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This paper presents the first sufficient conditions that guarantee the stability and almost sure convergence of multi-timescale stochastic approximation (SA) iterates. It extends the existing results on one-timescale and two-timescale SA iterates to general $N$-timescale stochastic recursions, for any $N \geq 1$, using the ordinary differential equation (ODE) method. As an application, we study SA algorithms augmented with heavy-ball momentum in the context of Gradient Temporal Difference (GTD) learning. The added momentum introduces an auxiliary state evolving on an intermediate timescale, yielding a three-timescale recursion. We show that with appropriate momentum parameters, the scheme fits within our framework and converges almost surely to the same fixed point as baseline GTD. The stability and convergence of all iterates including the momentum state follow from our main results without ad hoc bounds. We then study off-policy actor-critic algorithms with a baseline learner, actor, and critic updated on separate timescales. In contrast to prior work, we eliminate projection steps from the actor update and instead use our framework to guarantee stability and almost sure converge

Uncertainty in renewable energy generation and load consumption is a great challenge for microgrid operation, especially in islanded mode as the microgrid may be small in size and has limited flexible resources. In this paper, a multi-timescale, two-stage robust unit commitment and economic dispatch model is proposed to optimize the microgrid operation. The first stage is a combination of day-ahead hourly and real-time sub-hourly model, which means the day-ahead dispatch result must also satisfy the real-time condition at the same time. The second stage is to verify the feasibility of the day-ahead dispatch result in worst-case condition considering high-level uncertainty in renewable energy dispatch and load consumptions. In the proposed model, battery energy storage system (BESS) and solar PV units are integrated as a combined solar-storage system. The BESS plays an essential role to balance the variable output of solar PV units, which keeps the combined solar-storage system output unchanged on an hourly basis. In this way, it largely neutralizes the impact of solar uncertainty and makes the microgrid operation grid friendly. Furthermore, in order to enhance the flexibility and r

We derive the timescale for two initially pure subsystems to become entangled with each other through an arbitrary Hamiltonian that couples them. The entanglement timescale is inversely proportional to the "correlated uncertainty" between the two subsystems, a quantity which we will define and analyze in this paper. Our result is still applicable when one of the subsystems started in an arbitrarily mixed state, thus it generalizes the well-known "decoherence timescale" while coupled to a thermal state.

The timescale for star formation, a measure of how quickly neutral gas is being converted to stars, is considerably longer than typical dynamical timescales associated with a galactic disk. For purposes of modeling galaxy evolution, however, it would be extremely attractive if the star formation timescale were proportional to an easily derived dynamical timescale. We compare estimates of the star formation timescale within nearby galaxies, based on the work of Leroy et al. (2008) and existing BIMA SONG CO data, with three simple forms of the dynamical time: the orbital time, the free-fall time at the midplane density, and the disk Jeans time (the growth time for gravitational instabilities in a disk). When taking into account the gravity of the stellar disk in an approximate way, all three timescales show correlations with the star formation timescale, though none of the correlations can be accurately described as linear. Systematic errors in estimating appropriate gas masses and the stellar velocity dispersion may obscure an underlying correlation, but we focus instead on a model where the timescale for H_2 formation from HI is decoupled from the timescale for star formation from

If two physical timescales are independent, i.e. they depend on different physics, then (statistically) there is no reason to believe that their values should be equal, even to an order of magnitude. The timescale for abiogenesis $τ_{AB}$, which depends primarily on prebiotic-chemistry, is expected to be independent of the planetary habitability timescale $τ_{Hab}$, which depends primarily on the sun and therefore on nuclear forces and gravity. Therefore, we expect that either $τ_{AB} \ll τ_{Hab}$ or $τ_{AB} \gg τ_{Hab}$. The correct inequality is universal and a single example should suffice to resolve this binary choice. Here I argue that, contrary to a well known anthropic selection effect, our existence (which entails life on Earth) can be considered evidence that the correct choice is the former. A Bayesian analysis, taking into account that our existence is old evidence, implies that the probability of the hypothesis $τ_{AB} \ll τ_{Hab}$ is > 0.91, assuming equal priors. The Bayes factor, which depends only on the evidence, is > 10 and suggests strong to decisive support for the short abiogenesis timescale hypothesis, according to the Jeffreys interpretation.

