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Tasks of different nature and difficulty levels are a part of people's lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable and justifiable tasks. [ABSTRACT FROM AUTHOR]

This paper establishes a new fractional frSIS model utilizing a continuous time random walk method. There are two main innovations in this paper. On the one hand, the model is analyzed from a mathematical perspective. First, unlike the classic SIS infectious disease model, this model presents the infection rate and cure rate in a fractional order. Then, we proved the basic regeneration number R0 of the model and studied the influence of orders a and b on R0. Second, we found that frSIS has a disease-free equilibrium point E0 and an endemic equilibrium point E∗. Moreover, we proved frSIS global stability of the model using R0. If R0<1, the model of E0 is globally asymptotically stable. If R0>1, the model of E∗ is globally asymptotically stable. On the other hand, from the perspective of infectious diseases, we discovered that appropriately increasing a and decreasing b are beneficial for controlling the spread of diseases and ultimately leading to their disappearance. This can help us provide some dynamic adjustments in prevention and control measures based on changes in the disease. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts. [ABSTRACT FROM AUTHOR]

The Lindley distribution is useful in a wide variety of fields, such as biology and astronomy. Many generalizations of the Lindley distribution have been introduced in the literature, with various motivations. Inspired by the concept of superstatistics in nonequilibrium statistical mechanics, we introduce here a novel generalization of the Lindley distribution, by regarding its shape parameter as a stochastic variable. This procedure is particularly well motivated in astronomy applications, since this parameter may fluctuate over a large spatiotemporal scale. By appealing to the central limit theorem, two families of distributions are constructed, associated with additive and multiplicative random processes. We discuss in detail these distributions, we study their asymptotic behavior, and compute their moments. We briefly discuss their possible use in astronomy, biology, and elsewhere. [ABSTRACT FROM AUTHOR]

A mathematical model of tumor growth as a random process is studied, in which there is an interaction between tumor cells and normal cells. A set of two coupled stochastic differential equations defines the dynamics of tumor cells and normal cells. The evolution contains driven growth, therapy, and Gaussian white noises terms. The corresponding Fokker–Planck equation is used to study the large-time behavior of the system. Specifically, for periodic therapy terms, the ratio of the probability of large tumor cells number to the probability of small tumor cells number is investigated. It is seen that increasing the interaction between the tumor cells and normal cells decreases this ratio. [ABSTRACT FROM AUTHOR]

Starting from the notion of algebraic hidden processes introduced in Ref.9, we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs. [ABSTRACT FROM AUTHOR]

This paper introduces a methodology for computing expected values associated with compression networks resulting from the application of compression algorithms to independent and identically distributed random time series. Our analysis establishes a robust correspondence between the calculated expected values and empirically derived results obtained from constructing networks using nondeterministic time series. Notably, the ratio of the average indegree of a network to the computed expected indegree for stochastic time series serves as a versatile metric. It enables the assessment of inherent randomness in time series and facilitates the distinction between nondeterministic and chaotic systems. The metric demonstrates high sensitivity to nondeterminism in both synthetic and real-world datasets, highlighting its capacity to detect subtle disturbances and high-frequency noise, even in series characterized by a deficient sample rate. Our results extend and confirm previous findings in the field. [ABSTRACT FROM AUTHOR]

When the rate of shot noise is controlled by on–off states we speak of intermittent shot noise. The on–off states lead to alternately occurring clusters of events and intermissions, respectively. We derive the power spectrum of the intermittent shot noise by applying the Wiener-Khinchin theorem. Besides reduced shot noise, we obtain excess noise, which depends on the parameters of the on–off states. We calculate the excess noise for power-law distributed on-states; within the scaling region, the excess noise is excellently approximated by C∕fb. The behavior of the slope b and of the amplitude C in dependence of the on–off times is investigated. For large scaling regions, we find a preference for a pure 1/f shape. Finally, we regard the variance of events occurring within a time interval. In the presence of 1/f fluctuations, the variance of counts attains extreme values which are accompanied by an extreme property of slope b. [ABSTRACT FROM AUTHOR]

