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In this research, the mass stopping powers, projected range, effective atomic numbers, effective electron density, radiation yield and stopping time of light and heavy charged particles such as electron, proton and alpha in the Rene alloys which consist of Rene N4, Rene N5, Rene 41, Rene 65, Rene 80, Rene 88, Rene 95 and Rene 104 were calculated for the first time. The calculations were performed by using the programs, which are called as ESTAR, SRIM and PAGEX codes. Moreover, all the results were compared with each other, and a comprehensive and comparative study of the theoretical results were presented. [ABSTRACT FROM AUTHOR]

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ( N ≥ 2 ) with stopping times as strategies (or pure strategies). We prove existence of an ε -Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples in the specific case of finite horizon. [ABSTRACT FROM AUTHOR]

The determination of stopping power and the range of protons present in CH3OH, CH2O, and CO2 was accomplished within the energy range of 0.01–1,000 MeV by employing the Bethe equation and Ziegler equation. The latter two equations utilize quantum mechanics to derive an approximation of the stopping force formula, which applies to highly charged particles. The software MATLAB was utilized in the calculation of the results. A suitable equation for computing the halting power of protons in the range of 0.01–1,000 MeV has been identified. A correlation coefficient of 0.999 was determined, suggesting a high degree of concordance between the derived results and those of the SRIM 2013 and P-STAR programs. Proton stopping power analyses of this nature will aid the scientific community in selecting the optimal formulation of stopping power.

A functional calculus for an order complete vector lattice E was developed by Grobler (Indag Math (NS) 25(2):275–295, 2014) using the Daniell integral. We show that if one represents the universal completion of E as C ∞ (K) , where K is an extremally disconnected compact Hausdorff topological space, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in C ∞ (K) . This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in C ∞ (K) . We obtain a representation that is analogous to what is expected in probability theory. [ABSTRACT FROM AUTHOR]

Stopping power, range, and time of proton in water, lung, bladder, and intestinal human tissues are calculated using density functional theory and Beth’s relativistic equation in range of proton energy (0.01–1,000 MeV). The experimental data extracted from SRIM-2013 program were used to proton to the same human tissues applied in the MATLAB-2021 program, and the mean ionization potential of water and the studied tissues is calculated using Gaussian 09W program. A good agreement has been found between our calculations for stopping power, range, and time of protons in the studied human body tissues and SRIM-2013 program calculations.

In this study, the energy loss, range, and stopping time were calculated using the Bethe theory for protons in germanium and copper materials over an energy range of 0.01–1,000 MeV. The mathematical equations were programmed and implemented using MATLAB2013 to obtain the results for the relative mass stopping power and range, illustrated in tables and graphs. The computed results from the Bethe and Ziegler equations were compared with the experimental results from Pstar. The results showed a difference between the experimental and theoretical results at the beginning of the range used in the Bethe equation because it is a quantized equation that is not suitable for low energies. Hence, a correction term was proposed for the Bethe equation to correct the results at low energies, which produced negative values. On the other hand, the Ziegler equation showed good agreement along the range used as it is a semi-empirical equation. The range and stopping time of the incident particle increased with increasing incident particle energy in the mentioned materials.

The study aimed to clarify the legal regulation of the volunteer employment contract in the Kingdom of Saudi Arabia, through the Saudi volunteer work law and its executive regulations. To achieve this goal, the research was divided into an introduction and three sections: The introduction dealt with the concept of volunteer work, while the first section was devoted to explaining the concept of the volunteer work contract. The second section dealt with the effects of the volunteer work contract, and finally, the third section dealt with the termination of the volunteer work contract. The researcher used the descriptive and analytical approach as well as the comparative approach. The study concluded with many results including that the essence of volunteer work is to achieve public benefit without compensation, and that if the work or effort is in exchange for a financial reward, then it does not have the status of volunteer work. The results also showed that the expiration of a volunteer work contract is subject to the general causes of expiration and dissolution of contracts in general. In addition to being subject to special reasons related to it due to its characteristics. As for the recommendations, the study recommended several recommendations, the most prominent of which is: that volunteer work in the Saudi system be within the framework of a written contract between the volunteer or volunteer team and the beneficiary. [ABSTRACT FROM AUTHOR]

