Time Hierarchy Research Articles

Time of relaxation in dusty plasma model

Dust particles in plasma may have different values of average kinetic energy for vertical and horizontal motion. The partial equilibrium of the subsystems and the relaxation processes leading to this asymmetry are under consideration. A method for the relaxation time estimation in nonideal dusty plasma is suggested. The characteristic relaxation times of vertical and horizontal motion of dust particles in gas discharge are estimated by analytical approach and by analysis of simulation results. These relaxation times for vertical and horizontal subsystems appear to be different. A single hierarchy of relaxation times is proposed.

Molecular kinetic theory of strongly nonequilibrium processes of mass, momentum, and energy transfer: Local equilibrium criteria

Consequences of the complete system of transfer equations of the properties (momentum, energy, and mass) of particles and their pairs are considered under local equilibrium conditions with regard to the Bogoliubov hierarchy of relaxation times between the first and second distribution functions (DFs) and distinctions in the characteristic relaxation times of particle momentum, energy, and mass. It is found that even under the local equilibrium condition in the Bogoliubov hierarchy of relaxation times between the first and second DFs, pair correlations are maintained between all dynamic variables (velocity, temperature, and density) whose values are proportional to the gradients of transferable properties. A criterion is introduced requiring there be no local equilibrium condition upon reaching the critical value at which the description of the transfer process becomes incorrect in classical nonequilibrium thermodynamics. External forces are considered in the equations for strongly nonequilibrium processes. Along with allowing for intermolecular potentials, it becomes possible to discuss the concept of passive forces (introduced in thermodynamics by Gibbs) from the standpoint of the kinetic theory. It is shown that use of this concept does not reflect modern representations of real processes.

Modeling the complex dynamics of heterogeneous catalytic reactions with fast, intermediate, and slow variables

The paper presents some results of studying different types of the multi-peak oscillations and the chaotic behavior in the low-dimensional dynamical systems with a hierarchy of characteristic times. Under study is a kinetic model which consists of three nonlinear ordinary differential equations with fast, intermediate, and slow variables and illustrates that the influence of adsorbed species on the rate of catalytic reaction may lead to multi-peak oscillations and even to chaotic behavior under isothermal conditions. Such a situation occurs, for example, when the surface heterogeneity of the catalytic sites causes the activation energy of some reaction stage (or stages) to change with the surface coverage by one of the intermediate substances. Numerical simulations are used to demonstrate different types of the multi-peak oscillations and chaos in the kinetic model of hydrogen oxidation on nickel catalyst.

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Elementary linear logic is a simple variant of linear logic due to Girard and which characterizes in the proofs-as-programs approach the class of elementary functions, that is to say functions computable in time bounded by a tower of exponentials of fixed height. Other systems like light and soft linear logics have then been defined to characterize in a similar way the more interesting complexity class of polynomial time functions, but at the price of either a more complicated syntax or of more sophisticated encodings. These logical systems can serve as the basis of type systems for λ-calculus ensuring polynomial time complexity bounds on well-typed terms.This paper aims at reviving interest in elementary linear logic by showing that, despite its simplicity, it can capture smaller complexity classes than that of elementary functions. For that we carry a detailed analysis of its normalization procedure, and study the complexity of functions represented by a given type. We then show that by considering a slight variant of this system, with type fixpoints and free weakening (elementary affine logic with fixpoints) we can characterize the complexity of functions of type !W⊸!k+2B, where W and B are respectively types for binary words and booleans. The key point is a sharper study of the normalization bounds. We characterize in this way the class P of polynomial time predicates, and more generally the hierarchy of classes k-EXP, for k≥0, where k-EXP is the union of DTIME(2kni), for i≥1.

Arbitrary sequence RAMs

It is known that in some cases a Random Access Machine (RAM) benefits from having an additional input that is an arbitrary number, satisfying only the criterion of being sufficiently large. This is known as the ARAM model. We introduce a new type of RAM, which we refer to as the Arbitrary Sequence RAM (ASRAM), that generalises the ARAM by allowing the generation of additional arbitrary large numbers at will during execution time. We characterise the power contribution of this ability under several RAM variants.In particular, we demonstrate that an arithmetic ASRAM is more powerful than an arithmetic ARAM, that a sufficiently equipped ASRAM can recognise any language in the arithmetic hierarchy in constant time (and more, if it is given more time), and that, on the other hand, in some cases the ASRAM is no more powerful than its underlying RAM.

Equation of longitudinal vibrations of a beam with coefficients periodically depending on the coordinate

The equation of small longitudinal vibrations of a beam is considered for the case in which the coefficients are periodic and have “gaps” (sharp decreases in the value) at some point in the space inside the periodicity interval. We suggest to describe the process locally in the domain of minimum of the coefficients by an equation of the boundary layer approximation. Attention is mainly paid to the analysis of this equation. A time hierarchy of simplified models for describing this process is established.

