WEIGHTED APPROXIMATION OF RANDOM FUNCTIONS

Let (Ω, A, P) be a probability space, X(t, ω) a random function continuous in probability for t∈[0,+∞) or (-∞,+∞)(ω∈Ω), and F(t) a positive function continuous for t∈[0,+∞) or (-∞, +∞). If X(t, ω) and F(t) verify certain conditions, then there exists a sequence {Qn(t,ω)} of random polynomials such that we have almost surely: for t∈[0,+∞) or (-∞, +∞), lim|X(t, ω)-Qn(t, ω)|/F(t)=0.

Mean-square Almost Periodic Random Functions

In this paper, we present a basic theory of mean-square almost periodicity, apply the theory in random differential equation, and obtain mean-square almost periodic solution of some types stochastic differential equation.

Underwater Image Classification Based on EfficientnetB0 and Two-Hidden-Layer Random Vector Functional Link

The ocean plays an important role in maintaining the equilibrium of Earth’s ecology and providing humans access to a wealth of resources.To obtain a high-precision underwater image classification model,we propose a classification model that combines an EfficientnetB0 neural network and a two-hidden-layer random vector functional link network(EfficientnetB0-TRVFL).The features of underwater images were extracted using the EfficientnetB0 neural network pretrained via ImageNet,and a new fully connected layer was trained on the underwater image dataset using the transfer learning method.Transfer learning ensures the initial performance of the network and helps in the development of a high-precision classification model.Subsequently,a TRVFL was proposed to improve the classification property of the model.Net construction of the two hidden layers exhibited a high accuracy when the same hidden layer nodes were used.The parameters of the second hidden layer were obtained using a novel calculation method,which reduced the outcome error to improve the performance instability caused by the random generation of parameters of RVFL.Finally,the TRVFL classifier was used to classify features and obtain classification results.The proposed EfficientnetB0-TRVFL classification model achieved 87.28%,74.06%,and 99.59%accuracy on the MLC2008,MLC2009,and Fish-gres datasets,respectively.The best convolutional neural networks and existing methods were stacked up through box plots and Kolmogorov-Smirnov tests,respectively.The increases imply improved systematization properties in underwater image classification tasks.The image classification model offers important performance advantages and better stability compared with existing methods.

THE ANALYTICAL PROPERTIES FOR HOMOGENEOUS RANDOM TRANSITION FUNCTIONS

The concepts of Markov process in random environment and homogeneous random transition functions are introduced. The necessary and sufficient conditions for homogeneous random transition function are given. The main results in this article are the analytical properties, such as continuity, differentiability, random Kolmogorov backward equation and random Kolmogorov forward equation of homogeneous random transition functions.

QUANTITATIVE THEOREM AND SATURATION THEOREM FOR MULTIVARIATE RANDOMFUNCTIONS

In this paper, we give the quantitative theorem and saturation theorem for random functions with compact convex parameter spaces.

ON THE DISTRIBUTION OF VALUES AND DEFICIENT FUNCTION OF A GENERAL RANDOM INTEGRAL FUNCTION

Under suitable conditions on {X-n}, the author obtains the important results: it is almost sure that the random integral function f(w) = Sigma (infinity)(n=0) X(n)z(n) (of finite positive order) has no deficient function, and any direction is a Borel direction (without finite exceptional value) of f(w).

Laplace Transformation and Ergodic Potential Kernel for q-Process in Random Environment

This paper introduces some concepts such as q- process in random environment, Laplace transformation, ergodic potential kernel, error function and some basic lemmas.We study the continuity and Laplace transformation of random transition function. Finally, we give the sufficient condition for the existence of ergodic potential kernel for homogeneous q- processes in random environments.

ON MULTIPLE WIENER INTEGRAL EXPANSIONS FOR D'-VALUED NONLINEAR RANDOM FUNCTIONALS

In this paper, the multiple stochastic integral with respect to a Wiener D'-process is defined. And also it is shown that for a D'-valued nonlinear random functional there exists a sequence of multiple integral kernels such that the nonlinear functional can be expanded by series of multiple Wiener integrals of the integral kernels with respect to the Wiener D'-process.

