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A deep learning-based Visual Inertial SLAM technique is proposed in this paper to ensure accurate autonomous localization of mobile robots in environments with dynamic objects. Addressing the limitations of real-time performance in deep learning algorithms and the poor robustness of pure visual geometry algorithms, this paper presents a deep learning-based Visual Inertial SLAM technique. Firstly, a non-blocking model is designed to extract semantic information from images. Then, a motion probability hierarchy model is proposed to obtain prior motion probabilities of feature points. For image frames without semantic information, a motion probability propagation model is designed to determine the prior motion probabilities of feature points. Furthermore, considering that the output of inertial measurements is unaffected by dynamic objects, this paper integrates inertial measurement information to improve the estimation accuracy of feature point motion probabilities. An adaptive threshold-based motion probability estimation method is proposed, and finally, the positioning accuracy is enhanced by eliminating feature points with excessively high motion probabilities. Experimental results demonstrate that the proposed algorithm achieves accurate localization in dynamic environments while maintaining real-time performance. [ABSTRACT FROM AUTHOR]

This paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing random variations in the economic and financial environments. We prove a sufficient condition on the distribution of jumps of the business process ensuring the smoothness of the ruin probability as a function of the initial capital and obtain for this function an integro-differential equation. [ABSTRACT FROM AUTHOR]

The reconfiguration of wireless channels with reconfigurable reflecting surface (RIS) technology offers new design options for future wireless networks. Due to its high altitude and increased probability of establishing line-of-sight linkages with ground source/destination nodes, aerial RIS (ARIS) has greater deployment flexibility than traditional terrestrial RIS. It also provides a wider-view signal reflection. To leverage the advantages of ARIS-enabled systems, this paper defines air-to-ground linkages via Nakagami-m small-scale fading and inverse-Gamma large-scale shadowing, considering realistic composite fading channels. To construct a tight approximate closed-form formula for the outage probability (OP), a new mathematical framework is proposed. Additionally, a deep-learning-based system called the BiLSTM model is deployed to evaluate OP performance in the 3D spatial movement of the ARIS system. In the offline phase, the proposed model is trained with real-value channel state estimation sets and enhances OP performance in the online phase by learning channel information in a bidirectional manner. Simulation results demonstrate that the proposed BiLSTM model outperforms all other models in analyzing OP for the ARIS system. [ABSTRACT FROM AUTHOR]

Dynamic microsimulations are tools to stochastically project (synthetic) microdata into the future. In spatial microsimulations, regional discrepancies are of particular interest and must be considered accordingly. In practice, the probabilities for state changes are unknown and must be estimated, usually from survey data. However, estimating such models on the regional level is often not feasible due to limited sample size and lack of geographic information. Simply applying the model estimated at the national level to all geographies leads to biased state transitions due to regional differences in level and distribution. In this paper, we introduce a model-based alignment method to adapt predicted probabilities obtained from a nationally estimated model to subregions by integrating known marginal distributions to re-introduce regional heterogeneity and create more realistic trajectories, particularly in small areas. We show that the model-adjusted transition probabilities can capture region-specific patterns and lead to improved projections. Our findings are useful to researchers who want to harmonise model outputs with external information, in particular for the field of microsimulation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Green measure for a class of non-Gaussian processes in R d . These measures are associated with the family of generalized grey Brownian motions B β , α , 0 < β ≤ 1 , 0 < α ≤ 2 . This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability 1 for d α > 2 and 1 < α ≤ 2 . The Green measure then generalizes those measures of all these classes. [ABSTRACT FROM AUTHOR]

In recent years, there has been increasing interest in the joint modelling of compound extreme events such as high temperatures and low rainfall. The increase in the frequency of occurrence of these events in many regions has necessitated the development of models for estimating the concurrent probabilities of such compound extreme events. The current study discusses an application of copula models in predicting the concurrent probabilities of compound low rainfall and high-temperature events using data from the Lowveld region of the Limpopo province in South Africa. The second stage discussed two indicators for monitoring compound high temperature and low rainfall events. Empirical results from the study show that elevations ranging from 100–350 m, 350–700 m and 700–1200 m exhibit varying probabilities of experiencing drought, with mild droughts having approximately 64%, 66%, and 65% chances, moderate droughts around 36%, 39%, and 38%, and severe droughts at approximately 16%, 19%, and 18%, respectively. Furthermore, the logistic regression models incorporating the southern oscillation index as a covariate yielded comparable results of copula-based models. The methodology discussed in this paper is robust and can be applied to similar datasets in any regional setting globally. These findings could be useful to disaster management decision makers, helping them formulate effective mitigation strategies and emergency response plans. [ABSTRACT FROM AUTHOR]