Modern computing and communication technologies can make data collection procedures very efficient, while our ability to analyze and extract information from large data sets is hard-pressed to keep up with our capacity for data collection. If we are interested in learning a classification/prediction rule that was not considered in the original data collection procedure, then there is usually a lack of label information in the original data. Thus, how to start with minimum number of labeled data and aggressively selected most informative samples for being labeled becomes an important issue. The thoughts of active learning, with subjects to be selected sequentially without using label information, is a possible outlet for this situation. In addition, if we can identify variables, from the lengthy variable list of the data set, for constructing an effective classification rule with good interpretation ability will be better. Here, we propose an active learning procedure targeted the maximum area under the receiver operating characteristic curve via a commonly use logistic model, which can sequentially select the effective informative subjects and variables, simultaneously. We discuss the asymptotic properties of the proposed procedure and illustrate it with some synthesized and real data sets. [ABSTRACT FROM AUTHOR]

We give a survey of the results related to extensions of the Kolmogorov inequality for the distribution of the absolute value of the maximum of the sum of centered independent random variables to the case of martingales considered at random stopping times. [ABSTRACT FROM AUTHOR]

We study the 3D compressible Navier–Stokes equations coupled with the Q-tensor equation perturbed by a multiplicative stochastic force, which describes the motion of nematic liquid crystal flows. The local existence and uniqueness of strong pathwise solution up to a positive stopping time is established where "strong" is in both PDE and probability sense. The proof relies on the Galerkin approximation scheme, stochastic compactness, identification of the limit, uniqueness and a cutting-off argument. In the stochastic setting, we develop an extra level approximation to overcome the difficulty arising from the random effect while constructing the approximate solution. Due to the complex structure of the coupled system, the estimate of the high-order items are also the challenging part in the article. [ABSTRACT FROM AUTHOR]

All the available modes of travel and their respective travel parameters must be known to the commuters before their trip. Otherwise they may either spend more money or more time for the trip. In addition to this, recent pandemic, rapidly spreading novel corona virus is demanding a smart solution for contactless commuting. This paper suggests a practical solution to make both the above possible and it emphasizes the applicability of two developed android applications, one for travel data collection and another to predict travel time for a multimodal trip within the study area. If the whole trip is by a single mode, the user can get the corresponding travel time estimate from “Google maps”. But, if the trip is by multiple modes, it is not possible to get the total travel time estimate for the whole trip at a time from “Google maps”. A separate travel mode for “auto” is unavailable in “Google maps” alongside drive, two-wheeler, train or bus and walk alternatives. It is also observed that the travel time estimate of “Google maps” for the city buses is inaccurate. Hence, the two modes (Buses and Autos) were chosen for the study. Unless and until the travel times and stopping times of the two modes are known, it is not possible to predict their trip times. Hence, the mobility analysis was performed for the two modes in the study area to find their respective average travel rate at peak hours, across 15 corridors and the results were presented.

In this paper, the stochastic finite‐time stability (SFTS) criteria for switched stochastic nonlinear systems (SSNS) with nonlocal Lipschitz coefficients are established. The main tools that we used here are mode‐dependent limiting average dwell time (MD‐LADT), stopping time technology and the stochastic analysis theory. The established criteria show that the diffusion coefficient, the drift coefficient and the switching law play crucial roles in obtaining the SFTS for the SSNS. Besides, our criteria provide some approaches on how to design the switching law to achieve the SFTS if a switched stochastic system contains SFTS subsystems and stochastic finite‐time instable (SFTI) subsystems simultaneously. Finally, an example is given to illustrate the importance and usefulness of the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the statistical concept of causality in continuous time within filtered probability spaces based on Granger's definition of causality. The given causality concept is connected to the optional and predictable processes important in stochastic integration. More precisely, we establish that the preservation of predictability with respect to larger filtrations is implied by the considered notion of (self-)causality. We also consider the connections between the given causality concept and the optional and predictable projections of a stochastic process, which play an important role in the general theory of stochastic processes, semimartingale theory, and stochastic calculus. Some results show that the (self-)causality implies indistinguishability of the optional (or predictable) projections with respect to considered filtrations from those with respect to larger filtrations. [ABSTRACT FROM AUTHOR]

What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10 − 4 . This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes. [ABSTRACT FROM AUTHOR]

