Random Function Research Articles

Comparing K-Means, Nearest Neighbor, and Lloyd's Clustering Algorithms

Clustering Organizing items into groups based on their properties such that the items in the same group are similar and those in other groups are distinct is known as clustering and is one method of unsupervised learning. The primary benefit of clustering is that, with little or no prior information, fascinating patterns and structures can be discovered directly from very large data sets. The most representative algorithms, the K-Means algorithm, the nearest neighbor algorithm, and Lloyd's algorithm, were explored and evaluated in this study based on their basic strategies. The proposed algorithms proved highly efficient in classifying data, as k-means results were high by creating data points and classifying that data into 10 groups, while the nearest neighbor algorithm proved highly effective in predicting new groups in light of pre-existing groups for new data points, and finally Lloyd's algorithm achieved high results through several iterations to reach the target groups. Using the random function, 100 random values for inputs as x and 100 for outputs as y are generated, and these values are grouped as points. Calculations of the mean and standard deviation for the data set indicate that 68% of the data points will fall within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations. In the nearest neighbor algorithm, confidence in data points within a specified standard deviation number is determined by the coverage factor, or k value. For k = 10, 97% of the data points are expected to fall within one standard deviation. At Lloyd's, a normal distribution curve appears when generating calibration or measurement data.

MODELS OF CORRELATION FUNCTIONS OF NONSTATIONARY PROCESSES AND SEQUENCES FOR TECHNOLOGICAL SYSTEMS

Precision technological processes for the production of modern microelectronic products require compliance with the quality of source materials, working environments, and precise adherence to regimes. Due to natural fluctuations in the properties of materials and the environment, and variable states of all technological processes, the parameters of the product and technological processes cannot be described by deterministic laws. Due to the inevitable natural properties of fluctuations in the parameters of technological equipment and its operating modes, the state variables of all technological processes are random functions of space-time coordinates. In most cases, these accidents cannot be neglected, since they all affect the output parameters of the products. The most complex mathematical models of technological systems and processes in modern theory are random spatiotemporal fields, representing both input and output characteristics, as well as parameters of the systems under consideration. The purpose of this work is to model real-valued values of correlation functions of non-stationary random processes and sequences. When constructing a correlation theory of random processes and sequences, a complex representation is widely used, i.e. random functions of the form are considered: continuous or discrete time. This approach made it possible to construct a correlation theory of nonstationary random functions using the spectral theory of non-self-adjoint or unitary operators and to introduce the concept of complex spectrum. For applications of the correlation theory of nonstationary random functions and their modeling, it is convenient to deal with real-valued correlation functions. The construction of real-valued correlation functions can be carried out using the well-known fact that the real part of complex-valued correlation functions is also a correlation function (for the imaginary part this statement is unfair, since the imaginary part is a cross-correlation function of the real and imaginary parts of the corresponding random process or sequence) . The resulting models of correlation functions of non-stationary random processes and sequences can be used to construct algorithms for forecasting and filtering non-stationary random functions

Reliability for single-unit systems via virtual ages under cyclic environments

ABSTRACT Systems commonly operate in varying environments which are harmful in most situations for system reliability. The usage of virtual ages in reliability increases gradually, which is a powerful tool for analyzing system reliability. In the paper, by employing virtual ages, a system reliability model is developed for a single-unit system operating in cyclic environments. The system reliability formula is given in an expectation form of a random survival function whose variables include a virtual age, and the asymptotic estimates of system reliability are derived in terms of the law of large numbers and the central limit theorem. Meanwhile, some special cases are discussed, and the simulation algorithm is established. We believe this work may shed light on the related researches in future not only in the area of reliability but also in the subject of maintenance.

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We study the effective behavior of random, heterogeneous, anisotropic, second-order phase transitions energies that arise in the study of pattern formations in physical–chemical systems. Specifically, we study the asymptotic behavior, as ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} goes to zero, of random heterogeneous anisotropic functionals in which the second-order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first-order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed.