In this paper, we present an unprecedented method based on Kohonen networks that is able to automatically recognize partial discharge (PD) classes from phase-resolved partial discharge (PRPD) diagrams with features of various simultaneous PD patterns. The PRPD diagrams were obtained from the stator windings of a real-world hydro-generator rotating machine. The proposed approach integrates classification probabilities into the Kohonen method, producing self-organizing probability maps (SOPMs). For building SOPMs, a group of PRPD diagrams, each containing a single PD pattern for training the Kohonen networks and single- and multiple-class-featured samples for obtaining final SOPMs, is used to calculate the probabilities of each Kohonen neuron to be associated with the various PD classes considered. At the end of this process, a self-organizing probability map is produced. Probabilities are calculated using distances, obtained in the space of features, between neurons and samples. The so-produced SOPM enables the effective classification of PRPD samples and provides the probability that a given PD sample is associated with a PD class. In this work, amplitude histograms are the features extracted from PRPDs maps. Our results demonstrate an average classification accuracy rate of approximately 90% for test samples. [ABSTRACT FROM AUTHOR]

The selection of multiple contingency scenarios is a key task to perform resilience-oriented long-term planning analyses. However, the identification of relevant multiple contingencies may easily lead to combinatorial explosion issues, even for relatively small systems. This paper proposes an effective methodology for the identification of relevant multiple contingencies and their probabilities, suitable for the long-term resilience analysis of large power systems. The methodology is composed of two main pillars: (1) the clustering of lines that are more likely to fail together, to reduce the computational complexity of the analysis exploiting historical weather data and (2) the probability-based identification of multiple contingencies within each cluster, where the contingency probability is computed applying the copula theory. Tests performed on a portion of the Italian EHV transmission system confirm the validity of the clustering results compared against historical failure events. Moreover, the copula-based algorithm for contingency probability estimation passes the tests carried out on relatively large clusters with very low error tolerance. The method successfully pinpoints critical multiple contingency scenarios and their likelihoods, making it valuable for assessing power system resilience over long-term horizons in support of resilience-oriented planning activities. [ABSTRACT FROM AUTHOR]

As with probability theory, uncertainty theory has been developed, in recent years, to portray indeterminacy phenomena in various application scenarios. We are concerned, in this paper, with the convergence property of state trajectories to equilibrium states (or fixed points) of time delayed uncertain cellular neural networks driven by the Liu process. By applying the classical Banach's fixed-point theorem, we prove, under certain conditions, that the delayed uncertain cellular neural networks, concerned in this paper, have unique equilibrium states (or fixed points). By carefully designing a certain Lyapunov–Krasovskii functional, we provide a convergence criterion, for state trajectories of our concerned uncertain cellular neural networks, based on our developed Lyapunov–Krasovskii functional. We demonstrate under our proposed convergence criterion that the existing equilibrium states (or fixed points) are exponentially stable almost surely, or equivalently that state trajectories converge exponentially to equilibrium states (or fixed points) almost surely. We also provide an example to illustrate graphically and numerically that our theoretical results are all valid. There seem to be rare results concerning the stability of equilibrium states (or fixed points) of neural networks driven by uncertain processes, and our study in this paper would provide some new research clues in this direction. The conservatism of the main criterion obtained in this paper is reduced by introducing quite general positive definite matrices in our designed Lyapunov–Krasovskii functional. [ABSTRACT FROM AUTHOR]

Translation of the seminal 1877 paper by Ludwig Boltzmann which for the first time established the probabilistic basis of entropy. Includes a scientific commentary. [ABSTRACT FROM AUTHOR]

This paper concentrates on the finite-time  H ∞  control problem for a type of stochastic discrete-time Markovian jump systems, characterized by time-delay and partly unknown transition probabilities. Initially, a stochastic finite-time (SFT)  H ∞  state feedback controller and an SFT  H ∞  observer-based state feedback controller are constructed to realize the closed-loop control of systems. Then, based on the Lyapunov–Krasovskii functional (LKF) method, some sufficient conditions are established to guarantee that closed-loop systems (CLSs) satisfy SFT boundedness and SFT  H ∞  boundedness. Furthermore, the controller gains are obtained with the use of the linear matrix inequality (LMI) approach. In the end, numerical examples reveal the reasonableness and effectiveness of the proposed designing schemes. [ABSTRACT FROM AUTHOR]

This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and nonparametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum". [ABSTRACT FROM AUTHOR]

After a transmission line is covered by ice in winter, ice-shedding and vibration occurs under special meteorological and external dynamic conditions, which leads to intense transmission line shaking. Transmission line ice-shedding and vibration often cause line flashover trips and outages. In January 2018, three 500 kV transmission lines, namely, the 500 kV Guanli line, the 500 kV Dushan line, and the 500 kV Guanqiao line, tripped and cut off due to ice-shedding and vibration in Anhui province, seriously threatening the safe operation of a large power grid. Current studies mainly focus on analyzing the influence factors and characteristics of line ice-shedding and investigating suppression measures, but they only analyze the correlation between each influencing factor and icing or shedding, and do not consider the coupling effects between multiple factors. In this paper, the key influencing factors and the probability distribution of transmission line ice-shedding were analyzed, and a multiple-factor coupling fault probability calculation model of line ice-shedding based on Copula function was proposed. The fault probability was calculated directly by considering multiple influence factors at the same time, which effectively overcame the error caused by multi-factor transformation in fuzzy membership degree and other methods. It provided an important decision-making basis for preventing and controlling transmission line ice-shedding faults. [ABSTRACT FROM AUTHOR]