Sequential change point tests aim at giving an alarm as soon as possible after a structural break has occurred while controlling the asymptotic false alarm rate. For such tests it is of particular importance to understand how quickly a break is detected. In this paper, we derive the asymptotic distribution of the delay times for sequential change point procedures based on U-statistics. This includes the difference-of-means (DOM) sequential test that has been discussed previously but also a new robust Wilcoxon sequential change point test. Based on these results we can compare the detection delay of both procedures instead of having to rely on simulations only. Indeed, the Wilcoxon sequential procedure has a smaller detection delay for heavier tailed distributions which is also confirmed by simulations. While in the previous literature for the DOM test only results for early change points were obtained, we derive the asymptotic distribution of the delay times for both early as well as late change points. Finally, we evaluate how well the asymptotic distribution approximates the actual stopping times for finite samples via a simulation study. • Asymptotic distribution of delay times for sequential tests of U -statistics. • This includes a robust Wilcoxon sequential procedure. • This includes late (linear and superlinear) changes. • For early changes the approximation improves upon existing approximations. [ABSTRACT FROM AUTHOR]

This paper studies the issue of coordinating equipment maintenance operations with capital investment strategy in the presence of random equipment failures. The traditional approach, developed by Kamien and Schwartz (KS) in their celebrated paper published in 1971, is to formulate the problem as a deterministic optimal control problem with the probability of machine failure as the state variable. With this approach, the optimal policy is deterministic. As a major departure from the KS approach, we explicitly model the underlying stochastic process of machine failures. Our analysis of the stochastic dynamic programming model offers new insights into the problem. Under a long planning horizon with a limited replacement opportunity, each individual machine serves as a revenue generator and contributes a significant amount to the profit of the system. In contrast, when the replacement budget is quite generous over a relatively short planning horizon, adding one extra machine only helps as a backup for unexpected failures of the machines purchased before it. An interesting result derived from this comparison is that a deterministic policy turns out to be optimal for the former, while a state-contingent policy must be applied to the latter. In other words, the deterministic KS approach does not work in general when a chain of machine replacement is considered. We further characterize the effects of the discount rate, productivity deterioration, learning, decision delay, and technology advancement on the optimal policy. [ABSTRACT FROM AUTHOR]

In the current paper, the existence of fixed point is surveyed for a function of standard Brownian motion. A sequence of stopping times is given which converges to the fixed point. Necessary conditions are given and limiting distribution of stopping times is proposed. Newton Raphson recursive results are given to verify theoretical results. Finally, a conclusion is given. [ABSTRACT FROM AUTHOR]

This paper is concerned with financial application of Monte Carlo simulation of infinite series. In fact, present value of a cash flow is represented as infinite series. Hence, the Monte Carlo may be applied to approximate the present value. It is seen that the Monte Carlo method is sample approximation of the mean of cash flow computed at a stopping time. The stopping time, first, is considered as an up-crossing time point of sequence of uniform random variables and then it is replaced by up-crossing of macro-economic variables (called as factor) which brings the dependency structure to the problem which is modeled by auto-regressive to any things (ARTA) models. Here, two different Monte Carlo simulation methods are presented. A financial application is also given. Continuous time extensions of results are also proposed. Finally, a concluding remark section is given. [ABSTRACT FROM AUTHOR]

In the present paper we address the real-time detection problem of a change-point in the coefficients of a linear model with the possibility that the model errors are asymmetrical and that the explanatory variables number is large. We build test statistics based on the cumulative sum (CUSUM) of the expectile function derivatives calculated on the residuals obtained by the expectile and adaptive LASSO expectile estimation methods. The asymptotic distribution of these statistics are obtained under the hypothesis that the model does not change. Moreover, we prove that they diverge when the model changes at an unknown observation. The asymptotic study of the test statistics under these two hypotheses allows us to find the asymptotic critical region and the stopping time, that is the observation where the model will change. The empirical performance is investigated by a comparative simulation study with other statistics of CUSUM type. Two examples on real data are also presented to demonstrate its interest in practice. [ABSTRACT FROM AUTHOR]

All the available modes of travel and their respective travel parameters must be known to the commuters before their trip. Otherwise they may either spend more money or more time for the trip. In addition to this, recent pandemic, rapidly spreading novel corona virus is demanding a smart solution for contactless commuting. This paper suggests a practical solution to make both the above possible and it emphasizes the applicability of two developed android applications, one for travel data collection and another to predict travel time for a multimodal trip within the study area. If the whole trip is by a single mode, the user can get the corresponding travel time estimate from "Google maps". But, if the trip is by multiple modes, it is not possible to get the total travel time estimate for the whole trip at a time from "Google maps". A separate travel mode for "auto" is unavailable in "Google maps" alongside drive, two-wheeler, train or bus and walk alternatives. It is also observed that the travel time estimate of "Google maps" for the city buses is inaccurate. Hence, the two modes (Buses and Autos) were chosen for the study. Unless and until the travel times and stopping times of the two modes are known, it is not possible to predict their trip times. Hence, the mobility analysis was performed for the two modes in the study area to find their respective average travel rate at peak hours, across 15 corridors and the results were presented. [ABSTRACT FROM AUTHOR]

