearfree normal cosmological models

Conditions for the existence of repeatable light paths (RLPs) in the shearfree normal cosmological models are investigated. It is found that in the conformally nonflat models the only RLPs are radial null geodesics (in the spherical case) and their analogues in the plane and hyperbolically symmetric cases. In the conformally flat Stephani models, there exist special spherically, plane, and hyperbolically symmetric subcases, in which all null geodesics are RLPs. They are slightly more general than the Friedmann-Lema\^{\i}tre-Robertson-Walker models of the corresponding symmetries: their curvature index function $k(t)$ and the scale factor $R(t)$ are expressed through a single function of time. In addition to that, there exist special cases of the Stephani solution in which some of the null geodesics are RLPs. All these special cases are identified.

Foundations of the functional approach for signal analysis

In this paper, the mathematical foundation of the functional (or nonstochastic) approach for signal analysis is established. The considered approach is alternative to the classical one that models signals as realizations of stochastic processes. The work follows the fraction-of-time probability approach introduced by Gardner. By applying the concept of relative measure used by Bochner, Bohr, Haviland, Jessen, Wiener, and Wintner and by Kac and Steinhaus, a probabilistic—but nonstochastic—model is built starting from a single function of time (the signal at hand). Therefore, signals are modeled without resorting to an underlying ensemble of realizations, i.e., the stochastic process model. Several existing results are put in a common, rigorous, measure-theory based setup. It is shown that by using the relative measure concept, a distribution function, the expectation operator, and all the familiar probabilistic parameters can be constructed starting from a single function of time. The new concept of joint relative measurability of two or more functions is introduced in this paper which is shown to be necessary for the joint characterization of signals. Moreover, by using such a concept, the independence of signals is defined. The joint relative measurability property is then used to prove the nonstochastic counterparts of several useful theorems for signal analysis. It is shown that the convergence of parameter estimators requires (analytical) assumptions on the single function of time that are much easier to verify than the classical ergodicity assumptions on stochastic processes. As an example of application, nonrelatively measurable functions are shown to be useful in the design of secure information transmission systems.

Moment tensor rate functions for the 1989 Loma Prieta Earthquake

The fault plane and overall coseismic slip of the 1989 Loma Prieta, California, earthquake (Ms=7.1) are well determined [Plafker and Galloway, 1989]. Teleseismic waves can be used to determine the time history of moment release. We invert a data set of ten broadband P and SH waves for the most general point source description: the five moment tensor rate functions. The linear inversion also provides formal estimates of model uncertainty. While the moment tensor rate functions suggest a different focal mechanism for the first few seconds of the rupture process, it is not statistically significant at the 95% level. We can thus proceed to invert for one single time function (the source time function), and five scalars (the moment tensor). The major double couple that we find (strike 138°±6° clockwise from North, dip 76°±5°, slip angle 120°±10°) agrees with the results of Plafker and Galloway [1989]. The minor double couple is small (1%). The best point source depth is about 10 km. Several broadband P waves show that a small precursor occurred a few seconds before the main pulse of moment release. The duration of the source time function is 9 s, with a moment of 2±0.5 ×1019 Nm. A bilateral rupture with velocity of 2.8 km/s and 9 s duration encompasses the aftershock region.

On the theory of the viscoelastic behavior of soft polymers in moderately large deformations

Restricting consideration to deformations in which time shift invariance is preserved, a series of models was developed for the prediction of the viscoelastic behavior of isotropic soft polymers. The models describe the time dependent stress through the Boltzmann superposition integral incorporating into it an appropriately chosen nonlinear measure of strain. The theory was developed in its most general threedimensional form. It requires a single time function, the relaxation modulus of linear viscoelastic theory. In its simplest form the strain measure contains a single material parameter.