The aging process of transmission lines has a direct impact on the reliability and safety of the power grid. Therefore, an accurate age estimation method is imperative for effective maintenance planning and infrastructure investment. This paper introduces a systematic methodology for estimating the age of overhead transmission lines, utilizing the percentage statistical health index (%SHI) and the failure probability curve-fitting (FPCF) method. The %SHI, employing a scoring and weighting approach derived from test results and inspections, is used to assess the actual condition of transmission line equipment. Additionally, the FPCF approach is applied to illustrate the connection between the SHI and the likelihood of failure, facilitating the assessment of transmission line age by fitting failure probability curves to the SHI data. This age is directly associated with the probability of experiencing a failure. The evaluation was conducted on 924 towers situated along four transmission lines connecting the 115 kV substations S1–S2, S3–S4, S5–S6, and S7–S8. These transmission lines are in four regions with diverse terrain and environments such as mountains, rice fields, and more. In the SHI calculation, practical testing results and historical failure data were applied. The results clearly indicate that there were notable disparities in the age estimations for transmission lines in diverse geographical regions of Thailand when compared to their actual ages. These discrepancies can be attributed to various factors, including the local environment, such as rainfall, flooding, and salt-laden air as well as specific geographical features like mountainous and coastal terrain. To mitigate the deterioration of transmission lines in all regions, it is essential to implement a proactive maintenance strategy. This strategy should involve more frequent inspections, the use of advanced monitoring technologies, and the establishment of robust maintenance procedures, with which it would become possible to enhance the accuracy of equipment condition assessments, ultimately resulting in an overall improvement in the reliability of transmission lines. [ABSTRACT FROM AUTHOR]

The paper is devoted to the formal description of the running time of the user task on some virtual nodes in the computing network. Based on the probability theory framework, this time represents a random value with a finite mean and variance. For any class of user task, these moments are the functions of the node resources, task numerical characteristics, and the parameters of the current node state. These functions of the vector arguments can be treated as some surfaces in the multidimensional Euclidean spaces, so the proposed models are called the stochastic time complexity surfaces. The paper also presents a class of functions suitable for the description of both the mean and variance. They contain unknown parameters which should be estimated. The article includes the statement of the parameter identification problem given the statistical results of the node stress testing, recommendations concerning the test planning, and preprocessing of the raw experiment data. To illustrate the performance of the proposed model, the authors design it for an actual database application—the prototype of the passengers' personal data anonymization system. Its application functions are classified into two user task classes: the data anonymization procedures and fulfillment of the statistical queries. The authors identify the stochastic time complexity surfaces for both task types. The additional testing experiments confirm the high performance of the suggested model and its applicability to the solution of the practical providers' problems. [ABSTRACT FROM AUTHOR]

In this paper, a novel strategy is employed in which a degradation model affects the implied distribution of lifetimes differently compared to the traditional method. It is recognized that an existing link between the degradation measurements and failure time constructs the underlying time-to-failure model. We assume in this paper that the conditional survival function of a device under degradation is a piecewise linear function for a given level of degradation. The multiplicative degradation model is used as the underlying degradation model, which is often the case in many practical situations. It is found that the implied lifetime distribution is a classical mixture model. In this mixture model, the time to failure lies with some probabilities between two first passage times of the degradation process to reach two specified values. Stochastic comparisons in the model are investigated when the probabilities are changed. To illustrate the applicability of the results, several examples are given in cases when typical degradation models are candidates. [ABSTRACT FROM AUTHOR]

The author comments on the studies on neural networks and sensors. The author wonders on the possibility of the study to be applied in addressing medical applications such as breast cancer. He knows several applications of neural networks in breast cancer in such an instance when a developed website was created that offer predictions of survival probability among cancer patients. The author hopes for an artificial neural network (ANN) expert to investigate on ductal carcinoma in-situ (DCIS).

In this paper, we suggest using a partial randomized response model using Poisson distribution to efficiently estimate a rare sensitive attribute by applying the probability proportional to size (PPS) sampling method when the population is composed of several different and sensitive clusters. We have obtained estimators for a rare and sensitive attribute and their variances and variance estimates by applying PPS sampling and two-stage equal probability sampling. We compare the efficiency between the estimators of the rare sensitive attribute, one obtained via PPS sampling with replacement and the other obtained using the two-stage equal probability sampling with replacement. As a result, it is confirmed that the estimate obtained via the PPS sampling with replacement is more efficient than the estimate provided by the two-stage equal probability sampling with replacement when the cluster sizes are different. [ABSTRACT FROM AUTHOR]