We study a class of non-linear Backward stochastic differential equations (BSDE) with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition ξ is allowed to take the value +∞, i.e., singular. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value ∞ at terminal time S. Our goal is to show existence of solutions to the BSDE for a range of singular terminal values under the assumption that S is solvable. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., it is continuous at time S and attains the terminal value with probability 1. We consider three types of terminal values: 1) Markovian: i.e., ξ is of the form ξ = g(ΞS) where Ξ is a continuous Markovian diffusion process, S is a hitting time of Ξ and g is a deterministic function 2) terminal conditions of the form ξ1 = ∞· 1(τ≥) and 3) ξ2 = ∞· 1(τ≥) where τ is another stopping time. For general ξ we prove that minimal supersolution has a limit at time S provided that F is left continuous at time S. Finally, we discuss the implications of our results about Markovian terminal conditions to the solution of non-linear elliptic PDE with singular boundary conditions. [ABSTRACT FROM AUTHOR]

In this study, we consider an optimal control problem driven by a stochastic differential system with a stopping time terminal cost functional. To solve the optimal control problem under the stopping time terminal cost functional, we introduce a multi-time state optimal control systems, and prove that the multi-time state optimal control systems is a near-optimal control problem. We use the near-optimal control problem to take place the original optimal control problem under the cost functional with a stopping time terminal. Then, we establish the stochastic maximum principle for this kind of optimal control problem by introducing a discrete terminal system. Finally, we provide an example to describe the main results of this study. [ABSTRACT FROM AUTHOR]

Cognitive radio (CR) is a revolutionary paradigm to migrate the spectrum scarcity problem in wireless networks. In view of the spectrum scarcity of unmanned aerial vehicles (UAVs) communication system, cooperative spectrum sensing (CSS) has emerged as a key function of CR to identify the available spectrum for UAV sensing nodes. However, the flexible locations of flying UAVs make CSS a difficult task, which in turn makes it difficult for CSS performance to meet UAV requirements. In this paper, we design an intra-frame cooperation way to achieve CSS for cognitive UAV networks (CUAVNs), which provides the same CSS performance as the traditional inter-frame cooperation and higher throughput for UAVs. Furthermore, considering certain performance requirements in CUAVNs, we propose sequential probability ratio test (SPRT)-based CSS method, and further make an in-depth analysis of relationship among performance indices of CSS and approximate the actual detection performance and the stopping time by ignoring the excess over the boundaries and the random walk. Finally, numerical results corroborate the effectiveness and correctness of our theoretical analysis, while comparison results also represent the superiority of our proposal. [ABSTRACT FROM AUTHOR]

We consider the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution. This problem is investigated in a sequential setup under mild assumptions on the underlying random process. The optimal method minimizes the expected number of samples while ensuring that the average detection/estimation errors do not exceed a certain level. After converting the constrained problem to an unconstrained one, we characterize the general solution by a nonlinear Bellman equation, which is parameterized by a set of cost coefficients. A strong connection between the derivatives of the cost function with respect to the coefficients and the detection/estimation errors of the sequential procedure is derived. Based on this fundamental property, we further show that for suitably chosen cost coefficients the solutions of the constrained and the unconstrained problem coincide. We present two approaches to finding the optimal coefficients. For the first approach, the final optimization problem is converted into a linear program, whereas the second approach solves it with a projected gradient ascent. To illustrate the theoretical results, we consider two problems for which the optimal schemes are designed numerically. Using Monte Carlo simulations, it is validated that the numerical results agree with the theory. [ABSTRACT FROM AUTHOR]

We analyse the reliability (survival function) of a duplex system characterized by hot standby and sustained by an auxiliary unit in cold standby. The entire system is attended by two heterogeneous repairmen. Our methodology is based on the theory of sectionally holomorphic functions combined with the notion of dual transforms. Finally, we also study the total occupational time of the repairman responsible for the repair of the failed priority unit during the survival time of the system.

