Exponential sampling type neural network Kantorovich operators based on Hadamard fractional integral

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ,m)-convex functions, m-harmonic harmonically convex functions, (s,r)-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α,m)-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ-convex functions, MT-convex functions, (s,m)-convex functions, p-convex functions, h-convex functions, σ-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ,s-convex functions, modified (p,h)-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α,h−m)−p-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k,s)-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

In this paper, we introduce some Simpson’s type integral inequalities via the Katugampola fractional integrals for functions whose first derivatives at certain powers are s-convex (in the second sense). The Katugampola fractional integrals are generalizations of the Riemann–Liouville and Hadamard fractional integrals. Hence, our results generalize some results in the literature related to the Riemann–Liouville fractional integrals. Results related to the Hadamard fractional integrals could also be derived from our results.

In this study, we introduce several new Hermite-Hadamard type general integral inequalities for exponentially (s,m)-convex functions via Katugampola fractional integral. The Katugampola fractional integral is a broader form of the Riemann–Liouville and Hadamard fractional integrals. We utilized the power mean integral inequality, the H¨older inequality and a few additional generalizations to derive these inequalities. Numerous limiting results are derived from the main results presented in the remarks. Furthermore, we provide an example illustrating our theoretical findings, supported by a graphical representation.

In this article, we apply M\onch and Engl's fixed point theorems associated with the technique of measure of weak noncompactness to investigate the existence of random solutions for a class of partial random integral equations via Hadamard's fractional integral, under the Pettis integrability assumption.

In this paper, we establish some new inequalities of expectation and variance of continuous random variables by using the Hadamard fractional integral operator.

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function mathcal{G} in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of omega (theta ) and mathcal{G}(theta ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of omega (theta ) and mathcal{G}(theta ).

In this study, we begin by deriving a two-term strong converse inequality that characterizes the rate of approximation for generalized exponential sampling operators, utilizing the logarithmic moduli of smoothness. Subsequently, we combine the direct and converse estimates to establish the saturation property and identify the class of these approximation operators. Finally, we provide the results for the generalized exponential sampling series using specific kernels that meet the necessary assumptions. It should be noted that in the present study we follow a direct approach within the framework of Mellin analysis in the processes obtained.

To solve the problem of low accuracy of soft sensor of monitoring variables due to strong nonlinear, multivariable coupling, parameters time-varying and large time delays in complex chemical process, a soft sensor modeling method based on self-organizing interval type-2 fuzzy neural network (SOIT2FNN) is proposed. Firstly, to solve the problem that the structure of interval type 2 neural network (IT2FNN) is difficult to determine, an algorithm for self-organizing generation rules that uses firing strength and rule similarity to define growth and deletion indicators is proposed. The algorithm uses the firing strength to determine whether to generate rules, and deletes the rules according to the similarity, thereby determining the structure of the interval type-2 fuzzy neural network. Secondly, the relevant parameters of the SOIT2FNN model are corrected by the gradient descent algorithm. Finally, SOIT2FNN is used as a soft sensor model to detect the tail oxygen concentration for uncatalysed oxidation of cyclohexane process. The experimental results show that the soft sensor method based on SOIT2FNN model has the advantages of timely detection and high detection accuracy.

With the weight of ‐Baskakov basis function, this article considers the approximation properties of the integral modification of exponential type operators. The moments are estimated, and the Voronovskaja‐type result is established in simultaneous approximation. Additionally, the power series method is used for estimating convergence and error estimation. For some functions, the convergence of these operators has also been graphically illustrated.

High-frequency trading (HFT) requires fast data processing without information lags for precise stock price forecasting. This high-paced stock price forecasting is usually based on vectors that need to be treated as sequential and time-independent signals due to the time irregularities that are inherent in HFT. A well-documented and tested method that considers these time irregularities is a type of recurrent neural network (NN), named long short-term memory (LSTM) NN. This type of NN is formed based on cells that perform sequential and stale calculations via gates and states without knowing whether their order, within the cell, is optimal. In this article, we propose a revised and real-time adjusted LSTM cell that selects the best gate or state as its final output. Our cell is running under a shallow topology, has a minimal look-back period, and is trained online. This revised cell achieves lower forecasting error compared to other recurrent NNs (RNNs) for online HFT forecasting tasks such as the limit order book (LOB) mid-price (MP) prediction as it has been tested on two high-liquid U.S. and two less-liquid Nordic stocks.

Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations

In the present article, we derive some approximation results concerning the order of convergence for a family of Durrmeyer type exponential sampling operators. Further, we improve the rate of approximation by constructing linear combinations of these operators. At the end, we provide a few examples of the kernel functions to which the presented theory can be applied along with the graphical representation.

In the last decade, accurate identification of motor unit (MU) firings received a lot of research interest. Different decomposition methods have been developed, each with its advantages and disadvantages. In this study, we evaluated the capability of three different types of neural networks (NNs), namely dense NN, long short-term memory (LSTM) NN and convolutional NN, to identify MU firings from high-density surface electromyograms (HDsEMG). Each type of NN was evaluated on simulated HDsEMG signals with a known MU firing pattern and high variety of MU characteristics. Compared to dense NN, LSTM and convolutional NN yielded significantly higher precision and significantly lower miss rate of MU identification. LSTM NN demonstrated higher sensitivity to noise than convolutional NN.Clinical Relevance-MU identification from HDsEMG signals offers valuable insight into neurophysiology of motor system but requires relatively high level of expert knowledge. This study assesses the capability of self-learning artificial neural networks to cope with this problem.

The purpose of this study is to familiarise the reader with the foundations of neural networks. Artificial Neural Networks (ANNs) are algorithm-based systems that are modelled after Biological Neural Networks (BNNs). Neural networks are an effort to use the human brain's information processing skills to address challenging real-world AI issues. The evolution of neural networks and their significance are briefly explored. ANNs and BNNs are contrasted, and their qualities, benefits, and disadvantages are discussed. The drawbacks of the perceptron model and their improvement by the sigmoid neuron and ReLU neuron are briefly discussed. In addition, we give a bird's-eye view of the different Neural Network models. We study neural networks (NNs) and highlight the different learning approaches and algorithms used in Machine Learning and Deep Learning. We also discuss different types of NNs and their applications. A brief introduction to Neuro-Fuzzy and its applications with a comprehensive review of NN technological advances is provided.