Simple Summary: This paper provides a review of statistical methods for tumor-agnostic clinical trials. In particular, the review focuses on basket trials and provides methodological insights into various Bayesian approaches. The key concept of borrowing information through Bayesian hierarchical models is emphasized, and some novel trial designs are introduced. The review is expected to provide oncology and biostatistics researchers with more exposure to powerful Bayesian methods for the design and analysis of tumor-agnostic clinical trials. Basket trials allow simultaneous evaluation of a single therapy across multiple cancer types or subtypes of the same cancer. Since the same treatment is tested across all baskets, it may be desirable to borrow information across them to improve the statistical precision and power in estimating and detecting the treatment effects in different baskets. We review recent developments in Bayesian methods for the design and analysis of basket trials, focusing on the mechanism of information borrowing. We explain the common components of these methods, such as a prior model for the treatment effects that embodies an assumption of exchangeability. We also discuss the distinct features of these methods that lead to different degrees of borrowing. Through simulation studies, we demonstrate the impact of information borrowing on the operating characteristics of these methods and discuss its broader implications for drug development. Examples of basket trials are presented in both phase I and phase II settings. [ABSTRACT FROM AUTHOR]

One of the main disadvantages of the traditional mean square error (MSE)-based constructive networks is their poor performance in the presence of non-Gaussian noises. In this paper, we propose a new incremental constructive network based on the correntropy objective function (correntropy-based constructive neural network (C2N2)), which is robust to non-Gaussian noises. In the proposed learning method, input and output side optimizations are separated. It is proved theoretically that the new hidden node, which is obtained from the input side optimization problem, is not orthogonal to the residual error function. Regarding this fact, it is proved that the correntropy of the residual error converges to its optimum value. During the training process, the weighted linear least square problem is iteratively applied to update the parameters of the newly added node. Experiments on both synthetic and benchmark datasets demonstrate the robustness of the proposed method in comparison with the MSE-based constructive network, the radial basis function (RBF) network. Moreover, the proposed method outperforms other robust learning methods including the cascade correntropy network (CCOEN), Multi-Layer Perceptron based on the Minimum Error Entropy objective function (MLPMEE), Multi-Layer Perceptron based on the correntropy objective function (MLPMCC) and the Robust Least Square Support Vector Machine (RLS-SVM). [ABSTRACT FROM AUTHOR]

The D-S evidence theory is extensively applied to manage uncertain information. However, the theory encounters challenges related to conflicts during the fusion process, impeding the precise identification of multi-subset focal elements. This paper introduces a novel method for conflicting evidence fusion that incorporates the Bray–Curtis dissimilarity, cosine distance of the included angle, and belief entropy. The method comprehensively evaluates three aspects—evidence similarity, evidence distance, and the amount of information—while considering factors like the credibility and uncertainty of evidence. Initially, the evidence undergoes conversion into single-subset focal element evidence through the improved Pignistic probability function. Subsequently, the credibility between pieces of evidence is established using the Bray–Curtis dissimilarity and angle cosine distance, while the uncertainty of the evidence is computed using belief entropy. The weighted correction coefficient of the evidence is determined by integrating the credibility and uncertainty of the evidence. Subsequently, the corrected evidence is fused using the D-S evidence theory to derive the final judgment. An analysis of two sets of arithmetic examples, considering both single-subset and multi-subset focal elements, demonstrates the faster convergence and enhanced accuracy and reliability of the proposed method in comparison to existing approaches. [ABSTRACT FROM AUTHOR]

The Monte Carlo neutron transport method is used to accurately estimate various quantities, such as k-eigenvalue and integral neutron flux. However, in the case of estimating a distribution of a desired quantity, the Monte Carlo method does not typically provide continuous distribution. Recently, the functional expansion tally (FET) and kernel density estimation (KDE) methods have been developed to provide a continuous distribution of a Monte Carlo tally. In this paper, we propose a method to estimate a continuous distribution of a quantity in all phase-space variables using a fully connected feedforward artificial neural network (ANN) model with Monte Carlo-based training data. As a proof of concept, a continuous distribution of iterated fission probability (IFP) was estimated by ANN models in two distinct fissile systems. The ANN models were trained on the training data created using the Monte Carlo IFP method. The estimated IFP distributions by the ANN models were compared with the Monte Carlo-based data that include the training data. Additionally, the IFP distributions by the ANN models were also compared with the adjoint angular neutron flux distributions obtained with the deterministic neutron transport code PARTISN. The comparisons showed varying degrees of agreement or discrepancy; however, it was observed that the ANN models learned the general trend of the IFP distributions from the Monte Carlo-based training data. [ABSTRACT FROM AUTHOR]

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension dim P (E) of the target set E. [ABSTRACT FROM AUTHOR]

Link prediction aims to identify unknown or missing connections in a network. The methods based on network structure similarity, known for their simplicity and effectiveness, have garnered widespread attention. A core metric in these methods is "proximity", which measures the similarity or linking probability between two nodes. These methods generally operate under the assumption that node pairs with higher proximity are more likely to form new connections. However, the accuracy of existing node proximity-based link prediction algorithms requires improvement. To address this, this paper introduces a Link Prediction Algorithm Based on Weighted Local and Global Closeness (LGC). This algorithm integrates the clustering coefficient to enhance prediction accuracy. A significant advantage of LGC is its dual consideration of a network's local and global features, allowing for a more precise assessment of node similarity. In experiments conducted on ten real-world datasets, the proposed LGC algorithm outperformed eight traditional link prediction methods, showing notable improvements in key evaluation metrics, namely precision and AUC. [ABSTRACT FROM AUTHOR]