In the field of molecular nonequilibrium transports, physical mechanisms of multiple reaction dynamics of these systems are the core of deep understanding complex reactions and transport mechanisms. In order to explore related mechanisms, establishing multiple systems coupled with tremendous exit dynamics and studying their exit dynamics properties are quite vital. Beyond previous researches, new stochastic transport processes are emphasized here. Multiple new exit dynamic systems are established, which are motivated by the multiplicity of paths and products of real biochemical processes in organisms. In order to ensure research universality, core system modeling factors are fully considered. Countable parallel orbits, uniform connection with external sources, countable parallel orbits as subsystems in middle lattices and influences of all lattices on transport trajectories on dynamic properties are analyzed. Dynamic properties of different particles located in orbits are explored by deeply studying average exit time and time scale. Quantitative spatiotemporal impacts are extensively studied. The rationality of average exit time as a time scale in the universal exit dynamic system is proved. Main findings and fruitful results can not only serve as theoretical bases for broadening reaction path modeling, but also be helpful to support understanding nonequilibrium transport mechanisms, especially stochastic biochemical processes. [ABSTRACT FROM AUTHOR]

We studied the first detection problem of the quantum particle transition on a one-dimensional Lévy crystal, which includes the long-range interactions and the transition process on the system can be described by the spatial fractional Schrödinger equation. The results demonstrate that the long-range interactions can be directly obtained for short time. The average detection times 〈 n 〉 and the corresponding variance var (n) are discontinuous with respect to the detection interval τ , which could be caused by the beat of the detection frequency 1 / τ and the oscillation frequency of the first detection probability F n . [ABSTRACT FROM AUTHOR]

The nonlinear response of the dynamics of the stochastic neuronal activity based on the leaky integrate-and-fire model was reformulated using the Loewner theory. We observed that the signal-to-noise ratio (SNR), which is an indicator of stochastic resonance (SR), and fluctuation-dissipation relation (FDR) for the neuronal dynamics were newly obtained in the theoretical framework of the Loewner evolution. Particularly, we focus on the role of the Loewner entropy S Loew , defined as the entropy of the Loewner driving force, while showing the efficacy of Loewner time conversion to theorize the nonlinear characteristic of the neuronal dynamics. The present results indicate a possible approach to the novel formulation of biophysical models. [ABSTRACT FROM AUTHOR]

In order to unravel the physical and mathematical mystery of synchronized-flow mechanism and to reveal the fundamental mechanism and origin of synchronized flow produced by nonlinear stochastic processes, we have produced simple stochastic traffic flow models (the gradual-NaSch, Phoenix and mPhoenix models) with nonlinear safe speeds. In the mNaSch model and our gradual-NaSch model, the i th vehicle's speed is the same as or different from the j th vehicle's speed because they are discrete values. In the mNaSch and gradual-NaSch models, the same discrete values of the i th and j th states make it easy to identify synchronized flow with speed-synchronized phase of the i th and j th states. On the other hand, when we deal with synchronized flow in continuous traffic flow models, we face a problem. Continuous values cause the difficulty in identification of synchronization. In order to definitely clarify whether or not synchronized flow occurs in continuous models, we have established a novel idea of Exchange approach to recognizing synchronized flow as speed-synchronized phase. Our idea is generated from the analogical image that the whole system of synchronized metronomes does not change even if the i th and j th synchronized metronomes are exchanged. The Exchange approach (exchanging the i th vehicle's state for any j th vehicle's state in the whole system of traffic flow), which causes distortion such as a collision if non-synchronized vehicle's states are exchanged and makes it possible to definitely clarify whether or not synchronized flow occurs in continuous models, is applied to our simple stochastic continuous model (the mPhoenix model). On the basis of the Exchange approach, we can recognize that the mPhoenix model surely reproduces synchronized flow. In addition, we have proposed mathematical approach to deriving nonlinear safe speeds which guarantee collision free driving and reproduce synchronized flow. [ABSTRACT FROM AUTHOR]

In this paper, we propose stochastic scheme models with two kinds of errors for a computer system. A highly reliable computer system needs greatly to be backed up successfully in case of computer system crashes due to hardware failures and malicious software infections. We aim to determine appropriate backup times in computer processing of constant and random tasks. For this purpose, the trade-off between backup times and data losses due to errors is considered. The total execution times until the computing processing completes are obtained, and optimal backup policies to minimize them are derived analytically. Furthermore, we take up extended models of random tasks with n sub-tasks and analyze them. [ABSTRACT FROM AUTHOR]