Temporal decomposition: A strategy for building mathematical models of complex metabolic systems

In ‘Discovering complexity’ Bechtel and Richardson (1993) highlighted the connection between how biologists investigate the world and the type of explanations they give. This paper extends their account of how we investigate the world by examining the strategies used by researchers to build mathematical models of complex metabolic systems between the 1970s and 1990s. Bechtel and Richardson analysed how researchers decompose complex systems by reducing the number of variables included in the model, thus simplifying them and making them suitable objects for research and understanding. Bechtel and Abrahamsen (2005) later distinguished two types of decomposition: 1) Structural decomposition, starting with the identification of the relevant component parts and 2) functional decomposition, starting with the identification of the relevant component operations. I use my case studies to argue that temporal decomposition should be recognised as an additional strategy for investigating complex metabolic systems. Temporal decomposition involves the identification of the relevant dynamic variables. Existing accounts of decomposition are based on the assumption of a spatial hierarchy which classifies modules according to the frequency of interactions between components. Temporal decomposition is based on the assumption of a time hierarchy which classifies variables as dynamic or constant according to the relative speed with which properties of the system change.

Low-temperature dynamics of long-ranged spin-glasses: full hierarchy of relaxation times via real-space renormalization

We consider the long-ranged Ising spin-glass with random couplings decaying as a power-law of the distance, in the region of parameters where the spin-glass phase exists with a positive droplet exponent. For the Metropolis single-spin-flip dynamics near zero temperature, we construct via real-space renormalization the full hierarchy of relaxation times of the master equation for any given realization of the random couplings. We then analyze the probability distribution of dynamical barriers as a function of the spatial scale. This real-space renormalization procedure represents a simple explicit example of the droplet scaling theory, where the convergence towards local equilibrium on larger and larger scales is governed by a strong hierarchy of activated dynamical processes, with valleys within valleys.

Aeg maast ikus. Ristikivi metafüüsiline kronotoop

The article examines two novels published during the early years of Karl Ristikivi’s exile, All That Ever Was (1946) and Nothing Happened (1947), and his poetry from 1950s. The main aim is to explain the poetic and semantic mechanism that transformed Ristikivi’s realism of the 1930s into a metaphysical outlook in the 1940s and 1950s. The Estonian landscape has long been a significant source of inspiration for Estonian poetry (and, frequently, prose) for the expression of an individual’s spatio-temporal relations. For Ristikivi, landscape plays an essential role in the conception of his literary chronotope. To gain a deeper understanding of the emergence of Ristikivi’s metaphysical chronotope in these two novels one must search for corresponding time and space imagery. Natural elements manifest the spatio-temporal relations that find expression in time-containing landscapes. Often the cyclical landscape reflects a character’s life, although the narrator’s perspective is always from the present to the past. In Ristikivi’s works, the present lacks a sense of place – a permanent relationship with a physical place. Regarding the time aspect, one could however refer to the so-called biographical inversion, biographical reversals, retreating to a long-gone golden era, where space too had different qualities. The spatio-temporal universality of nature imagery is vividly expressed in Ristikivi’s poetry. Metaphysical space is portrayed using concrete metaphors (sea, island, shore), but time-specific independent imagery is missing – time is portrayed as an organic part of landscape and of space as a whole. Here, what is important is not the realistic passage of time and its peripherally associated phenomena, but the experience of an earthly journey, the so-called destined time. The atmosphere is one of metaphysical flight from life, unreal landscapes in a bizarre world and a temporal vacuum. In the lyrics, Risikivi’s metaphysical chronotope is concentrated into a compact poetic whole. Ristikivi locates time in the landscape, thus creating a chronotope that develops a new and meaningful metaphysical unity. However, a temporal invariance endures, i.e., in the hierarchy of time, the past predominates, which recall and preserves the geneological, historical or societal-social subtext. While Estonian poetry is characterized by yearning for the past, thus implying a spatio-temporal infinity, this major Estonian artist names the infinite wandering as „a human journey”.

Time Hierarchies for Sampling Distributions

We show that “a little more time gives a lot more power to sampling algorithms.” We prove that for every constant $k\ge 2$, every polynomial time bound $t$, and every polynomially small $\epsilon$, there exists a family of distributions on $k$ elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance $1-1/k-\epsilon$ in time $t$. This implies the following general time hierarchy for sampling distributions on arbitrary-size domains such as $\{0,1\}^n$: For every polynomial time bound $t$ and every constant $\epsilon>0$, there exists a family of distributions that can be sampled exactly in polynomial time but cannot be sampled within statistical distance $1-\epsilon$ in time $t$. Our proof involves reducing the problem to a communication problem over a certain type of noisy channel. To solve the latter problem we use a type of list-decodable code for a setting where there is no bound on the number of errors but each error gives more information than an erasure. This type of code can be constructed using certain known traditional list-decodable codes, but we give a new construction that is elementary, self-contained, and tailored to this setting.