In recent years, the research on complex networks has created a boom. The objective of the present paper is to study a random cyclooctatetraene chain whose graph-theoretic mathematical properties arose scientists' interests. By applying the concept of symmetry and probability theory, we obtain the explicit analytical expressions for the variances of Schultz index, multiplicative degree-Kirchhoff index Gutman index, and additive degree-Kirchhoff index of a random cyclooctatetraene chain with n octagons, which plays a crucial role in the research and application of topological indices. [ABSTRACT FROM AUTHOR]

Consider the one-dimensional random walk X n : as it evolves (at each unit of time), it either increases by one with probability p or resets to 0 with probability 1 − p . In the present paper, we analyze the law of the height statistics H n , corresponding to our model X n . Also, we prove that the limiting distribution of the walk X n is a shifted geometric distribution with parameter 1 − p and find the closed forms of the mean and the variance of X n using the probability-generating function. [ABSTRACT FROM AUTHOR]

Physical layer secret key (SK) generation is known to be an efficient means to achieve a high secrecy rate, on the condition that dynamic channel state information (CSI) is provided. For this reason, the secrecy performance is highly degraded in a static environment. The intelligent reflecting surface (IRS) is a promising solution to create dynamic randomness, and thus lead to enhanced secrecy performance regardless of the user environments. This paper proposes an IRS-assisted physical layer SK generation scheme, by efficiently combining phase information of the direct and reflected channel information in a hybrid way. In particular, the initial SKs are obtained by adopting an efficient phase quantization method with symmetric bit allocation to complex numbered channel estimates. Simulation results show that the proposed hybrid phase quantization (PQ) can improve the SK generation rate and the key disagreement probability in a static environment. [ABSTRACT FROM AUTHOR]

The increasing popularity of portable smart devices has led to the emergence of vehicular crowdsensing as a novel approach for real-time sensing and environmental data collection, garnering significant attention across various domains. Within vehicular crowdsensing, task assignment stands as a fundamental research challenge. As the number of vehicle users and perceived tasks grows, the design of efficient task assignment schemes becomes crucial. However, existing research solely focuses on task deadlines, neglecting the importance of task duration. Additionally, the majority of privacy protection mechanisms in the current task assignment process emphasize safeguarding user location information but overlook the protection of user-perceived duration. This lack of protection exposes users to potential time-aware inference attacks, enabling attackers to deduce user schedules and device information. To address these issues in opportunistic task assignment for vehicular crowdsensing, this paper presents the minimum number of participants required under the constraint of probability coverage and proposes the User-Based Task Assignment (UBTA) mechanism, which selects the smallest set of participants to minimize the payment cost while measuring the probability of accomplishing perceived tasks by user combinations. To ensure privacy protection during opportunistic task assignment, a privacy protection method based on differential privacy is introduced. This method fuzzifies the sensing duration of vehicle users and calculates the probability of vehicle users completing sensing tasks, thus avoiding the exposure of users' sensitive data while effectively assigning tasks. The efficacy of the proposed algorithm is demonstrated through theoretical analysis and a comprehensive set of simulation experiments. [ABSTRACT FROM AUTHOR]

This expository paper advocates an approach to physics in which "typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions state that some property Φ (x) holds for P-almost all x ∈ X , where P is a probability measure on some space X. Their more refined (and typically more recent) formulations show that Φ (x) holds for all P-random x ∈ X . The computational notion of P-randomness used here generalizes the one introduced by Martin-Löf in 1966 in a way now standard in algorithmic randomness. Examples come from probability theory, analysis, dynamical systems/ergodic theory, statistical mechanics, and quantum mechanics (especially hidden variable theories). An underlying philosophical theme, inherited from von Mises and Kolmogorov, is the interplay between probability and randomness, especially: which comes first? [ABSTRACT FROM AUTHOR]

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition (D) for the ergodicity of a Markov chain is equivalent to condition (∗) : all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case. [ABSTRACT FROM AUTHOR]

In this paper, an underwater robot system with nonlinear characteristics is studied by a backstepping method. Based on the state preservation problem of an Autonomous Underwater Vehicle (AUV), this paper applies the backstepping probabilistic gain controller to the nonlinear system of the AUV for the first time. Under the comprehensive influence of underwater resistance, turbulence, and driving force, the motion of the AUV has strong coupling, strong nonlinearity, and an unpredictable state. At this time, the system's output feedback can solve the problem of an unmeasurable state. In order to achieve a good control effect and extend the cruising range of the AUV, first, this paper will select the state error to make it a new control objective. The system's control is transformed into the selection of system parameters, which greatly simplifies the degree of calculation. Second, this paper introduces the concept of a stochastic backstepping control strategy, in which the robot's actuators work discontinuously. The actuator works only when there is a random disturbance, and the control effect is not diminished. Finally, the backstepping probabilistic gain controller is designed according to the nonlinear system applied to the simulation model for verification, and the final result confirms the effect of the controller design. [ABSTRACT FROM AUTHOR]