The focus of this paper is to examine the dynamic factor of the suburban railway by utilizing the random vibration approach. Breaking through the previous methods of relying on huge amounts of measured data, herein, the dynamic factor essentially results from two major parts: the dynamic effect caused by moving train loads and that generated by track irregularity, which has clear physical significance. As the internal excitation of the vehicle–bridge system, track irregularity has strong randomness. Based on the dimension reduction method, the spatial domain power spectral density (PSD) of the track irregularity is transformed into the time-domain PSD. Therefore, the randomness of the random process is reduced by exploiting the constraint form of a random function, and then, the typical samples of the track irregularity considering randomness are constructed. Using the vehicle–bridge coupled vibration model, the standard deviation of the dynamic factor is evaluated accounting for the random track irregularity and 99.7% guarantee rate. Finally, the impact coefficient of the track irregularity on the bridge is methodically obtained. The sensitivity of the standard deviation of the dynamic factor to vehicle speed and bridge frequency is analyzed. The given solution methodology can fully take into account the randomness of the track irregularity. Thereby, it provides the dynamic factor formulas as a reference for the dynamic performance evaluation of suburban railway bridges and possible revision of current design specifications. [ABSTRACT FROM AUTHOR]

This paper explores memory mechanisms in complex socio-technical systems, using a mobility demand model as an example case. We simplify a large-scale agent-based mobility model, formulate the corresponding stochastic process, and observe that the mobility decision process is non-Markovian. This is due to its dependence on the system's history, including social structure and local infrastructure, which evolve based on prior mobility decisions. Complementing the mobility process with two history-determined components leads to an extended mobility process that is Markovian. Although our model is a very much reduced version of the original one, it remains too complex for the application of usual analytic methods. Instead, we employ simulations to examine the functionalities of the two history-determined components. We think that the structure of the analyzed stochastic process is exemplary for many socio-technical, -economic, -ecological systems. Additionally, it exhibits analogies with the framework of extended evolution, which has previously been used to study cultural evolution. [ABSTRACT FROM AUTHOR]

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures. [ABSTRACT FROM AUTHOR]

The nanofiltration (NF) process becomes the most recently used technologies for the desalination of seawater and brackish water. The porous media transport properties are first related to the geometrical complexity of the product. However, the membrane transport models used in desalination process constitute approximatively the less understood process. The objective of this work is to address modeling and numerical study of the desalination process of the water and ions fluxes by using a porous membrane with nanoparticles. Our filtration system used is constituted by two different zones that the membrane sheets are sandwiched. The fluid undergoes a first simple filtration and a second NF process by the injected nanoparticles. The impacts of the permeability K and porosity S of the membrane under the effect of a pressure P 0 were discussed. Our findings are obtained in the framework of the dynamic Langevin approach based on the competitiveness between the stochastic process and dissipation. The results show that the performance of the rejection membrane is significant as the nanoparticle concentration decreases, and increases as a power law with the ratio of the viscosity of the salty fluid to the pure fluid. [ABSTRACT FROM AUTHOR]

Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP T u that is the image of another GP u under a linear transformation T acting on the sample paths of u are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when T is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator T acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable. [ABSTRACT FROM AUTHOR]

We study genetic coalgebras whose comultiplications, equivalently, their associated cubic stochastic matrices, are skewed by the simultaneous action of three square stochastic matrices. A two-fold, algebraic and stochastic, approach is pursued. First, we detail the structural consequences on the algebraic structure, to then tackle the underlying stochastic processes. [ABSTRACT FROM AUTHOR]

The increasing uncertainties in power systems have brought non-negligible influences on the dynamic behaviors. An accurate and efficient simulation method considering the effects of stochastic disturbances is of critical importance for the analysis of the dynamic performance of power systems. In this paper, a methodology for the simulation of stochastic transients in power systems is developed based on frequency shifting theory. The stochastic differential equations describing the stochastic process of parameter migration are represented through analytic signals. The frequency shifting operation is introduced. The Fourier spectra of analytic signals are shifted to reduce their maximum frequency contents, thereby permitting a larger time-step in time-domain simulation in accordance with Shannon's sampling theory. Through the trapezoidal Milstein scheme, the stochastic differential equations with shifted analytic signals are discretized. Branch companion models that process analytic signals for the simulation of stochastic transients are formulated. The numerical properties of branch models are further examined. The analysis of test cases demonstrates that the proposed method can be used to represent stochastic transients caused by parameter migrations accurately and efficiently. [ABSTRACT FROM AUTHOR]