Recently it was shown that the growth of entanglement in an initially separable state, as measured by the purity of subsystems, can be characterized by a timescale that takes a universal form for any Hamiltonian. We show that the same timescale governs the growth of entanglement for all Rényi entropies. Since the family of Rényi entropies completely characterizes the entanglement of a pure bipartite state, our timescale is a universal feature of bipartite entanglement. The timescale depends only on the interaction Hamiltonian and the initial state.

Using observations of pulsars from the Parkes Pulsar Timing Array (PPTA) project we develop the first pulsar-based timescale that has a precision comparable to the uncertainties in international atomic timescales. Our ensemble of pulsars provides an Ensemble Pulsar Scale (EPS) analogous to the free atomic timescale Echelle Atomique Libre (EAL). The EPS can be used to detect fluctuations in atomic timescales and therefore can lead to a new realisation of Terrestrial Time, TT(PPTA11). We successfully follow features known to affect the frequency of the International Atomic Timescale (TAI) and we find marginally significant differences between TT(PPTA11) and TT(BIPM11). We discuss the various phenomena that lead to a correlated signal in the pulsar timing residuals and therefore limit the stability of the pulsar timescale.

The paradigm of layered networks is used to describe many real-world systems -- from biological networks, to social organizations and transportation systems. Recently there has been much progress in understanding the general properties of multilayer networks, our understanding of how to control such systems remains limited. One aspect that makes this endeavor challenging is that each layer can operate at a different timescale, thus we cannot directly apply standard ideas from structural control theory of individual networks. Here we address the problem of controlling multilayer and multi-timescale networks focusing on two-layer multiplex networks with one-to-one interlayer coupling. We investigate the case when the control signal is applied to the nodes of one layer. We develop a theory based on disjoint path covers to determine the minimum number of inputs ($N_\T i$) necessary for full control. We show that if both layers operate on the same timescale then the network structure of both layers equally affect controllability. In the presence of timescale separation, controllability is enhanced if the controller interacts with the faster layer: $N_\T i$ decreases as the timescale dif

Setting up a relativistic lunar reference frame is of a prime importance in the context of future exploration missions to the Moon. If the procedure for building a consistent reference frame within the framework of the general theory of relativity is well established (cf. resolutions B.3 of IAU 2000), there is still some freedom in the choice of the coordinate timescale to be adopted as reference in the lunar region. In this paper, we review the orders of magnitude of the relativistic effects resulting from (i) the gravitational redshift of a clock on the lunar surface and (ii) the time transformations between a clock on the surface of the Moon and a clock on the surface of the Earth. We then discuss possible options for a lunar reference timescale with their advantages and drawbacks, taking note that the solution which is adopted for the Moon shall then be reemployed for Mars and other planets. Finally, we propose possible realizations of the lunar reference timescale as well as its traceability to UTC.

Tasks that one wishes to have done by a computer often come with conditions that relate to timescales. For instance, the processing must terminate within a given time limit; or a signal processing computer must integrate input information across several timescales; or a robot motor controller must react to sensor signals with a short latency. For classical digital computing machines such requirements pose no fundamental problems as long as the physical clock rate is fast enough for the fastest relevant timescales. However, when digital microchips become scaled down into the few-nanometer range where quantum noise and device mismatch become unavoidable, or when the digital computing paradigm is altogether abandoned in analog neuromorphic systems or other unconventional hardware bases, it can become difficult to relate timescale conditions to the physical hardware dynamics. Here we explore the relations between task-defined timescale conditions and physical hardware timescales in some depth. The article has two main parts. In the first part we develop an abstract model of a generic computational system that admits a unified discussion of computational timescales. This model is genera