AFFINE TRANSFORMATION IN RANDOM ITERATED FUNCTION SYSTEMS

Random iterated function systems (IFSs) is discussed, which is one of the methods for fractal drawing. A certain figure can be reconstructed by a random IFS. One approach is presented to determine a new random IFS, that the figure reconstructed by the new random IFS is the image of the origin figure reconstructed by old IFS under a given affine transformation. Two particular examples are used to show this approach.

Robust H_∞ control of piecewise-linear chaotic systems with random data loss

This paper studies the problem of robust H∞ control of piecewise-linear chaotic systems with random data loss. The communication links between the plant and the controller are assumed to be imperfect （that is, data loss occurs intermittently, which appears typically in a network environment）. The data loss is modelled as a random process which obeys a Bernoulli distribution. In the face of random data loss, a piecewise controller is designed to robustly stabilize the networked system in the sense of mean square and also achieve a prescribed H∞ disturbance attenuation performance based on a piecewise-quadratic Lyapunov function. The required H∞ controllers can be designed by solving a set of linear matrix inequalities （LMIs）. Chua＇s system is provided to illustrate the usefulness and applicability of the developed theoretical results.

Parametric characteristic of the random vibration response of nonlinear systems

Volterra series is a powerful mathematical tool for nonlinear system analysis,and there is a wide range of nonlinear engineering systems and structures that can be represented by a Volterra series model.In the present study,the random vibration of nonlinear systems is investigated using Volterra series.Analytical expressions were derived for the calculation of the output power spectral density（PSD） and input-output cross-PSD for nonlinear systems subjected to Gaussian excitation.Based on these expressions,it was revealed that both the output PSD and the input-output crossPSD can be expressed as polynomial functions of the nonlinear characteristic parameters or the input intensity.Numerical studies were carried out to verify the theoretical analysis result and to demonstrate the effectiveness of the derived relationship.The results reached in this study are of significance to the analysis and design of the nonlinear engineering systems and structures which can be represented by a Volterra series model.

THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT(I):MODEL,MARKOV PROPERTY,MOMENTS

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment （BCDSTRE）, i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.

A Machine Learning Approach for Artifact Removal from Brain Signal

Electroencephalography(EEG),helps to analyze the neuronal activity of a human brain in the form of electrical signals with high temporal resolution in the millisecond range.To extract clean clinical information from EEG signals,it is essential to remove unwanted artifacts that are due to different causes including at the time of acquisition.In this piece of work,the authors considered the EEG signal contaminated with Electrocardiogram(ECG)artifacts that occurs mostly in cardiac patients.The clean EEG is taken from the openly available Mendeley database whereas the ECG signal is collected from the Physionet database to create artifacts in the EEG signal and verify the proposed algorithm.Being the artifactual signal is non-linear and non-stationary the Random Vector Functional Link Network(RVFLN)model is used in this case.The Machine Learning approach has taken a leading role in every field of current research and RVFLN is one of them.For the proof of adaptive nature,the model is designed with EEG as a reference and artifactual EEG as input.The peaks of ECG signals are evaluated for artifact estimation as the amplitude is higher than the EEG signal.To vary the weight and reduce the error,an exponentially weighted Recursive Least Square(RLS)algorithm is used to design the adaptive filter with the novel RVFLN model.The random vectors are considered in this model with a radial basis function to satisfy the required signal experimentation.It is found that the result is excellent in terms of Mean Square Error(MSE),Normalized Mean Square Error(NMSE),Relative Error(RE),Gain in Signal to Artifact Ratio(GSAR),Signal Noise Ratio(SNR),Information Quantity(IQ),and Improvement in Normalized Power Spectrum(INPS).Also,the proposed method is compared with the earlier methods to show its efficacy.

Adaptive Control of Discrete-time Nonlinear Systems Using ITF-ORVFL

Random vector functional ink(RVFL)networks belong to a class of single hidden layer neural networks in which some parameters are randomly selected.Their network structure in which contains the direct links between inputs and outputs is unique,and stability analysis and real-time performance are two difficulties of the control systems based on neural networks.In this paper,combining the advantages of RVFL and the ideas of online sequential extreme learning machine(OS-ELM)and initial-training-free online extreme learning machine(ITFOELM),a novel online learning algorithm which is named as initial-training-free online random vector functional link algo rithm(ITF-ORVFL)is investigated for training RVFL.The link vector of RVFL network can be analytically determined based on sequentially arriving data by ITF-ORVFL with a high learning speed,and the stability for nonlinear systems based on this learning algorithm is analyzed.The experiment results indicate that the proposed ITF-ORVFL is effective in coping with nonparametric uncertainty.