In the complex mountain wind environment, especially in the strong wind, fluctuating wind speed is a non-stationary and non-Gaussian random process. With the increase of suspension bridges' span, they are more sensitive to wind-induced vibration response. In this paper, a non-stationary and non-Gaussian buffeting reliability analysis method for long-span bridges is presented. Firstly, the non-stationary wind speed is simulated by the modulation function based on the stationary wind speed model. Secondly, the load is obtained by simulated wind speed, the obtained loads are loaded on the finite element model by ANSYS and the whole bridge time-domain analysis is performed, the time history of displacement response is obtained. Thirdly, based on the first excursion failure criterion of random vibration, the samples of the non-stationary and non-Gaussian displacement time history are transformed into a standard Gaussian process through a modified Fleishman approximation method, and then the non-stationary Poisson distribution method is used for structural reliability analysis. Finally, a mountainous long-span suspension bridge is used as the engineering background, and the proposed reliability analysis method is applied to analyze the reliability of the bridge. The influence of different wind attack angles and modulation functions on the dynamic reliability of the bridge is further studied. The results indicate that the displacement response obtained by the transformation of non-stationary and non-Gaussian random process is less reliable than that obtained by traditional analysis method. Its dynamic reliability is maximum at 1 ∘ wind attack angle, the reliability of negative wind attack angle is lower than that of positive wind attack angle, and decreases with the increase of wind attack angle. The reliability obtained by using different modulation functions is lower than that of traditional method. If the traditional analysis method is still used for reliability analysis, it will produce unsafe consequences and reduce the engineering safety reserve. [ABSTRACT FROM AUTHOR]

In this paper, we propose a pairs trading strategy using the theory of topological data analysis (TDA). The proposed strategy is model-free. We propose a TDA-based distance to measure dependence between a pair of stochastic processes. We derive an upper bound of this distance in terms of a function of the canonical correlation of the processes, which allows for interpretability of this distance. We also study Karhunen–Loève expansions of certain processes to qualitatively explore their shape properties. We check the performance of the strategy on simulated data from correlated geometric Brownian motion, correlated Ornstein–Uhlenbeck process and DCC-GARCH. We also examine the profitability of the proposed strategy on high-frequency data from the National Stock Exchange of India in 2018. We compare the method to a Euclidean distance-based method for pairs trading. We propose a pairs trading strategy evaluation framework using a Bayesian model for comparing gains from these two strategies. We find that the proposed approach based on TDA is more profitable and trades more frequently than the Euclidean distance-based strategy. [ABSTRACT FROM AUTHOR]

This paper is concerned with the general theme of relating the Large Deviation Principle (LDP) for the invariant measures of stochastic processes to the associated trajectorial LDP. It is shown that if the trajectorial deviation function has certain structure and if the invariant measures are exponentially tight, then the LDP for the invariant measures is implied by the trajectorial LDP, no other properties of the stochastic processes in question being material. As an application, we obtain an LDP for the stationary distributions of jump diffusions. Methods of large deviation convergence and idempotent probability play an integral part. [ABSTRACT FROM AUTHOR]

By installing the visco-elastic (VE) decouplers between the floor slab and the beam, the floors of a building are isolated from the primary structure. Such a structural system, termed VE decoupled floors, has an increased damping and a reduced inertial force for mitigating the floor seismic response. These features lead to a significantly improved seismic performance of the important facilities resting (or anchored) on floors. This paper focuses on the floor seismic response spectrum (FSRS) of the building equipped with VE decoupled floors. First, the frequency-dependent property of the structure is dealt with, whereas the time-variant stiffness and damping of the VE decoupler are replaced by a couple of stabilized substitutions, and the appropriation of this substitution is numerically validated. Secondly, given the stabilized substitutive model, the FSRS is derived based on the classic complex modal decomposition process and the stochastic dynamics of linear oscillators. In this process, the input seismic excitation is considered via a specific model that accounts for its non-stationary nature. Lastly, a parametrical analysis is conducted to validate the proposed FSRS model and to reveal some special features of the VE decoupled flooring system. It is shown that the FSRS can be better mitigated if a more flexible VE decoupler and a larger mass (isolation) ratio are incorporated. A VE decoupled floor installed at the level of the roof is more effective than the ones installed at the lower levels. Basically, the peak of FSRS can be reduced by 70% if the VE decoupling system is properly designed. [ABSTRACT FROM AUTHOR]