In view of the spectrum scarcity of unmanned aerial vehicles' (UAVs) communication systems, cooperative spectrum sensing (CSS) is used to identify the available spectrum for cognitive UAV networks (CUAVNs). Due to the flexible locations of flying UAVs, we first design an intraframe cooperation way to replace the traditional interframe cooperation to achieve CSS in this article. Considering certain performance requirements in CUAVNs, we perform an in-depth analysis of the relationship among performance indices of sequential probability ratio test (SPRT)-based CSS and approximate the actual detection performance and the stopping time by ignoring the excess over the boundaries and the random walk. Finally, numerical results corroborate the effectiveness and correctness of our theoretical analysis, and comparison results represent the superiority of our proposal. [ABSTRACT FROM AUTHOR]

Bias approximation has played an important rôle in statistical inference, and numerous bias calculation techniques have been proposed under different contexts. We provide a unified approach to approximating the bias of the maximum likelihood estimator and the l2 penalized likelihood estimator for both linear and nonlinear models, where the design variables are allowed to be random and the sample size can be a stopping time. The proposed method is based on the Woodroofe–Stein identity and is justified by very weak approximations. The accuracy of the derived bias formulas is assessed by simulation for several examples. The bias of the ridge estimator in high‐dimensional settings is also discussed. [ABSTRACT FROM AUTHOR]

The paper considers the problem of optimal control in models with disorder and provides the computational scheme of its solution. The proposed method is based on replacing the disorder with its estimate by the stopping time. The calculations take place on the binary tree with nodes divided into two classes. The division of tree nodes into two classes is carried out using a decision rule associated either with testing two hypothesis, or with the optimal stopping of a Markov process developing on a binary tree. The estimate of the disorder is the moment at which the walk on the tree is placed in a node of the first class for the first time. The example of the optimal control problem is related to the problem of quantile hedging of a payoff. [ABSTRACT FROM AUTHOR]

A specific function f(r) involving a ratio of complicated gamma functions depending upon a real variable r(> 0) is handled. Details are explained regarding how this function f(r) appeared naturally for our investigation with regard to its behavior when r belongs to R+. We determine explicitly where this function attains its unique minimum. In doing so, quite unexpectedly the customary Cramér-Rao inequality comes into play in order to nail down a valid proof of the required lower bound for f(r) and locating where is that lower bound exactly attained. [ABSTRACT FROM AUTHOR]

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Abstract Background The Elliot wave principle commonly characterizes the impulsive and corrective wave trends for both financial market trends and electrocardiograms. The impulsive wave trends of electrocardiograms can annotate several wave components of heart-beats including pathological heartbeat waveforms. The stopping time inquires which ordinal element satisfies the assumed mathematical condition within a numerical set. The proposed work constitutes several algorithmic states in reinforcement learning from the stopping time decision, which determines the impulsive wave trends. Each proposed algorithmic state is applicable to any relevant algorithmic state in reinforcement learning with fully numerical explanations. Because commercial electrocardiographs still misinterpret myocardial infarctions from extraordinary electrocardiograms, a novel algorithm needs to be developed to evaluate myocardial infarctions. Moreover, differential diagnosis for right ventricle infarction is required to contraindicate a medication such as nitroglycerin. Methods The proposed work implements the stopping time theory to impulsive wave trend distribution. The searching process of the stopping time theory is equivalent to the actions toward algorithmic states in reinforcement learning. The state value from each algorithmic state represents the numerically deterministic annotated results from the impulsive wave trend distribution. The shape of the impulsive waveform is evaluated from the interoperable algorithmic states via least-first-power approximation and approximate entropy. The annotated electrocardiograms from the impulsive wave trend distribution utilize a structure of neural networks to approximate the isoelectric baseline amplitude value of the electrocardiograms, and detect the conditions of myocardial infarction. The annotated results from the impulsive wave trend distribution consist of another reinforcement learning environment for the evaluation of impulsive waveform direction. Results The accuracy to discern myocardial infarction was found to be 99.2754% for the data from the comma-separated value format files, and 99.3579% for those containing representative beats. The clinical dataset included 276 electrocardiograms from the comma-separated value files and 623 representative beats. Conclusions Our study aims to support clinical interpretation on 12-channel electrocardiograms. The proposed work is suitable for a differential diagnosis under infarction in the right ventricle to avoid contraindicated medication during emergency. An impulsive waveform that is affected by myocardial infarction or the electrical direction of electrocardiography is represented as an inverse waveform.

