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The purpose of this study is to explore the potential of trainable activation functions to enhance the performance of deep neural networks, specifically ResNet architectures, in the task of image classification. By introducing activation functions that adapt during training, the authors aim to determine whether such flexibility can lead to improved learning outcomes and generalization capabilities compared to static activation functions like ReLU. This research seeks to provide insights into how dynamic nonlinearities might influence deep learning models' efficiency and accuracy in handling complex image data sets.

This research integrates three novel trainable activation functions – CosLU, DELU and ReLUN – into various ResNet-n architectures, where “n” denotes the number of convolutional layers. Using CIFAR-10 and CIFAR-100 data sets, the authors conducted a comparative study to assess the impact of these functions on image classification accuracy. The approach included modifying the traditional ResNet models by replacing their static activation functions with the trainable variants, allowing for dynamic adaptation during training. The performance was evaluated based on accuracy metrics and loss profiles across different network depths.

The findings indicate that trainable activation functions, particularly CosLU, can significantly enhance the performance of deep learning models, outperforming the traditional ReLU in deeper network configurations on the CIFAR-10 data set. CosLU showed the highest improvement in accuracy, whereas DELU and ReLUN offered varying levels of performance enhancements. These functions also demonstrated potential in reducing overfitting and improving model generalization across more complex data sets like CIFAR-100, suggesting that the adaptability of activation functions plays a crucial role in the training dynamics of deep neural networks.

This study contributes to the field of deep learning by introducing and evaluating the impact of three novel trainable activation functions within widely used ResNet architectures. Unlike previous works that primarily focused on static activation functions, this research demonstrates that incorporating trainable nonlinearities can lead to significant improvements in model performance and adaptability. The introduction of CosLU, DELU and ReLUN provides a new pathway for enhancing the flexibility and efficiency of neural networks, potentially setting a new standard for future deep learning applications in image classification and beyond.

This study offers practical insights into how generative artificial intelligence (AI) can enhance responsible manufacturing within the context of Industry 5.0. It explores how manufacturers can strategically maximize the potential benefits of generative AI through a synergistic approach.

The study developed a strategic roadmap by employing a mixed qualitative-quantitative research method involving case studies, interviews and interpretive structural modeling (ISM). This roadmap visualizes and elucidates the mechanisms through which generative AI can contribute to advancing the sustainability goals of Industry 5.0.

Generative AI has demonstrated the capability to promote various sustainability objectives within Industry 5.0 through ten distinct functions. These multifaceted functions address multiple facets of manufacturing, ranging from providing data-driven production insights to enhancing the resilience of manufacturing operations.

While each identified generative AI function independently contributes to responsible manufacturing under Industry 5.0, leveraging them individually is a viable strategy. However, they synergistically enhance each other when systematically employed in a specific order. Manufacturers are advised to strategically leverage these functions, drawing on their complementarities to maximize their benefits.

This study pioneers by providing early practical insights into how generative AI enhances the sustainability performance of manufacturers within the Industry 5.0 framework. The proposed strategic roadmap suggests prioritization orders, guiding manufacturers in decision-making processes regarding where and for what purpose to integrate generative AI.

A system for solving a wide variety of inverse and optimization problems in solid mechanics is introduced. The system consists of a general purpose finite element method (FEM) analysis system “Elfen” and a shell which controls this system. The shell functions as a stand‐alone programme, so the system is physically divided into two separated parts. The “optimization part”, which corresponds to the shell, possesses optimization and inverse problem solution algorithms. The “analysis part”, which corresponds to an FEM system, serves for the definition of the objective function to which these algorithms are applied. The shell has a user interface implemented in the form of file interpreter which imposes a great flexibility at the definition of various optimization and inverse problems, including parameter identification in constitutive modelling, frictional contact problems and heat transfer. Concepts of the shell are discussed in detail.