Many long-span bridges are located at typhoon prone regions. With the continuous increase of the bridge span, the typhoon-induced buffeting becomes more and more prominent. In this study, based on the structural health monitoring system installed in the Sutong Bridge, the recorded buffeting responses of the main girder during typhoons Damrey and Haikui were analyzed. The run test method demonstrated that the recorded acceleration responses can be regarded as zero-mean non-stationary random processes. Hence, to capture the energy distribution of the recorded data in the time-frequency domain, the evolutionary power spectral density (EPSD) estimation was conducted using efficient generalized harmonic wavelet (GHW) and filtered harmonic wavelet (FHW), respectively. Compared with the GHW, narrower wavelet bandwidth is required by the FHW to yield a compromise between the time and frequency resolution. For the FHW-based method, the power spectral density amplitudes of the averaging EPSDs are slightly larger for certain major frequency components than those obtained by the Pwelch method. Results show that the non-stationary features of the buffeting of long-span bridges during Typhoon events should be considered. This study can also provide references for non-stationary buffeting analysis of other long-span bridges during extreme wind events. [ABSTRACT FROM AUTHOR]

In recent years, the epidemic model with anomalous diffusion has gained popularity in the literature. However, when introducing anomalous diffusion into epidemic models, they frequently lack physical explanation, in contrast to the traditional reaction–diffusion epidemic models. The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable. Specifically, based on the continuous-time random walk (CTRW), starting from two stochastic processes of the waiting time and the step length, time-fractional space-fractional diffusion, time-fractional reaction–diffusion and fractional-order diffusion can all be naturally introduced into the SIR (S: susceptible, I: infectious and R: recovered) epidemic models, respectively. The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays. Distributed time delay systems can also be reduced to existing models, such as the standard SIR model, the fractional infectivity model and others, within the proper bounds. Meanwhile, as an application of the above stochastic modeling method, the physical meaning of anomalous diffusion is also considered by taking the SEIR (E: exposed) epidemic model as an example. Similar methods can be used to build other types of epidemic models, including SIVRS (V: vaccine), SIQRS (Q: quarantined) and others. Finally, this paper describes the transmission of infectious disease in space using the real data of COVID-19. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a more generalized class of convex stochastic processes and explore some of their algebraic properties. This new class of stochastic processes is called the n -polynomial s -type convex stochastic process. We demonstrate that this new class of stochastic processes leads to the discovery of novel Hermite–Hadamard type inequalities. These inequalities provide upper bounds on the integral of a convex function over an interval in terms of the moments of the stochastic process and the convexity parameter s. To compare the effectiveness of the newly discovered Hermite–Hadamard type inequalities, we also consider other commonly used integral inequalities, such as Hölder, Hölder–Ïşcan, and power-mean, as well as improved power-mean integral inequalities. We show that the Hölder–Ïşcan and improved power-mean integral inequalities provide a better approach for the n -polynomial s -type convex stochastic process than the other integral inequalities. Finally, we provide some applications of the Hermite–Hadamard type inequalities to special means of real numbers. Our findings provide a useful tool for the analysis of stochastic processes in various fields, including finance, economics, and engineering. [ABSTRACT FROM AUTHOR]

We consider a type of stochastic Newton equations in multi-dimension, with single-well potential functions, and study the limiting behaviors of their solution processes when the coefficients of the potentials diverge to infinity. The explicit formulations of the limiting processes are also given, by introducing several new stochastic processes. Especially, different from the one-dimensional case, the limiting processes are not deterministic. [ABSTRACT FROM AUTHOR]