Spatio-temporal hierarchy in the dynamics of a minimalist protein model

A method for time series analysis of molecular dynamics simulation of a protein is presented. In this approach, wavelet analysis and principal component analysis are combined to decompose the spatio-temporal protein dynamics into contributions from a hierarchy of different time and space scales. Unlike the conventional Fourier-based approaches, the time-localized wavelet basis captures the vibrational energy transfers among the collective motions of proteins. As an illustrative vehicle, we have applied our method to a coarse-grained minimalist protein model. During the folding and unfolding transitions of the protein, vibrational energy transfers between the fast and slow time scales were observed among the large-amplitude collective coordinates while the other small-amplitude motions are regarded as thermal noise. Analysis employing a Gaussian-based measure revealed that the time scales of the energy redistribution in the subspace spanned by such large-amplitude collective coordinates are slow compared to the other small-amplitude coordinates. Future prospects of the method are discussed in detail.

An interpolation-based fast-multipole accelerated boundary integral equation method for the three-dimensional wave equation

A new fast multipole method (FMM) is proposed to accelerate the time-domain boundary integral equation method (TDBIEM) for the three-dimensional wave equation. The proposed algorithm is an enhancement of the interpolation-based FMM for the time-domain case, adopting the notion of the plane-wave time-domain algorithm. With the application being targeted at a low-frequency regime, the proposed time-domain interpolation-based FMM can reduce the computational complexity of the TDBIEM from O(Ns2Nt) to O(Ns1+δNt) (where δ=1/3 or 1/2) with the help of multilevel space–time hierarchy, where Ns and Nt are the spatial and temporal degrees of freedom, respectively. The computational accuracy and speed of the proposed accelerated TDBIEM are verified in comparison with those of the conventional (direct) TDBIEM via numerical experiments.

A win ratio approach to comparing ordinal or non-normal outcomes in clinical trials

Clinical trials are often designed to compare an ordinal outcome such as New York Heart Association (NYHA) functional classification or a non-Normal outcome such as duration of hospital stay in cardiovascular trials. The conventional statistical method for such a comparison is a non-parametric Mann-Whitney test, which provides a P-value for testing the hypothesis that the distributions of both treatment groups are identical but does not provide an estimate of treatment effect. For that, Hodges and Lehmann proposed estimating the shift parameter between two populations and its confidence interval. However, such a shift parameter does not have a straightforward interpretation and its confidence interval contains zero in some cases when Mann-Whitney test produces significant result. To overcome the above problems, we introduce the concept of the win ratio for analysing such data. The original use of the win ratio was for a hierarchy of composite time to event outcomes (Pocock et al 2012). Patients in the new and control treatment are formed into all possible pairs. For each pair the new treatment patient is labelled a winner or a loser if it is known who had a favourable outcome. The win ratio is the total number of winners divided by the total numbers of losers. A 95% CI and P-value for the win ratio are readily obtained. This method is demonstrated by analyses of NYHA in PARTNER B trial and hospital stay in PLACIDE trial. Statistical properties of the win ratio statistic are investigated using simulation studies of non-Normal outcomes under various scenarios.

How Microscopic Characteristics of the Adsorption Kinetics Impact Macroscale Transport in Chromatographic Beds

Chromatography aims at maximizing the differential migration (separation) of analytes relative to their dispersive transport (peak broadening). This dynamics involves a complex transport scheme with a hierarchy of characteristic time and length scales, from analyte–surface interactions to effective macroscale transport behavior. Here we report on results obtained with a comprehensive modeling approach that combines three-dimensional pore-scale simulations of fluid flow and mass transport, i.e., advection–diffusion in the interstitial void space of a random packing of polydisperse hard spheres (representing a chromatographic bed) with the stochastic model of analyte sorption at the solid–liquid interface of the packing. The retention factor of an analyte is constructed by independent adjustment of the adsorption probability and the mean adsorption time. The approach can realize any microscopic model of the adsorption kinetics based on a distribution of adsorption sojourn times expressed in analytical or nu...