This Special Issue (SI), titled "Probability Theory and Stochastic Modeling with Applications", is concerned with the theory and applications of stochastic models. Footnotes 1 B Disclaimer/Publisher's Note: b The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). It consists of sixteen papers, and we would like to thank all the authors for their positive answers to our call for papers for the SI, as well as for their efforts to provide a high-quality contribution. Using fuzzy variables and robust optimization, the authors provide the electricity transaction policy under different expected costs so that the expectation of the risk-averse distribution system operator is fulfilled. [Extracted from the article]

A general sampling problem can be described by an Ellsberg urn, which is a mathematical model that assumes that balls are randomly drawn from an urn with an uncertain numbers of colored balls. This means that the Ellsberg urn is essentially an intricate model with simultaneous randomness and epistemic uncertainty, and this is the core problem discussed in this paper. Since practical sampling is usually processed in an intricate environment, the solution for an equivalent mathematical problem is necessary. Suppose an Ellsberg urn contains three unknown numbers of colored balls (i.e., a two-degrees-of-freedom Ellsberg urn), and three balls are randomly drawn from the urn. Compared to the published papers, this paper first constructs a chance space with two-dimensional uncertainty space and three-dimensional probability space to rigorously calculate the color distributions for those drawn balls by uncertainty theory, probability theory, and chance theory. Moreover, it is interesting to find that all cases of the drawn balls are symmetric in such a specific situation of a sample problem with epistemic uncertainty. [ABSTRACT FROM AUTHOR]

Simple Summary: Cancer is basically a tough condition on a patient's body where cell grows uncontrollably. Normal cells are affected, which destroys the health of the patient. The main problem in cancer is spreading from one part to another. Therefore, the mathematical modeling of cancerous tumors integrates to check overall stability. A novel approach is introduced such as Bernstein polynomial with combination of genetic algorithm, sliding mode controller, and synergetic control. The proposed solution has easily eliminated cancerous cells within five days using synergetic control. In addition, five cases are incorporated to evaluate error function. In addition, a brief comparative study is added to contrast the simulation results with theoretical modeling. Cancerous tumor cells divide uncontrollably, which results in either tumor or harm to the immune system of the body. Due to the destructive effects of chemotherapy, optimal medications are needed. Therefore, possible treatment methods should be controlled to maintain the constant/continuous dose for affecting the spreading of cancerous tumor cells. Rapid growth of cells is classified into primary and secondary types. In giving a proper response, the immune system plays an important role. This is considered a natural process while fighting against tumors. In recent days, achieving a better method to treat tumors is the prime focus of researchers. Mathematical modeling of tumors uses combined immune, vaccine, and chemotherapies to check performance stability. In this research paper, mathematical modeling is utilized with reference to cancerous tumor growth, the immune system, and normal cells, which are directly affected by the process of chemotherapy. This paper presents novel techniques, which include Bernstein polynomial (BSP) with genetic algorithm (GA), sliding mode controller (SMC), and synergetic control (SC), for giving a possible solution to the cancerous tumor cells (CCs) model. Through GA, random population is generated to evaluate fitness. SMC is used for the continuous exponential dose of chemotherapy to reduce CCs in about forty-five days. In addition, error function consists of five cases that include normal cells (NCs), immune cells (ICs), CCs, and chemotherapy. Furthermore, the drug control process is explained in all the cases. In simulation results, utilizing SC has completely eliminated CCs in nearly five days. The proposed approach reduces CCs as early as possible. [ABSTRACT FROM AUTHOR]

In order to describe human uncertainty more precisely, Baoding Liu established uncertainty theory. Thus far, uncertainty theory has been successfully applied to uncertain finance, uncertain programming, uncertain control, etc. It is well known that the limit theorems represented by law of large numbers (LLN), central limit theorem (CLT), and law of the iterated logarithm (LIL) play a critical role in probability theory. For uncertain variables, basic and important research is also to obtain the relevant limit theorems. However, up to now, there has been no research on these limit theorems for uncertain variables. The main results to emerge from this paper are a strong law of large numbers (SLLN), a weak law of large numbers (WLLN), a CLT, and an LIL for Bernoulli uncertain sequence. For studying these theorems, we first propose an assumption, which can be regarded as a generalization of the duality axiom for uncertain measure in the case that the uncertainty space can be finitely partitioned. Additionally, several new notions such as weakly dependent, Bernoulli uncertain sequence, and continuity from below or continuity from above of uncertain measure are introduced. As far as we know, this is the first study of the LLN, the CLT, and the LIL for uncertain variables. All the theorems proved in this paper can be applied to uncertain variables with symmetric or asymmetric distributions. In particular, the limit of uncertain variables is symmetric in (c) of the third theorem, and the asymptotic distribution of uncertain variables in the fifth theorem is symmetrical. [ABSTRACT FROM AUTHOR]