We propose optimal checkpointing models for dual and majority decision structures with imperfect checkpoints. In systems with high reliability, the process returns to the previous checkpoint when errors occur. However, in real-time systems, if the process always returns to the previous checkpoint, its process might not end in time. In this paper, we propose extended checkpoint models in which when errors occur, the process makes forward and backward recoveries, i.e., the process makes the same one again or returns to the first one with some probability. Using dual and majority decision structures as processing modules, these models are applied to constant and random tasks for the processing of objective works. Furthermore, it is shown that processing models are given by the more general forms using a K-out-of-n system. We formulate simultaneous renewal equations for imperfect models and solve them skillfully, using mathematical methods. The mean execution times until the process succeeds are obtained, and optimal policies to minimize them are derived analytically. These results include the former ones and would be useful for some practical checkpoint models. [ABSTRACT FROM AUTHOR]

Almost every natural process is stochastic due to the basic consequences of nature's existence and the dynamical behavior of each process that is not stationary but evolves with the passage of time. These stochastic processes not only exist and appear in the fields of biological sciences but are also evident in industrial, agricultural and economical research datasets. Stochastic processes are challenging to model and to solve as well. The stochastic patterns when repeated result into random fractals and are very common in natural processes. These processes are usually simulated with the aid of smart computational and optimization tools. With the progress in the field of artificial intelligence, smart tools are developed that can model the stochastic processes by generalization and genetic optimization. Based on the basic theoretical description of the stochastic optimization algorithms, the stochastic learning tools, stochastic modeling, stochastic approximation and stochastic fractals, a comparative analysis is presented with the aid of the stochastic fractal search, multi-objective stochastic fractal search and pattern search algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate specific least action principles for laws of stochastic processes within a framework which stands on filtrations preserving variations. The associated Euler–Lagrange conditions, which we obtain, exhibit a deterministic process in the dynamics aside the canonical martingale term. In particular, taking specific action functionals, extremal processes with respect to those variations encompass specific laws of continuous semi-martingales whose drift characteristic is integrable with independent increments. Then, we relate extremal processes of classical cost functions, in particular of specific entropy functions, to a class of forward-backward systems of Mckean–Vlasov stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this work, we present the dynamics of the one dimension fractional-order Rulkov map of biological neurons. The one-dimensional neuron map shows all the dynamical behaviors observed in the real-time experiment. The integer order one-dimensional Rulkov map exhibits chaotic dynamics in the presence of time-dependent external stimuli like periodic sinusoidal force or random Gaussian process. When we construct a large complex network of neurons, the higher system dimension, as well as the external forcing, is always an obstacle. Interestingly, our study shows even with constant external stimuli, the fractional-order one-dimensional neuron shows a rich variety of complex dynamics including chaotic dynamics. We present our results based on the Lyapunov exponent of the fractional-order systems. [ABSTRACT FROM AUTHOR]

This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz König, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of Lévy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels. [ABSTRACT FROM AUTHOR]

It has been recently discovered that some random processes may satisfy limit theorems even though they exhibit intermittency, namely an unusual growth of moments. In this paper, we provide a deeper understanding of these intricate limiting phenomena. We show that intermittent processes may exhibit a multiscale behavior involving growth at different rates. To these rates correspond different scales. In addition to a dominant scale, intermittent processes may exhibit secondary scales. The probability of these scales decreases to zero as a power function of time. For the analysis, we consider large deviations of the rate of growth of the processes. Our approach is quite general and covers different possible scenarios with special focus on the so-called supOU processes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a quantum decomposition algorithm (QDA) that decomposes the problem ∂ ρ ∂ t = ℒ ρ = λ ρ into a summation of eigenvalues times phase–space variables. One interesting feature of QDA stems from its ability to simulate damped spin systems by means of pure quantum harmonic oscillators adjusted with the eigenvalues of the original eigenvalue problem. We test the proposed algorithm in the case of undriven qubit with spontaneous emission and dephasing. [ABSTRACT FROM AUTHOR]

For stochastic processes of non-commuting random variables, we formulate a Cox–Ingersoll–Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by D. Voiculescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive spectrum), we show the global existence for a free CIR equation. The main challenge lies in the transition from a stochastic differential equation driven by a classical Brownian motion to a stochastic differential equation driven by the free analogue to the classical Brownian motion, the so-called free Brownian motion. [ABSTRACT FROM AUTHOR]