A single jump filtration ${({\mathcal{F}_{t}})_{t\in {\mathbb{R}_{+}}}}$ generated by a random variable γ with values in ${\overline{\mathbb{R}}_{+}}$ on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$ is defined as follows: a set $A\in \mathcal{F}$ belongs to ${\mathcal{F}_{t}}$ if $A\cap \{\gamma >t\}$ is either ∅ or $\{\gamma >t\}$. A process M is proved to be a local martingale with respect to this filtration if and only if it has a representation ${M_{t}}=F(t){\mathbb{1}_{\{t0\}$. This result seems to be new even in a special case that has been studied in the literature, namely, where $\mathcal{F}$ is the smallest σ-field with respect to which γ is measurable (and then the filtration is the smallest one with respect to which γ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.

In many aero gliding vehicles, achieving the maximum gliding range is a challenging task. A frequent example is the breakdown of an engine during flight or the use of unpowered stand-off weapons. When an unpowered stand-off weapon begins gliding at a given height, it eventually strikes the ground after some distance, and height is considered a stopping constraint in this general condition. To avoid the time-scaling approach for the free time optimal problem, the maximum stoppable time with a stopping constraint is addressed to attain the maximum glide range. This problem can be chosen as an optimal gliding range problem which can be solved by direct or indirect methods. In this paper, the inverted Y-tail joint stand-off weapon is selected as the subsonic unpowered gliding vehicle (SUGV). After being released from dispersion points, the SUGV has to face fluctuating gliding flight because of flight phase transition that causes gliding range reduction. To achieve a damped and steady gliding flight while maximizing the gliding range, we propose a non-uniform control vector parameterization (CVP) approach that uses the notion of exponential spacing for the time vector. When compared with the maximum step input and conventional uniform CVP approach, simulations of the proposed non-uniform CVP approach demonstrate that the SUGV exhibits superior damping and steady gliding flight, with a maximum gliding range of 121.278 km and a maximum horizontal range of 120.856 km.

We investigate the asymptotic properties of the stopping times based on the monitoring tests for a bubble, that is, ADF‐type and CUSUM‐type tests. We derive the stochastic orders of these stopping times and find that they depend on the relative position of the emerging date of a bubble. We show that when a bubble emerges early in the monitoring period, the diverging rate of the stopping time based on the CUSUM‐type detector is slower than that based on the ADF‐type detector, which implies that the former tends to find a bubble earlier than the latter, whereas this advantage disappears for the late emerging bubble. We also derive the limiting distribution of the stopping time based on the CUSUM detector. The finite sample simulations support the theoretical result. [ABSTRACT FROM AUTHOR]

The paper proposes a method for constructing models based on the analysis of birth and death processes with linear growth in semimartingale terms. Based on this method, stochastic models of simple just-in-time systems (analyzed in the theory of productive systems) and windows of vulnerability (widely discussed in risk theory) are considered. The main results obtained in the work are presented in terms of the average values of the time during which the processes reach zero values. At the same time, they are considered and used in the study of assessment models for local times of the processes. Here, simple Markov processes with a linear growth of intensities (perhaps, depending on time) are analyzed. At the same time, the obtained and used estimates are of theoretical interest. Thus, for example, the average value of the stopping time, at which the process reaches zero, depends on functions such as the harmonic number and the remainder term for the logarithmic function in the Taylor theorem. As the main result, the method of mathematical modeling of just-in-time systems and windows of vulnerability is proposed. The semimartingale description method used here should be considered as the first step of such a modeling, since, being a trajectory method, it allows diffusion (including non-Markov processes) generalizations when constructing stochastic models of windows of vulnerability and just-in-time. In the theoretical part of the article, we formulate statements for the average values of the local time and the stopping times when the birth and death processes reach a given value. This allows us to uniformly present estimates for the models of the just-in-time system and for windows of vulnerability, the result for which is given in the form of a limit theorem. The main results are formulated as theorems and lemmas. The proofs use semimartingale methods.

Braking distances play an important role in the organization of rail traffic and exploitation of rail vehicles. The braking distances of rail vehicles affect several factors that cannot be defined and determined precisely. In this paper, a mathematical model, for train moving through braking, is presented by taking into account the relevant train braking resistance and a procedure is given for solving a differential equation of movement of the rail vehicle during braking. The procedure allows determining influential factors that have an effect on the length of braking distances. The influence and the changes of specific resistance during the movement of the train, braking force, and adhesion between wheel and rail during the braking process are presented. The results obtained with the presented mathematical model are accurate and match with the results obtained experimentally. [ABSTRACT FROM AUTHOR]