Stochastic Meaning Research Articles

Power spectral density analysis of a scaled model simulation of an optical tweezer

The majority of microscale and nanoscale mechanical systems encountered in the world are stiff. One example of such a system is a microbead trapped in an optical tweezer. For this example, the ratio of a body’s mass to the viscous damping coefficient is negligible. Apart from being stiff, this system is also stochastic meaning that the differential equations representing the system include a random variable. This type of model can be solved with existing stochastic differential equation (SDE) solvers. Addressing the stochastic nature of the model usually requires averaging the results of multiple simulations. The computational time required to run the hundreds of simulations that may be necessary to obtain a useful average can be prohibitive. This paper presents a scaling approach that significantly reduces the computational time required to obtain these averages. However, this scaling changes the model and therefore the power spectral density (PSD) of the predicted motion. This work provides a general analysis of a mass-spring-damper model, such as an optical tweezer, that clearly shows the effect of scaling on the frequency content of the stochastic system. This analysis shows that the scaling technique can be used effectively without much loss of information. An experimental dataset of a 2 μm diameter microbead falling into an optical trap is compared with the simulation for validating the scaling method.

Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

A Note on the Generalized Relativistic Diffusion Equation

We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning.

PROJECT MANAGEMENT AND DEVELOPMENT

Determinant systems - are systems in which the state of the system can be fully identified at any time even if the implementation of a specific control act or entry leads to a false state, which can be foreseen, then such a system is defined. Namely, the determinant system is a system whose behavior is fully known. Such systems consist of the elements between which precisely defined links exist, and their interconnections are known, so that when we know the prior state and the information processing algorithm we can predict the following state of the system. Governance with defined systems implies the existence of a specific purpose and criterion of governance. With these systems we mean real-material systems that exist in a given environment. Once determined, the timing breaks down to a small or large extent, depending on the changing environment, just as under the influence and influence of the system itself. There is no 100% defined system for this. Stochastic systems - are those in which the application of the controlling or influential action of inputs to the system transforms the known state of the system into one possible state, but not the only action. Stochastic meaning is the opposite of determination. Stochastic meaning is used when we want to mark phenomena (processes) which do not perform according to the prescribed law but contain a random character. Their prediction is based on false probability experience. In the stochastic set of systems belong composite systems, hence systems consisting of a large number of interconnected elements. Often with stochastic systems we refer to systems that cannot be described accurately in the language of mathematics either because the system has a large number of related elements unknown to us or because we know enough about the nature of the phenomena that occur. in the system itself, and therefore we cannot quantitatively describe them. Systems with stochastic behavior are thought of as systems where processes and changes in them are driven by insufficient causes for us. To govern these systems we need to know the probability of the origin of the particular state of the system in which they can be found after the implementation of the particular action (experiments).Equilibrium - is once and for all the right condition. When the system sets a state of equilibrium, then it tends to contain that state regardless of the circumstances that prevail in the surroundings. If the system with external or internal impact is removed from the equilibrium state and if it returns to equilibrium state even after the effects of those triggers are stopped, then it is a stable system. The stable system has the specified equilibrium time and does not change location and condition without the influence of external forces. Labile system - after exerting force from outside it returns to the previous state again with the help of another force. The system with large entropy does not return on its own. Label system processes usually progress and we stop them as needed. Indifferent systems - take the position and remain n and remain in it depending on the force acting on it.

An Entropy-Like Estimator for Robust Parameter Identification

This paper describes the basic ideas behind a novel prediction error parameter identification algorithm exhibiting high robustness with respect to outlying data. Given the low sensitivity to outliers, these can be more easily identified by analysing the residuals of the fit. The devised cost function is inspired by the definition of entropy, although the method in itself does not exploit the stochastic meaning of entropy in its usual sense. After describing the most common alternative approaches for robust identification, the novel method is presented together with numerical examples for validation.

Study of fuzzy stochastic limited cutting width on chatter

Fuzzy mathematical theory is applied to drawing the fuzzy stability lobes in which each lobe is characterized by a membership grade of experiential distribution of testing data in the theoretical distribution set of chatter signal. The judgement of limit value of free-chatter cutting width is spread over the fuzzy domain in this paper. The fuzzy combination relationship between the spindle speed and the depth of cut in milling is also addressed. According to the limit width, a safety criterion on which the cutting process is stable is developed. Also, the concept and definition of safety criterion for the cutting process stability operation for fuzzy stochastic meaning are given. Analysis indicates that the fuzzy stability lobes have definite physical significance. First, they can tell us in which status the cutting process is for the drawn lobe. Second, they reflect the probability distribution of the limit value of cut width in the fuzzy domain with respect to the identification of chatter status (fuzzy event). Meanwhile, it indicates that there is a transition between unstable lobes and stable lobes in a stability threshold graph with the influence of both fuzzy stochastic parametric excitation and fuzzy stochastic external excitation. Testing value curves of the fuzzy allowed domain of the limited cutting width are developed via experiment.

New Stochastic Approach to the Renormalization of the Supersymmetric ϕ4 with Ultrametric

We present a new real space renormalization-group map, on the space of probabilities, to study the renormalization of the SUSY ϕ4. In our approach we use the random walk representation on a lattice labeled by an ultrametric space. Our method can be extended to any ϕn. New stochastic meaning is given to the parameters involved in the flow of the map and results are provided.

On a two-level multigrid solution method for finite Markov chains

We study a two-level algebraic multigrid scheme for computing the stationary distribution of a homogeneous Markov chain with a finite state space. We reveal its relationship to an iterative aggregation-disaggregation method, provide an error analysis, and prove its local convergence. Moreover, we discuss the stochastic meaning of a block SOR procedure that can be used as iterative smoother of the multigrid scheme.

Stochastic differential calculus, the Moyal *-product, and noncommutative geometry

A reformulation of the Ito calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d2 = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Ito calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.