This study supposes a single unit and investigates cumulative damage and catastrophic failure models for the unit, in situations where the interarrival times between the shocks, and the magnitudes of the shocks, involve two different stochastic processes. In order to consider two essentially different stochastic processes, integer gamma and Weibull distributions are treated as distributions with two parameters and extensions of exponential distributions. With respect to the cumulative damage models, under the assumption that the interarrival times between shocks follow exponential distributions, the case in which the magnitudes of the shocks follow integer gamma distributions is analyzed. With respect to the catastrophic failure models, the respective cases in which the interarrival times between shocks follow integer gamma and Weibull distributions are discussed. Finally, the study provides some characteristic values for reliability in such models. [ABSTRACT FROM AUTHOR]

New results are obtained with a modified experimental setup, in order to reduce the residual detected signal when the unmodulated beam is strongly attenuated. In the light of these results, the transfer of modulation, from the modulated beam to the unmodulated one, can be attributed to a photon–photon scattering mechanism, operating in the crossing area of the two beams. Even the stochastic character of the process becomes more plausible. [ABSTRACT FROM AUTHOR]

In this expository paper, we consider quantum Gaussian signal + noise models with time-continuous stationary colored noise and discuss the coding theorem for the classical capacity of quantum broadband gauge-covariant Gaussian channels. [ABSTRACT FROM AUTHOR]

A model of the space of random joint events is constructed. In the space, along with the existence of a symmetric difference of joint events, a new postulate about the existence of a symmetric sum of random joint events is introduced. In the generated space, a stochastic equation of the system motion and an equation for the probabilities of the system transition between two events are constructed. The probability of transition depends on the probabilities of compatibility of two, three, etc. events. The equation is equivalent to the equation of quantum theory if the events are compatible only in pairs. The proposed equation for incompatible events passes into the equation of the Markov chain. [ABSTRACT FROM AUTHOR]

A study of the distribution of the value of traded goods under the Harmonized System is presented. The ramifications of this classification system are found to exhibit an approximate power law decay, indicating complexity and self-organization in the nomenclature of traded merchandises. For almost all countries with available data, log-values of annually imported and exported goods are well described by three-parameter Weibull distributions. This distribution commonly appears in particles size distributions, suggesting a connection between random fragmentation processes and the mechanisms behind the international trade of merchandises. Analysis of the resulting values for the fitting parameters from 1995 to 2018 shows a nearly constant linear relationship between the parameters of the Weibull distributions, so that, for each country, the distribution of log-values can be approximately characterized by a single shape parameter β. The empirical findings of this paper suggest that specialization on trading a constant set of goods prevents the values of all traded merchandises from growing/decreasing simultaneously. [ABSTRACT FROM AUTHOR]

In general, a system deteriorates due to internal physical degradation and external random shocks. Previous studies mainly focused on establishing dependent competing risk models to evaluate mutual effects of degradation processes and random shocks affecting system health. However, there is a lack of consideration about the magnitude of impacts caused by random shocks on degradation processes. Thus, a shock-loading-based degradation model is proposed to classify the magnitude of impacts from random shocks on degradation processes based on the threshold of the cumulative shock loading. Copula methods are utilized to derive joint reliability function from multiple marginal distributions of degradation processes. Two numerical examples are utilized to demonstrate the reliability prediction performance of the proposed model. First, a simulated example is used. The second example employs the turbofan engine degradation data from the NASA Prognostic Data Repository to show the performance of the proposed shock-loading-based degradation process and its corresponding system reliability model. [ABSTRACT FROM AUTHOR]

We prove the existence of a pathwise weak solution to the single-phase, miscible displacement of one incompressible fluid by another in a porous medium with random forcing. Our system is described by a parabolic concentration equation driven by an additive noise coupled with an elliptic pressure equation. We use a pathwise argument combined with Schauder's fixed point theorem. [ABSTRACT FROM AUTHOR]

This study introduces a stochastic multi-period dividend discount model (DDM) that includes (i) a compound nonhomogenous Poisson process for dividend growth and (ii) the probability of firm default. We obtain maximum likelihood (ML) estimators and confidence interval formulas of our model parameters. We apply the model to a set of firms from the S&P 500 index using historical dividend and price data over a 42-year period. Interestingly, stock price estimations calculated with the model are close to the observable prices. Overall, we prove that the model can be a useful tool for stock pricing. [ABSTRACT FROM AUTHOR]