Extraction rates of algorithmically random continuous functionals

Extraction rates of algorithmically random continuous functionals

Towards a new roughness parametrization through the effective distribution function

Turbulent flows over rough surfaces can be encountered in a wide range of engineering applications. Despite the progress made after several decades of studies, the prediction of drag and roughness function from the surface geometrical parameters remains an open question. Several methods have shown encouraging results. However, they lack generality and present some scatter in the data. In this paper we propose a new parameter, the effective distribution ( $ED$ ), which lays foundation on the effective slope with some changes to take into account the sheltering effect of large roughness elements and the drag induced by pinnacles higher than the average roughness elements. To develop this new correlation between geometrical features of the wall and the drag, we performed a set of simulations of the turbulent flow over a rough surface made of triangular elements varying their height and spatial distribution. The $ED$ correlates quite well both with the drag and the roughness function for a wide range of cases having different mean roughness height, skewness and kurtosis. To further validate the $ED$ , and assessing how it can be generalized to real rough wall, an irregular wall made from the superposition of random sinusoidal function was considered. Results were consistent with the correlation here presented.

The effectiveness of multimedia mental health self-care program based on cyber space on the mental health of infertile women: a randomized controlled trial

BackgroundSo far, some training interventions have been carried out to improve the mental health in women with infertility, but designing the need and evidence-based, as well as multimedia mental health self-care interventions based on cyber space has received less attention. Due to the spread of the internet and the role of self-care in improving mental disorders, this study was conducted to evaluate the effectiveness of the multimedia mental health self-care program on mental health and to assess the users' satisfaction.MethodsThis study is a randomized controlled trial with pretest–posttest follow-up design. The sample was selected using a convenience sampling method (n = 90). The random number function was used to assign random numbers. The research instruments include a demographic, psychological Well-being, depression, anxiety, perceived stress, fertility problems and satisfaction with training questionnaire. Six weeks of intervention was conducted following the pre-test and the link of each session's content was sent to the participants, based on the training schedule, through Eitaa Messenger. The post-test and follow-up were conducted 1 week and 1 month post intervention. The data were analyzed using independent t-test and repeated measures ANOVA.ResultsA statistically significant difference was observed between the intervention and control group in the mean score of psychological well-being, perceived stress and infertility stress 1 week and 1 month post intervention and in the mean score of depression and anxiety 1 month post intervention. The intervention group scored higher than the control in psychological well-being but lower in perceived stress, depression, anxiety and infertility stress. The intervention had a positive effect and reduced the score of perceived stress, depression, anxiety and infertility in the intervention group over time. The score reduction continued until the follow-up stage. No significant time-interaction effect was observed on psychological well-being and on the control group. Satisfaction with the program and subscales was desirable.ConclusionThis program could significantly reduce the depression, anxiety, perceived stress and infertility stress and desirable satisfaction with the program was observed among users. This program can be used in designing the experimental and therapeutic interventions to improve mental-health self-care behaviors.Trial registrationRCT Registry: Iranian Registry of Clinical Trials; RCT registration number: IRCT20210526051410N1; Registration date: 2022–11-06. Last update: 2023–01-28 https://irct.behdasht.gov.ir/trial/57874

On the strong stability of ergodic iterations

Abstract We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists for which the process is stationary and ergodic, and for any other initialization the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

Water invasion and residual gas distribution in partially filled fractures via phase-field method

Water invasion is a significant factor affecting the conductivity of fractures in coal seams. The partially contact characteristics of deep coal seam fractures are pronounced, and surface wettability varies significantly. However, there is a limited understanding of how water invasion behavior in partially filled fractures affects the gas produced by these fractures. In this study, a high-temperature and high-pressure contact angle testing device was employed to assess the wettability of coal seams under in situ conditions. The geometry of partially filled fractures was reconstructed using random functions, while the phase field method was employed to calculate the interactions at the two-phase interface during water invasion. The results indicate that the deep coal seams in the Ordos Basin demonstrate weak air-wetting properties under in situ conditions. The partially contact characteristics of the filled fractures in the deep coal seams categorize the fractures into distinct pore and throat regions. The variations in connectivity levels lead to the gas exhibiting uninvaded, clustered, and fully invaded characteristics following water invasion. The change in gas saturation during water invasion is more sensitive to larger values of lgCa and higher cos(θ). A larger displacement pressure difference and a smaller contact angle enable the invasion fluid to penetrate smaller throats, resulting in a higher number of clusters of residual gas and a smaller cluster radius. The results enhance our understanding of water invasion behavior, and the variability of fracture surface properties and gas-water two-phase flow in deep coal seams deserves further investigation.

Arbitrary Random Variables and Wiman’s Inequality

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

Humanoid Robot Path Planning Using Rapidly Explored Random Tree and Motion Primitives

Path planning is an essential part of the control system of any mobile robot. In this article the path planner for a humanoid robot is presented. The short description of an universal control framework and the Motion Generation System is also presented. Described path planner utilizes a limited number of motions called the Motion Primitives. They are generated by Motion Generation System. Four different algorithms, namely the: Informed RRT, Informed RRT with random bias, and RRT with A* likeheuristics were tested. For the last one the version with biased random function was also considered. All mentioned algorithms were evaluated considering three dif ferent scenarios. Obtained results are described and discussed.