Accurate estimation of the merger timescale of galaxy clusters is important to understand the cluster merger process and further the formation and evolution of the large-scale structure of the universe. In this paper, we explore a baryonic effect on the merger timescale of galaxy clusters by using hydrodynamical simulations. We find that the baryons play an important role in accelerating the merger process. The merger timescale decreases with increasing the gas fraction of galaxy clusters. For example, the merger timescale is shortened by a factor of up to 3 for merging clusters with gas fractions 0.15, compared with the timescale obtained with zero gas fractions. The baryonic effect is significant for a wide range of merger parameters and especially more significant for nearly head-on mergers and high merging velocities. The baryonic effect on the merger timescale of galaxy clusters is expected to have impacts on the structure formation in the universe, such as the cluster mass function and massive substructures in galaxy clusters, and a bias of "no-gas" may exist in the results obtained from the dark matter-only cosmological simulations.

Multiscale phenomena that evolve on multiple distinct timescales are prevalent throughout the sciences. It is often the case that the governing equations of the persistent and approximately periodic fast scales are prescribed, while the emergent slow scale evolution is unknown. Yet the course-grained, slow scale dynamics is often of greatest interest in practice. In this work we present an accurate and efficient method for extracting the slow timescale dynamics from signals exhibiting multiple timescales that are amenable to averaging. The method relies on tracking the signal at evenly-spaced intervals with length given by the period of the fast timescale, which is discovered using clustering techniques in conjunction with the dynamic mode decomposition. Sparse regression techniques are then used to discover a mapping which describes iterations from one data point to the next. We show that for sufficiently disparate timescales this discovered mapping can be used to discover the continuous-time slow dynamics, thus providing a novel tool for extracting dynamics on multiple timescales.

We propose the timescale-resolved spectroscopy (TRS) as a new method to combine the timing and spectral study. TRS is based on the time domain power spectrum and reflects the variable amplitudes of spectral components on different timescales. We produce the TRS with the RXTE PCA data for Cyg X-1 and studied the spectral parameters (the power law photon index and the equivalent width of the iron fluorescent line) as a function of timescale. The results of TRS and frequency-resolved spectra (FRS) have been compared, and similarities have been found for the two methods with the identical motivations. We also discover the correspondences between the evolution of photon index with timescale and the evolution of the equivalent width with timescale. The observations can be divided into three types according to the correspondences and different type is connected with different spectral state.

The microlensing effective timescale t_eff=beta*t_E is used frequently in high-magnification (beta << 1) microlensing events, because it is better constrained than either the impact parameter beta or the Einstein timescale t_E separately. It also facilitates intuitive understanding of lightcurves prior to determination of a model. Similar considerations may apply to very low magnification events. I therefore provide a generalization of this quantity to all events: t_eff = beta*t_E*sqrt((1+beta^2/2)(1+beta^2/4)).

In the human brain, sequences of language input are processed within a distributed and hierarchical architecture, in which higher stages of processing encode contextual information over longer timescales. In contrast, in recurrent neural networks which perform natural language processing, we know little about how the multiple timescales of contextual information are functionally organized. Therefore, we applied tools developed in neuroscience to map the "processing timescales" of individual units within a word-level LSTM language model. This timescale-mapping method assigned long timescales to units previously found to track long-range syntactic dependencies. Additionally, the mapping revealed a small subset of the network (less than 15% of units) with long timescales and whose function had not previously been explored. We next probed the functional organization of the network by examining the relationship between the processing timescale of units and their network connectivity. We identified two classes of long-timescale units: "controller" units composed a densely interconnected subnetwork and strongly projected to the rest of the network, while "integrator" units showed the long