Biological, social and man-made systems often harbor various types of higher-order interaction, whose presence may substantially impact the dynamics on these systems. As a special random walk, paradigmatic trapping problem is still scarcely explored on networks encoding higher-order interactions. In this paper, applying simplicial networks produced by edge corona product as test bed, we explore how the higher-order interaction impact trapping process. Specifically, an immobile trap is first deployed at an initial node of the networks and average trapping time (ATT) is derived analytically to measure trapping efficiency of standard weight-dependent random walk. Then, delayed random walk incorporating delay phenomenon into random walk process is introduced into the networks and closed form solution of ATT is still obtained for quantifying trapping efficiency. The obtained analytical solutions of ATT in both scenarios show that ATT grows sub-linearly with network size but its leading scaling is quantitatively dependent on the parameter specifying building block of networks. Besides, analytical expression of ATT obtained in the latter scenario shows that the parameter p manipulating delayed random walk does not affect the leading scaling of ATT but can quantitatively modify its prefactor. Therefore, introduction of delay phenomenon still can modulate trapping efficiency quantitatively in the presence of higher-order interaction. This work may pave the way for modulating trapping process and related dynamical processes on general networks modeling higher-order interactions. [ABSTRACT FROM AUTHOR]

In the uncertain vibration analysis of fractionally-damped beams whose damping characteristic is described using fractional derivative model, the uncertain excitation is usually modeled as a stochastic process. However, it is often difficult to obtain sufficient samples of the excitation to establish a precise probability distribution function for the stochastic process model in practical engineering problems. Hence, in this paper, a nonrandom vibration analysis method for fractionally-damped beams is proposed to obtain the dynamic displacement response bounds of the beams under the uncertain excitation. Specifically, the uncertain excitation applied to the fractionally-damped beam is treated as a spatial-time interval field, so that the dynamic displacement response of the beam is also a space-time interval field. The middle point function and the radius function of the displacement response of the fractionally-damped beam can be derived based on the modal superposition method and the Laplace transform, through which the bound functions of the dynamic displacement response can be obtained. In addition, several numerical examples are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Lazy random walks have been used in many scientific fields such as image segmentation and optimal transport; however, related theoretical results are much less for this dynamical process. In this paper, we study lazy random walks in a pseudofractal scale-free web, where the self-loop jumps on graph vertexes are considered. For a special random walks with one trap fixed at a hub node, also known as the trapping problem, we derive the exact analytic formulas of the average trapping time (ATT), an indicator measuring the efficiency of the trapping process, by using two different methods. The results obtained by the two methods are consistent. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule with the self-loop jumps. Specifically, adding the self-loop to change the walking rule can affect the coefficient of the ATT formula, but it cannot change the leading scaling of the trapping efficiency. We hope that these results in this paper can help us better understand the biased random walk process in complex systems. [ABSTRACT FROM AUTHOR]

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays. [ABSTRACT FROM AUTHOR]

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen–Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description. [ABSTRACT FROM AUTHOR]

Uncovering the impact of special phenomena on dynamical processes in more distinct weighted network models is still needed. In this paper, we investigate the impact of delay phenomenon on random walk by introducing delayed random walk into a family of weighted m-triangulation networks. Specifically, we introduce delayed random walk into the networks. Then one and three traps are deployed, respectively, on the networks in two rounds of investigation. In both rounds of investigation, average trapping time (ATT) is applied to measure trapping efficiency and derived analytically by harnessing iteration rule of the networks. The analytical solutions of ATT obtained in both investigations show that ATT increases sub-linearity with the size of the network no matter what value the parameter p manipulating delayed random walk takes. But p can quantitatively change both its leading scaling and prefactor. So, introduction of delay phenomenon can control trapping efficiency quantitatively. Besides, parameters h and m governing networks' evolution quantitatively impact both the prefactor and leading scaling of ATT simultaneously. In summary, this work may provide incremental insight into understanding the impact of observed phenomena on special trapping process and general random walks in complex systems. [ABSTRACT FROM AUTHOR]