Proof of Work with Random Selection (PoWR): An Energy Saving Consensus Algorithm with Proof of Work and the Random Selection Function

Bitcoin, which has been used for 13 years, has a role in transactions and investments as a major cryptocurrency. However, as the number of users increases, Bitcoin faces difficulties, such as scalability for transaction throughput and energy-consumption problems due to the concentration of the mining pool. When Bitcoin first started to come out, it began to develop gradually through the mining of individuals. Nevertheless, as the price of the cryptocurrency gradually climbed, large mining corporation groups entered the mining competition with integrated circuit (IC) chips. Consequently, the substantial increase in power consumption is raising concerns regarding energy expenditure. This paper confirms that the verifiable random selection consensus protocol based on proof of work facilitates a fair and efficient system, enabling the participation of numerous individual miners in the mining competition while counteracting the monopolization of the hash rate by large mining corporations, thereby preserving the decentralization of mining. The protocol demonstrates the potential to mitigate substantial energy consumption. Moreover, it embodies features that create barriers to the adoption of high-energy-consuming application-specific integrated circuit equipment, significantly diminishing the principal factors contributing to extensive power utilization.

Bifurcation control and analysis of traffic flow model based on driver prediction effect.

Based on the measured data, in this paper, we find that there are significant differences in the ideal speed of drivers with different attributes during their driving process. Through wavelet analysis, it is proved that the regularity of the car-following behavior of the driver is poor in a short time, and it cannot be predicted and analyzed at a small time scale. As one of the important participants in the traffic system, it is necessary to consider the driving behavior factors in the traffic flow model. Therefore, in this paper, we propose a nonuniform continuous traffic flow model. Based on the difference in the expected headway of the driver, the model considers the self-stabilization effect of the prediction time of the driver on the optimal speed difference and considers control theory to further improve the model. Through this model, the bifurcation theory can be applied to the stability analysis of the traffic system to study the stability mutation behavior of the traffic system at the bifurcation point. Through linear and nonlinear analysis methods, the stability conditions of the model can be derived, and the type of equilibrium point of the model can be judged. In addition, the random function is applied to the traffic model, and the bifurcation control is carried out for the bifurcation behavior in the traffic; that is, the bifurcation characteristics of the traffic system are changed by designing linear and nonlinear random feedback controllers. In this paper, we theoretically prove the existence conditions and types of Hopf bifurcation at the equilibrium point of the model and analyze the internal causes of the stability mutation of the traffic flow through the Hopf bifurcation point. Then a feedback controller is designed to control the amplitude of the Hopf bifurcation and the limit cycle formed by the Hopf bifurcation. Finally, the theoretical derivation results are verified by experimental numerical simulation. In this paper, we show that, by adjusting the control parameters of the feedback controller, the Hopf bifurcation of the stochastic system can be delayed or even eliminated, and the amplitude of the limit cycle formed by the Hopf bifurcation can be adjusted to achieve the purpose of controlling the stability of the traffic system and prevent or alleviate traffic congestion.

Blockchain for video watermarking: An enhanced copyright protection approach for video forensics based on perceptual hash function.

As a direct result of advancements in digital technology and the Internet, the copyright protection and information integrity of multimedia that are being published across the Internet have emerged as a major and urgent issue that needs to be addressed. The technique of digital watermarking may be used to protect intellectual property. In terms of authentication, resilience, storage, and capacity of digital watermarking information, there is still room for development. Blockchain's potential in video copyright protection and management applications has motivated researchers. Copyright owners and consumers may now communicate directly via copyright protection apps built on the blockchain, eliminating the need for distributers and the associated fees. Nonetheless, the current blockchain-based video watermarking solutions require storing a significant number of coordinates depending on the watermark size and are susceptible to video frame attacks on the video frame texture region. This study proposes an enhanced video copyright management approach that incorporates digital watermarking, the blockchain, and a perceptual hash function. Watermark information is stored on a blockchain structure, which also acts as a timestamp for verification purposes. To verify watermark data without the source video, a perceptual hash function is employed to compute a hash value based on the structural information of video frames. The contribution is in learning how to extract a hash function from a small number of video frames that still adequately represent a large amount of video content while also reducing the number of unnecessary video frames and the amount of computation required to summarize and index that content in a blockchain. This expedites the dissemination of copyrighted works and increases their security and readability, hence facilitating their circulation on the Internet. Our experimental results demonstrate that this approach is memory efficient, as it only needs to store one key for each key frame, regardless of the size of the watermark. Additionally, the overall robustness is greatly improved by using the blockchain's random hash function. Therefore, new and important advancements in video watermarking have been realized because of this effort.

