Set Of Time Instants Research Articles

Модели времени для сложных распределенных дискретных систем

The impetus for writing this article was the thesis about the inadequacy of using linearly-ordered time models in describing processes in complex discrete systems characterized by the presence of components with independent behavior between the moments of their interaction, with recursively defined, hierarchically organized, varying (dynamic) structure unlimited in complexity, and with non-deterministic behavior. The cause-and-effect relationships between possible events in a distributed multicomponent system leading to changes in its state determine, among other things, the minimally necessary partial order in the set of time instants corresponding to possible events in the system. The time universe is understood to mean the class of isomorphic finite partially-strictly ordered sets as a necessary basic formalism. The above-mentioned concept differs from other well-known formalizations (such as acyclic digraphs, Hasse diagrams, etc.) in that it uses a strictly partial order under the assumption that the set of simultaneous events in the system can be considered as one complex event. The presence of the above-mentioned features of complex systems generates the need to analyze the sets of time universes both in view of possible non-determinism in the behavior of systems and as a consequence of changes resulting from events associated with the system structure itself and the cause-and-effect relationships acting in it. The problem of developing a constructive description—with different degrees of complexity—of time universe sets, primarily infinite ones, which is similar to specifying formal languages as sets of words in a certain alphabet. The authors have solved a similar problem in the theory of directed relationships - the basic formalism for constructing declarative programming languages using the network representation of directed relationships. The key point of such solution is to correctly define the operation of substituting a network into another network instead of some element thereof, which plays the role of entering the network variable into the network structure and has the same specifications as the substituted network. The T-network and T-network language notions introduced in the article, i.e., sets of T-networks, are used to constructively specify the sets of time universes in modeling parallel processes in systems. The main syntactic difference between T-networks and networks of directed relationships is the requirement that T-networks shall not contain cycles, which is ensured by fulfilling certain proposed restrictions posed to network components. The correctness of the substitution operation with respect to these restrictions is proven. The substitution operation can be used both for hierarchically constructing T-networks and for multivariate and recursive assignment of T-network sets. Application of the substitution operation for defining a representative class of context-free T-network languages using context-free T-network grammars is described. A simple grammar example describing the time model for parallel recursive calculation of a definite integral is given, and further research lines are pointed out.

Trading-network order formation with the help of the aggregation of specialized forecast algorithms

The problem concerning the aggregating of the forecasts of specialized expert strategies is examined using the mathematical theory of machine learning. Expert strategies are understood as the algorithms capable of successively predicting the components of a time series in the online mode. The specialized strategies can refrain from predictions at certain time instants—they make forecasts in compliance with the application area of the specific model of an object region forming their basis. An optimal algorithm whereby the forecasts of such expert strategies are aggregated into the single forecast is proposed. The algorithmic optimality consists in that, on average, its total losses are asymptotically less than those of any active prediction strategies on a set of time instants. The uppermost estimated error of the given mixing of predictions, i.e., the regret of aggregating strategies, is determined. The errors are estimated in the worst situation where no assumptions are made about the mechanism underlying the initial data source. The proposed algorithm is tested using the real information on the commodity circulation of a trading network. The numerical results and estimates of the regret are presented.

On an asymptotic analysis problem related to the construction of an attainability domain

Problems of constructing and analyzing the properties of attainability domains play an important role in control theory and its applications. In particular, this applies to control under impulse constraints that reflect the energetics of a process. The situation is complicated by the possible instability of the process under variation (in particular, under relaxation) of constraints related to boundary and intermediate conditions. Stability of the problem is also missing, in general, under relaxation of state constraints. In these cases, it is natural to focus on the asymptotic variant of the statement; this is especially expedient when one has to deal with initially asymptotic requirements. In all these cases, it seems expedient to use analogs of J. Warga’s approximate solutions. At the same time, to seek necessary approximate (and, in fact, asymptotic) solutions, it is natural to use generalized modes. For problems with impulse constraints and discontinuity in the coefficients of control actions, such modes lead to phenomena described by products of discontinuous functions and generalized functions even in the class of linear systems. In a large series of his studies, to overcome the arising difficulties, one of the authors used constructions of extension in the class of finitely additive measures. The present paper follows this approach and is ideologically relevant to the engineering problem of controlling the thrust of an engine under conditions of a given program of variation of its orientation; it is postulated that energy resources are completely consumed in a natural (for a number of impulse control problems) mode of short-duration impulses: the set of time instants at which the instantaneous control is different from zero can be embedded in an interval of vanishingly small length. Within these short periods of time, the engine should consume all energy resources while obeying some other constraints (making the sense of moment constraints) to a high degree of accuracy. In addition, one should take into account the possible discontinuity of the functions defining the coefficients of control actions. As a natural analog of the attainability domain, we use an attraction set, whose construction, together with the subsequent study of its main properties, constitutes the goal of the present study.

MIMICKING FINITE DIMENSIONAL MARGINALS OF A CONTROLLED DIFFUSION WITH JUMPS

Given a controlled diffusion with jumps, it is shown under some conditions that there exists a Markov controlled diffusion which has the same cost under any criterion which only depends on the one-dimensional distributions. This result is then used to prove that the finite dimensional marginal distribution of a controlled diffusion with jumps at a prescribed set of time instants can also be attained by using a control from a much smaller class of controls called "nearly Markov controls", extending the work of Borkar [4] to the discontinuous setting.

Encoding Bandpass Signals Using Zero/Level Crossings: A Model-Based Approach

A new approach to representing a time-limited, and essentially bandpass signal x(t) , by a set of discrete frequency values is proposed. The set of discrete frequency values is the set of locations along the frequency axis at which (real and/or imaginary parts of) the Fourier transform of the signal x(t) cross certain levels (especially zero level). Analogously, invoking time-frequency duality, a set of time instants denoting the zero/level crossings of a waveform x(t) can be used to represent a bandlimited spectrum X(f) . The proposed signal representation is based on a simple bandpass signal model that exploits our prior knowledge of the bandwidth/timewidth of the signal. We call it a Sum-of-Sincs (SOS) model, where Sinc stands for the familiar sinx/x function. Given the discrete fequency/time locations, we can accurately reconstruct the signal x(t) or the spectrum X(f) by solving a simple eigenvalue or a least squares problem. Using this approach as the basis, we propose an analysis/synthesis algorithm to decompose and represent complex multicomponent signals like speech over the entire time-frequency region. The proposed signal representation is an alternative to standard analog to discrete conversion based on the sampling theorem, and in principle, possesses some of the desirable attributes of signal representation in natural sensory systems.