Let { ξ 1 , ξ 2 , ... } be a sequence of independent possibly differently distributed random variables, defined on a probability space (Ω , F , P) with distribution functions { F ξ 1 , F ξ 2 , ... } . Let η be a counting random variable independent of sequence { ξ 1 , ξ 2 , ... } . In this paper, we find conditions under which the distribution function of randomly stopped sum S η = ξ 1 + ξ 2 + ... + ξ η belongs to the class of generalized subexponential distributions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system's performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account. [ABSTRACT FROM AUTHOR]

In the middle and late stages of gas field development, the establishment of a fine reservoir lithological model is an important basis for drilling well pattern adjustment and potential exploitation. The SU 36-11 area of the Ordos basin in China is developing braided channel sediment with rich gas resources. However, the success rate of drilling wells is low due to the complex reservoir heterogeneity and the lack of a fine reservoir lithological model. In this paper, the complex internal structure of the reservoir sand body is revealed using the architectural element analysis method. Three sand body models, that is, isolated channel, superimposed channel, and cut superimposed channel, can be recognized. The effective sand body is mainly the channel bar deposit with a thickness of 2–5 m, a width of 200–500 m, a length of 400–700 m, a width ratio of 50–120, and a length-to-width ratio of 1.5–2. The 2D maps of the lithofacies (architectural elements) were then digitized to create 2D training images (TI) for the construction of the 3D model. The 2D data template was selected to scan the TI to obtain the 2D multi-point probability. The 3D multi-point probability was then generated using the probability fusion theory. The Monte Carlo sampling was used to predict the lithological type between wells. Finally, the 3D reservoir lithological model was built directly using the 2D lithological profiles. From the model, the geometry of the braided channel, channel bar, and flood plain was well revealed, and the spatial distribution of effective reservoir sand bodies was accurately predicted. The cross-validation test shows that the error of the channel bar is 6.5% on average, which improves the accuracy of the prediction of lithology in the sub-surface and can be used to guide the subsequent development of residual gas. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive mathematical analysis of the hallucination phenomenon in generative pretrained transformer (GPT) models. We rigorously define and measure hallucination and creativity using concepts from probability theory and information theory. By introducing a parametric family of GPT models, we characterize the trade-off between hallucination and creativity and identify an optimal balance that maximizes model performance across various tasks. Our work offers a novel mathematical framework for understanding the origins and implications of hallucination in GPT models and paves the way for future research and development in the field of large language models (LLMs). [ABSTRACT FROM AUTHOR]

The shift toward electric mobility in Germany is a major component of the German climate protection program. In this context, public charging is growing in importance, especially in high-density urban areas, which causes an additional load on the distribution grid. In order to evaluate this impact and prevent possible overloads, realistic models are required. Methods for implementing such models and their application in the context of grid load are research topics that are only minorly addressed in the literature. This paper aims to demonstrate the entire process chain from the selection of a modelling method to the implementation and application of the model within a case study. Applying a stochastic approach, charging points are modelled via probabilities to determine the start of charging, plug-in duration, and charged energy. Subsequently, load profiles are calculated, integrated into an energy system model and applied in order to analyze the effects of a high density of public charging points on the urban distribution grid. The case study highlights a possible application of the implemented probabilistic load profile model, but also reveals its limitations. The primary results of this paper are the identification and evaluation of relevant criteria for modelling the load profiles of public charging points as well as the demonstration of the model and its comparison to real charging processes. By publishing the determined probabilities and the model for calculating the charging load profiles, a comprehensive tool is provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-parametric regression model relying on Riesz estimators. This linear regression model is similar to the usual linear regression model since they both rely on projection operators. We indicate that Riesz estimator regression relies on the positive basis elements of the finite-dimensional sub-lattice generated by the rows of some design matrix. A strong motivation for using the Riesz estimator model is that the data of explanatory variables may come from categorical variables. Calculations related to Riesz estimator regression are very easy since they arise from the measurability in finite-dimensional probability spaces. Moreover, we show that the fitted model of Riesz estimators is an ordinary least squares model. Any vector of some Euclidean space is supposed to be a rendom variable under the objective probability values, being used in expected utility theory and its applications. Finally, the reader may notice that goodness-of-fit measures are similar to those defined for the usual linear regression. Due to the fact that this model is non-parametric, it may include samples relevant to finance and actuarial science variables. [ABSTRACT FROM AUTHOR]

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results. [ABSTRACT FROM AUTHOR]

This paper generalizes the structural Markov properties for undirected decomposable graphs to arbitrary ones. This helps us to exploit the conditional independence properties of joint prior laws to analyze and compare multiple graphical structures, while being able to take advantage of the common conditional independence constraints. This work provides a theoretical support for full Bayesian posterior updating about the structure of a graph using data from a certain distribution. We further investigate the ratio of graph law so as to simplify the acceptance probability of the Metropolis–Hastings sampling algorithms. [ABSTRACT FROM AUTHOR]