Coulomb fission mechanism may take place if the maximum Coulomb-excitation energy transfer in a reaction exceeds the fission barrier of either the projectile or target. This condition is satisfied by all the reactions used for the earlier blocking measurements except one reaction 208 Pb + Natural Ge crystal, where the measured timescale was below the measuring limit of the blocking measurements < 1 as. Hence, inclusion of the Coulomb fission in the data analysis of the blocking experiments leads us to interpret that the measured time longer than a few attoseconds (about 2-2.5 as) is nothing but belonging to the Coulomb fission timescale and shorter than 1 as are due to the quasifission. Consequently, this finding resolves the critical discrepancies between the fission timescale measurements using the nuclear and blocking techniques. This, in turn, validates the fact that the quasifission timescale is indeed of the order of zeptoseconds in accordance with the nuclear experiments and theories. It thus provides a radical input in understanding the reaction mechanism for heavy element formation via fusion evaporation processes

We present $Γ$-nets, a method for generalizing value function estimation over timescale. By using the timescale as one of the estimator's inputs we can estimate value for arbitrary timescales. As a result, the prediction target for any timescale is available and we are free to train on multiple timescales at each timestep. Here we empirically evaluate $Γ$-nets in the policy evaluation setting. We first demonstrate the approach on a square wave and then on a robot arm using linear function approximation. Next, we consider the deep reinforcement learning setting using several Atari video games. Our results show that $Γ$-nets can be effective for predicting arbitrary timescales, with only a small cost in accuracy as compared to learning estimators for fixed timescales. $Γ$-nets provide a method for compactly making predictions at many timescales without requiring a priori knowledge of the task, making it a valuable contribution to ongoing work on model-based planning, representation learning, and lifelong learning algorithms.

A technique for timescale analysis of spectral lags performed directly in the time domain is developed. Simulation studies are made to compare the time domain technique with the Fourier frequency analysis for spectral time lags. The time domain technique is applied to studying rapid variabilities of X-ray binaries and $γ$-ray bursts. The results indicate that in comparison with the Fourier analysis the timescale analysis technique is more powerful for the study of spectral lags in rapid variabilities on short time scales and short duration flaring phenomena.

Employing multiple pulsars and using an appropriate algorithm to establish ensemble pulsar timescale can reduce the influences of various noises on the long-term stability of pulsar timescale, compared to a single pulsar. However, due to the low timing precision and the significant red noises of some pulsars, their participation in the construction of ensemble pulsar timescale is often limited. Inspired by the principle of solving non-stationary sequence modeling using co-integration theory, we puts forward an algorithm based on the co-integration theory to establish ensemble pulsar timescale. It is found that this algorithm can effectively suppress some noise sources if a co-integration relationship between different pulsar data exist. Different from the classical weighted average algorithm, the co-integration method provides the chances of the pulsar with significant red noises to attend the establishment of ensemble pulsar timescale. Based on the data from the North American Nanohertz Observatory for Gravitational Waves, we found that the co-integration algorithm can successfully reduce several timing noises and improve the long-term stability of the ensemble pulsar timescale.

Thixotropy is characterized by an increase in viscosity when a material is subjected to no flow (quiescent) or weak flow conditions and a decrease in viscosity when it is subjected to strong flow conditions. The characteristic timescale associated with the thixotropic phenomenon, particularly how the viscosity increases with time, has been termed the thixotropic timescale. In the literature, several approaches have been suggested for estimating the thixotropic timescale. The most prominent approach, however, infers it from a specific form of a kinetic expression for structure parameter evolution. In this paper, we study the various kinds of structural kinetic models, and by carrying out a careful analysis of the same, we propose a parameter for the thixotropic timescale that is associated with the most generic form of the kinetic expression for structure parameter evolution. We observe that when the viscosity of the structural kinetic model undergoes continuous increase with time and eventually diverges under quiescent conditions, which we believe is the most practical scenario, our analysis suggests that increasing the thixotropic timescale weakens the thixotropic character of a s