Subdiffusive rocking ratchets in viscoelastic media: Transport optimization and thermodynamic efficiency in overdamped regime

We study subdiffusive overdamped Brownian ratchets periodically rocked by an external zero-mean force in viscoelastic media within the framework of a non-Markovian generalized Langevin equation approach and associated multidimensional Markovian embedding dynamics. Viscoelastic deformations of the medium caused by the transport particle are modeled by a set of auxiliary Brownian quasiparticles elastically coupled to the transport particle and characterized by a hierarchy of relaxation times which obey a fractal scaling. The most slowly relaxing deformations which cannot immediately follow to the moving particle imprint long-range memory about its previous positions and cause subdiffusion and anomalous transport on a sufficiently long time scale. This anomalous behavior is combined with normal diffusion and transport on an initial time scale of overdamped motion. Anomalously slow directed transport in a periodic ratchet potential with broken space inversion symmetry emerges due to a violation of the thermal detailed balance by a zero-mean periodic driving and is optimized with frequency of driving, its amplitude, and temperature. Such optimized anomalous transport can be low dispersive and characterized by a large generalized Peclet number. Moreover, we show that overdamped subdiffusive ratchets can sustain a substantial load and do useful work. The corresponding thermodynamic efficiency decays algebraically in time since the useful work done against a load scales sublinearly with time following to the transport particle position, but the energy pumped by an external force scales with time linearly. Nevertheless, it can be transiently appreciably high and compare well with the thermodynamical efficiency of the normal diffusion overdamped ratchets on sufficiently long temporal and spatial scales.

Rigorous incorporation of tautomers, ionization species, and different binding modes into ligand-based and receptor-based 3D-QSAR methods.

Speciation of drug candidates and receptors caused by ionization, tautomerism, and/or covalent hydration complicates ligandand receptor-based predictions of binding affinities by 3-dimensional structure-activity relationships (3D-QSAR). The speciation problem is exacerbated by tendency of tautomers to bind in multiple conformations or orientations (modes) in the same binding site. New forms of the 3D-QSAR correlation equations, capable of capturing this complexity, can be developed using the time hierarchy of all steps that lie behind the monitored biological process - binding, enzyme inhibition or receptor activity. In most cases, reversible interconversions of individual ligand and receptor species can be treated as quickly established equilibria because they are finished in a small fraction of the exposure time that is used to determine biological effects. The speciation equilibria are satisfactorily approximated by invariant fractions of individual ligand and receptor species for buffered experimental or in vivo conditions. For such situations, the observed drug-receptor association constant of a ligand is expressed as the sum of products, for each ligand and receptor species pair, of the association microconstant and the fractions of involved species. For multiple binding modes, each microconstant is expressed as the sum of microconstants of individual modes. This master equation leads to new 3D-QSAR correlation equations integrating the results of all molecular simulations or calculations, which are run for each ligand-receptor species pair separately. The multispecies, multimode 3D-QSAR approach is illustrated by a ligand-based correlation of transthyretin binding of thyroxine analogs and by a receptor-based correlation of inhibition of MK2 by benzothiophenes and pyrrolopyrimidines.

Dimensionality Reduction of Bistable Biological Systems

Time hierarchies, arising as a result of interactions between system's components, represent a ubiquitous property of dynamical biological systems. In addition, biological systems have been attributed switch-like properties modulating the response to various stimuli across different organisms and environmental conditions. Therefore, establishing the interplay between these features of system dynamics renders itself a challenging question of practical interest in biology. Existing methods are suitable for systems with one stable steady state employed as a well-defined reference. In such systems, the characterization of the time hierarchies has already been used for determining the components that contribute to the dynamics of biological systems. However, the application of these methods to bistable nonlinear systems is impeded due to their inherent dependence on the reference state, which in this case is no longer unique. Here, we extend the applicability of the reference-state analysis by proposing, analyzing, and applying a novel method, which allows investigation of the time hierarchies in systems exhibiting bistability. The proposed method is in turn used in identifying the components, other than reactions, which determine the systemic dynamical properties. We demonstrate that in biological systems of varying levels of complexity and spanning different biological levels, the method can be effectively employed for model simplification while ensuring preservation of qualitative dynamical properties (i.e., bistability). Finally, by establishing a connection between techniques from nonlinear dynamics and multivariate statistics, the proposed approach provides the basis for extending reference-based analysis to bistable systems.

Cascade dispersion of metal drops continuously charged in electron flow

The main effects determining the efficiency of cascade dispersion of metal drops continuously charged in an electron beam are theoretically analyzed. A hierarchy of the characteristic times of charging, deformation, and dispersion of drops is established. Experimental data show that the distribution of particle sizes is multimodal and consists of three separate peaks corresponding to various steps of the process of cascade dispersion. This character of distribution agrees well with predictions of theoretical analysis.

Multicanonical distribution: Statistical equilibrium of multiscale systems

A multicanonical formalism is introduced to describe the statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.

Applicative theories for the polynomial hierarchy of time and its levels

In this paper we introduce applicative theories which characterize the polynomial hierarchy of time and its levels. These theories are based on a characterization of the functions in the polynomial hierarchy using monotonicity constraints, introduced by Ben-Amram, Loff, and Oitavem.