Random nonequilibrium Green's function method for large-scale quantum transport simulation

Random nonequilibrium Green's function method for large-scale quantum transport simulation

Deterministic equivalent and error universality of deep random features learning*

Abstract This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture. In light of our results, we investigate the interplay between architecture design and implicit regularization.

A shaping two-stage anomaly data recovery method based on multi-norm joint optimization under energy internet

With the deep integration of energy networks and information communication technology, the involvement of hardware and environmental factors intensifies the challenges to data security. Prior research focuses predominantly on interpolating random missing data under the assumption of no data damage, without providing strategies for addressing continuous data loss. To overcome, we proposes a novel two-stage data recovery method, that includes three-fold ideas: (1) it introduces the matrix shaping differentiation strategy to realize the differentiated recovery of random and continuous missing data; (2) it designs a two-stage data recovery approach, in which the first stage constructs a random missing data recovery objective function by eliminating the consecutive missing data columns, and the second stage performs temporal pre-interpolation backfilling on the eliminated columns and introduces the temporal correlation to construct the temporal smoothing objective function; and (3) it develops an alternating direction method of multipliers-based iterative solution algorithm. Experimental results on four public datasets demonstrate that our model is superior to five state-of-the-art and classical methods in terms of random missing error rate, random column missing error rate, root mean square error, and mean absolute error.

An efficient Proof-of-Authority consensus scheme against cloning attacks

Proof-of-Authorization (PoA) consensus algorithms are widely used in permissioned blockchain networks due to their high throughput, security, and efficiency. However, PoA is susceptible to cloning attacks, where attackers copy the authenticator identity and key, thereby compromising the consensus integrity. This study proposes a novel randomized authenticator within the PoA framework to mitigate cloning attacks and solve the leader selection bottleneck. The main contributions include 1) Introducing unpredictability in leader selection through Verifiable Random Functions (VRFs) to prevent identity duplication.2) Dynamic group management using a hierarchical decentralized architecture of distributed ledgers that balances security and performance.3) Using threshold signatures to avoid a single point of failure among validators.4) Comprehensively analyzing attacks, security, randomness, and availability.5) Evaluating the effectiveness of a randomized authenticator by means of OMNET++ simulations to assess efficiency. By integrating randomness into leader selection and robust consensus design, the approach enables reliable and secure dynamic group management in decentralized networks.

A physical stochastic model of near-surface fluctuating wind fields

Modeling and simulation of atmospheric turbulence in coastal areas has emerged as a prominent area of research in recent years. This study presents a physical stochastic model to describe the horizontal coherence of fluctuating wind fields within the physical stochastic modeling frame. Based on the one-dimensional stochastic Fourier spectrum and isotropic turbulence theory, the horizontal coherence for fluctuating wind fields is expressed as a random function, with its probability information determined by the physical mechanism/background. The proposed physical stochastic model directly depicts the stochastic time series thereby enabling it to capture all probability information in detail comprehensively. The proposed model is numerically validated with the measured data obtained from an observation array constructed in Southeast China. This investigation holds significant potential for its application in wind-resistance design and reliability assessment of long-span structures.

Performance analysis and evaluation of postquantum secure blockchained federated learning

As the field of quantum computing progresses, traditional cryptographic algorithms such as RSA and ECDSA are becoming increasingly vulnerable to quantum-based attacks, underscoring the need for robust post-quantum security in critical systems like Federated Learning (FL) and Blockchain. In light of this, we propose a novel hybrid approach for blockchain-based FL (BFL) that integrates a stateless signature scheme, such as Dilithium or Falcon, with a stateful hash-based scheme like XMSS. This combination leverages the complementary strengths of both schemes to provide enhanced security. To further optimize performance, we introduce a linear formula-based device role selection method that takes into account key factors such as computational power and stake accumulation. This selection process is reinforced by a verifiable random function (VRF), which strengthens the blockchain consensus mechanism. Our extensive experimental results demonstrate that this hybrid approach significantly enhances both the security and efficiency of BFL systems, establishing a robust framework for the integration of post-quantum cryptography as we transition into the quantum computing era.