Multiplicative Functions Resembling the Mobius Function

A multiplicative function f is said to be resembling the Mobius function if f is supported on the square-free integers,and f(p)=±1 for each prime p.We prove O-and Ω-results for the summatory function ∑_(n)≤x f(n)for a class of these f,and the point is that these O-results demonstrate cancellations better than the square-root saving.It is proved in particular that the summatory function is O(x^(1/3+ε))under the Riemann Hypothesis.On the other hand it is proved to be Ω(x^(1/4))unconditionally.It is interesting to compare these with the corresponding results for the Mobius function.

Improved deep mixed kernel randomized network for wind speed prediction

Forecasting wind speed is an extremely complicated and challenging problem due to its chaotic nature and its dependence on several atmospheric conditions.Although there are several intelligent techniques in the literature for wind speed prediction,their accuracies are not yet very reliable.Therefore,in this paper,a new hybrid intelligent technique named the deep mixed kernel random vector functional-link network auto-encoder(AE)is proposed for wind speed prediction.The proposed method eliminates manual tuning of hidden nodes with random weights and biases,providing prediction model generalization and representation learning.This reduces reconstruction error due to the exact inversion of the kernel matrix,unlike the pseudo-inverse in a random vector functional-link network,and short-ens the execution time.Furthermore,the presence of a direct link from the input to the output reduces the complexity of the prediction model and improves the prediction accuracy.The kernel parameters and coefficients of the mixed kernel system are optimized using a new chaotic sine–cosine Levy flight optimization technique.The lowest errors in terms of mean absolute error(0.4139),mean absolute percentage error(4.0081),root mean square error(0.4843),standard deviation error(1.1431)and index of agreement(0.9733)prove the efficiency of the proposed model in comparison with other deep learning models such as deep AEs,deep kernel extreme learning ma-chine AEs,deep kernel random vector functional-link network AEs,benchmark models such as least square support vector machine,autoregressive integrated moving average,extreme learning machines and their hybrid models along with different state-of-the-art methods.

Quasimonotone random and stochastic functional differential equations with applications

In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random dynamical system(RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems,such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given.

Pseudorandomness of Camellia-Like Scheme

Luby and Rackoff idealized DES by replacing each round function with one large random function. In this paper, the author idealizes Camellia by replacing each S-box with one small random function, which is named Camellialike scheme. It is then proved that five-round Camellia-like scheme is pseudorandom and eight-round Camellia-like scheme is super-pseudorandom for adaptive adversaries. Further the paper considers more efficient construction of Camellia-like scheme, and discusses how to construct pseudorandom Camellia-like scheme from less random functions.

Various Expressions for Modulus of Random Convexity

We first prove various kinds of expressions for modulus of random convexity by using an L^0（F, R）-valued function＇s intermediate value theorem and the well known Hahn-Banach theorem for almost surely bounded random linear functionals, then establish some basic properties including continuity for modulus of random convexity. In particular, we express the modulus of random convexity of a special random normed module L^0（F, X） derived from a normed space X by the classical modulus of convexity of X.

Workload-based randomization byzantine fault tolerance consensus protocol

This paper introduces a new Byzantine fault tolerance protocol called workload-based randomization Byzantine fault tolerance protocol(WRBFT).Improvements are made to the Practical Byzantine Fault Tolerance(PBFT),which has an important position in the Byzantine Fault consensus algorithm.Although PBFT has numerous ad-vantages,its primary node selection mechanism is overly fixed,the communication overhead of the consensus process is also high,and nodes cannot join and exit dynamically.To solve these problems,the WRBFT proposed in this paper combines node consensus workload and verifiable random function(VRF)to randomly select the more reliable primary node that dominates the consensus.The selection of the nodes involved in the consensus is based on the node workload,and the optimization of the agreement protocol of the PBFT is also based on this.Simulation results show that the WRBFT has higher throughput,lower consensus latency,and higher algorithmic efficiency compared to the PBFT.