The Quantitative Structure of Exponential Time

Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E = DTIME(2^linear) and E2 = DTIME(2^polynomial) are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed

The Quantitative Structure of Exponential Time

Department of Computer Science Iowa State University Ames, Iowa 50010 Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed

Decay estimates for evolution equations with classical and fractional time-derivatives

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators. Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviors, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation

Modelling IEEE 802.11 CSMA/CA RTS/CTS with stochastic bigraphs with sharing

Stochastic bigraphical reactive systems (SBRS) is a recent formalism for modelling systems that evolve in time and space. However, the underlying spatial model is based on sets of trees and thus cannot represent spatial locations that are shared among several entities in a simple or intuitive way. We adopt an extension of the formalism, SBRS with sharing, in which the topology is modelled by a directed acyclic graph structure. We give an overview of SBRS with sharing, we extend it with rule priorities, and then use it to develop a model of the 802.11 CSMA/CA RTS/CTS protocol with exponential backoff, for an arbitrary network topology with possibly overlapping signals. The model uses sharing to model overlapping connectedness areas, instantaneous prioritised rules for deterministic computations, and stochastic rules with exponential reaction rates to model constant and uniformly distributed timeouts and constant transmission times. Equivalence classes of model states modulo instantaneous reactions yield states in a CTMC that can be analysed using the model checker PRISM. We illustrate the model on a simple example wireless network with three overlapping signals and we present some example quantitative properties

Spin dynamics simulations of the magnetic dynamics of RbMnF3_3 and direct comparison with experiment

Spin-dynamics techniques have been used to perform large-scale simulations of the dynamic behavior of the classical Heisenberg antiferromagnet in simple cubic lattices with linear sizes $L\leq 60$. This system is widely recognized as an appropriate model for the magnetic properties of RbMnF$_3$. Time-evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins using a new algorithm implemented by Krech {\it etal}, which is based on fourth-order Suzuki-Trotter decompositions of exponential operators. The dynamic structure factor was calculated from the space- and time-displaced spin-spin correlation function. The crossover from hydrodynamic to critical behavior of the dispersion curve and spin-wave half-width was studied as the temperature was increased towards the critical temperature. The dynamic critical exponent was estimated to be $z=(1.43\pm 0.03)$, which is slightly lower than the dynamic scaling prediction, but in good agreement with a recent experimental value. Direct, quantitative comparisons of both the dispersion curve and the lineshapes obtained from our simulations with very recent experimental results for RbMnF$_3$ are presented.Comment: 30 pages, RevTex, 9 figures, to appear in PR

Complexity of Quantum States and Reversibility of Quantum Motion

We present a quantitative analysis of the reversibility properties of classically chaotic quantum motion. We analyze the connection between reversibility and the rate at which a quantum state acquires a more and more complicated structure in its time evolution. This complexity is characterized by the number ${\cal M}(t)$ of harmonics of the (initially isotropic, i.e. ${\cal M}(0)=0$) Wigner function, which are generated during quantum evolution for the time $t$. We show that, in contrast to the classical exponential increase, this number can grow not faster than linearly and then relate this fact with the degree of reversibility of the quantum motion. To explore the reversibility we reverse the quantum evolution at some moment $T$ immediately after applying at this moment an instant perturbation governed by a strength parameter $\xi$. It follows that there exists a critical perturbation strength, $\xi_c\approx \sqrt{2}/{\cal M}(T)$, below which the initial state is well recovered, whereas reversibility disappears when $\xi\gtrsim \xi_c(T)$. In the classical limit the number of harmonics proliferates exponentially with time and the motion becomes practically irreversible. The above results are illustrated in the example of the kicked quartic oscillator model.Comment: 15 pages, 13 figures; the list of references is update

Quantification of LV Volumes with 4D Real-Time Echocardiography

This paper presents a new 4D (3D+Time) expansion of echocardiographic volumes on complex exponential wavelet-like basis functions called Brushlets. Brushlet functions offer good localization in time and frequency and decompose a signal into distinct patterns of oriented textures, invariant to intensity and contrast range. Automatic left ventricle (LV) endocardial border detection is carried out in the transform domain where speckle noise is attenuated while cardiac structure location is preserved. Quantitative validation and clinical applications of this new spatio-temporal analysis tool are reported with results on phantoms and clinical data sets to quantify LV volumes and ejection fraction

Scale and structure of time-averaging (age mixing) in terrestrial gastropod assemblages from Quaternary eolian deposits of the eastern Canary Islands

Quantitative estimates of time-averaging (age mixing) in gastropod shell accumulations from Quaternary (the late Pleistocene and Holocene) eolian deposits of Canary Islands were obtained by direct dating of individual gastropods obtained from exceptionally well-preserved dune and paleosol shell assemblages. A total of 203 shells of the gastropods Theba geminata and T. arinagae, representing 44 samples (= strati graphic horizons) from 14 sections, were dated using amino acid (isoleucine) epimerization ratios calibrated with 12 radiocarbon dates. Most samples reveal a substantial variation in shell age that exceeds the error that could be generated by dating imprecision, with the mean within-sample shell age range of 6670 years and the mean standard deviation of 2920 years. Even the most conservative approach (Monte Carlo simulations with a non-sequential Bonferroni correction) indicates that at least 25% of samples must have undergone substantial time-averaging (e.g., age variations within those samples cannot be explained by dating imprecision alone). Samples vary in shell age structure, including both left-skewed (17 out of 44) and right-skewed distributions (26 out of 44) as well as age distributions with a highly variable kurtosis. Dispersion and shape of age distributions of samples do not show any notable correlation with the stratigraphic age of samples, suggesting that the structure and scale of temporal mixing is time invariant. The statistically significant multi-millennial time-averaging observed here is consistent with previous studies of shell accumulations from various depositional settings and reinforces the importance of dating numerous specimens per horizon in geochrono logical studies. Unlike in the case of marine samples, typified by right-skewed age distributions (attributed to an exponential-like shell loss from older age classes), many of the samples analyzed here displayed leftskewed distributions, suggestive of different dynamics of age mixing in marine versus terrestrial shell accumulations