Automated vehicles are expected to greatly boost traffic efficiency. However, how to estimate traffic breakdown probability for the mixed flow of autonomous vehicles and human driven vehicles around ramping areas remains to be answered. In this paper, we propose a stochastic temporal queueing model to reliably depict the queue dynamics of mixed traffic flow at ramping bottlenecks. The new model is a specified Newell's car-following model that allows two kinds of vehicle velocities and first-in-first-out (FIFO) queueing behaviors. The jam queue join time is supposed to be a random variable for human driven vehicles but a constant for automated vehicles. Different from many known models, we check the occurrence of significant velocity drop along the road instead of examining the duration of the simulated jam queue so as to avoid drawing the wrong conclusions of traffic breakdown. Monte Carlo simulation results show that the generated breakdown probability curves for pure human driven vehicles agree well with empirical observations. Having noticed that various driving strategy of automated vehicles exist, we carry out further analysis to show that the chosen car-following strategy of automated vehicles characterizes the breakdown probabilities. Further tests indicate that when the penetration rate of automated vehicles is larger than 20%, the traffic breakdown probability curve of the mixed traffic will be noticeably shifted rightward, if an appropriate car-following strategy is applied. This indicates the potential benefit of automated vehicles in improving traffic efficiency. [ABSTRACT FROM AUTHOR]

Inhomogeneous random graphs are commonly used models for complex networks where nodes have varying degrees of connectivity. Computing the degree distribution of such networks is a fundamental problem and has important applications in various fields. We define the inhomogeneous random graph as a random graph model where the edges are drawn independently and the probability of a link between any two vertices can be different for each node pair. In this paper, we present an exact and an approximation method to compute the degree distribution of inhomogeneous random graphs using the Poisson binomial distribution. The exact algorithm utilizes the DFT-CF method to compute the distribution of a Poisson binomial random variable. The approximation method uses the Poisson, binomial, and Gaussian distributions to approximate the Poisson binomial distribution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a random entire function of the form f (z , ω) = ∑ n = 0 + ∞ ε n (ω 1) × ξ n (ω 2) f n z n , where (ε n) is a sequence of independent Steinhaus random variables, (ξ n) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers f n ∈ C is such that lim ¯ n → + ∞ | f n | n = 0 and # { n : f n ≠ 0 } = + ∞. We investigate asymptotic estimates of the probability P 0 (r) = P { ω : f (z , ω) has no zeros inside r D } as r → + ∞ outside of some set E of finite logarithmic measure, i.e., ∫ E ∩ [ 1 , + ∞) d ln r < + ∞ . The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions. [ABSTRACT FROM AUTHOR]

The Simeck family of lightweight block ciphers was proposed by Yang et al. in 2015, which combines the design features of the NSA-designed block ciphers Simon and Speck. Previously, we proposed the use of linear cryptanalysis using super-rounds to increase the efficiency of implementing Matsui's second algorithm and achieved good results on all variants of Simon. The improved linear attacks result from the observation that, after four rounds of encryption, one bit of the left half of the state of the cipher depends on only 17 key bits (19 key bits for the larger variants of the cipher). We were able to follow a similar approach, in all variants of Simeck, with an improvement in Simeck 32 and Simeck 48 by relaxing the previous constraint of a single active bit, using multiple active bits instead. In this paper we present improved linear attacks against all variants of Simeck: attacks on 19-rounds of Simeck 32/64, 28-rounds of Simeck 48/96, and 34-rounds of Simeck 64/128, often with the direct recovery of the full master key without repeating the attack over multiple rounds. We also verified the results of linear cryptanalysis on 8, 10, and 12 rounds for Simeck 32/64. [ABSTRACT FROM AUTHOR]

A two-party authenticated key exchange (AKE) protocol allows each of the two parties to share a common secret key over insecure channels, even in the presence of active adversaries who can actively control and modify the exchanged messages. To capture the malicious behaviors of the adversaries, there have been many efforts to define security models. Amongst them, the extended Canetti–Krawczyk (eCK) security model is considered one of the strongest security models and has been widely adopted. In this paper, we present a simple construction of a pairing-based eCK -secure AKE protocol in the standard model. Our protocol can be instantiated with a suitable signature scheme (i.e., an existentially unforgeable signature scheme against adaptive chosen message attacks). The underlying assumptions of our construction are the decisional bilinear Diffie–Hellman assumption and the existence of a pseudorandom function. Note that the previous eCK -secure protocol constructions either relied on random oracles for their security or used somewhat strong assumptions, such as the existence of strong-pseudorandom functions, target collision-resistant functions, etc., while our protocol construction uses fewer and more-standard assumptions in the standard model. Furthermore, preserving the same security argument, our protocol can be instantiated with any appropriate signature scheme that comes in the future with better efficiency. [ABSTRACT FROM AUTHOR]

Bell's polynomials are used in many different fields of mathematics, ranging from number theory to operator theory. This paper shows a relevant application in probability theory aimed at computing the moments of generalized Gaussian distributions. To this end, a table containing the first values of the complete Bell's polynomials is provided. Furthermore, a dedicated code for approximating the moments of the general distributions in terms of complete Bell's polynomials is detailed. Several test cases concerning different nested functions are discussed. [ABSTRACT FROM AUTHOR]