Presents a procedure for obtaining an improved finite element solution of boundary problems by estimating the principle of exact displacement method in the finite element technique. The displacement field is approximated by two types of functions: the shape functions satisfying the homogeneous differential equilibrium equation and the full clamping element functions as a particular solution of the differential equation between the nodes. The full clamping functions represent the solution of the full clamping state on finite elements. An improved numerical solution of displacements, strains, stresses and internal forces, not only at nodes but over the whole finite element, is obtained without an increase of the global basis, because the shape functions are orthogonal with the full clamping functions. This principle is generally applicable to different finite elements. The contribution of introducing two types of functions based on the principle of the exact displacement method is demonstrated in the solution procedure of frame structures and thin plates.

Defines and investigates fuzzy symmetric functions with don't‐care conditions and most‐unsymmetric functions. Represents and illustrates by examples algorithms for finding the grade of membership function and the number of most unsymmetric functions. Also presents applications to function representation, data reduction and error correction. The results may have useful applications to fuzzy logics, finding most‐unsymmetric images, fuzzy neural networks and related areas.

The purpose of this paper is to discuss the necessary condition of the relative error between continuous function transformation after inverse transformation and original sequence is not larger than the relative error between transformed sequence and its corresponding simulation sequence.

First, explore the function transformation feature of after inverse transformation the relative error not enlarged, then combine this feature with the function transformation feature of not enlarge the class ratio dispersion, not reduce the smoothness which author have got, and obtain a kind of special transformation that not enlarge class ratio dispersion, not reduce the smoothness and after inverse transformation keep the relative not enlarged. Meanwhile, offer the concrete form of this special function type to monotone increasing continuous function transformation and monotone decreasing continuous function transformation, respectively, and study its properties.

This paper finds the concise and important feature of monotonically increasing function transformation after inverse transformation whether the relative error enlarge or not is at first, the concise and important feature of monotonically decreasing function transformation after inverse transformation relative error not enlarged is. And find that the ideal function transformation which both reduces class ratio dispersion strictly and keeps error of inverse transformation not enlarged is non-exist for monotone increasing function transformation and monotone decreasing function transformation.

Use the necessary condition given by this paper, it may use to judge whether function transformation can keep relative error of inverse transformation not enlarged by easy data calculation before build modeling, therefore, choose the best function transformation. These results tell the authors: the paper cannot treat any functions as the same that whether the relative error of inverse transformation will not enlarge (or not reduced), but the authors should divide them into two parts to discuss that it will be expanded in some range or be reduced in some range. It will affect the future direction of the research, not to find the function transform both satisfies the class ratio dispersion reduced and keep the relative error of inverse transformation not enlarged, but to study which kind of function transform will narrow class ratio dispersion in some range, after the modeling accuracy improvement, but after the inverse transformation the relative error enlarged, and at this time the simulation accuracy is still higher than the simulation accuracy of original data modeling directly. Which kind of function transform will expand class ratio dispersion in some range, after the modeling accuracy diminution, but after the inverse transformation the relative error not enlarged, and now the simulation accuracy is still higher than the simulation accuracy of original data modeling directly, too.

Let peers no longer spend energy in seeking the function transformation which both reduce class ratio dispersion and keep relative error of inverse transformation not enlarged. At the same time, also remind peers that even if a function transformation reduces class ratio dispersion greatly, the data modeling accuracy improves a lot after transformation, but the error of inverse transformation is may quite large, still. Besides, even if function transformation increases class ratio dispersion, the data modeling accuracy is not good after transformation, the ideal situation after inverse transformation would occur, and the possibility cannot be excluded.

The purpose of this paper is to provide the general method for the solution of the Boolean function's primary function, and its application in encryption systems.

In the Boolean function theory, there is always a lack in the general method for the solution of primary function. This paper uses a very useful tool – e‐derivative, and by the help of this tool, obtains some very useful theorems.

The paper not only finds the method of solving the primary function of the Boolean function but also provides its application in encryption systems.

The Boolean function's primary function is also very useful in the detection of circuitry, not introduced in this paper.

This paper is very useful for information security.

The paper solves the initial‐value problem of the Boolean function's primary function, and constructs some Boolean functions with many cryptographical characters which are very useful in encryption systems.

The purpose of this paper is to answer the question that what the best shape of fuzzy sets is in fuzzy systems for function approximation which is essential in many applications of fuzzy systems.

The uniform approximation rates indicate the approximating capabilities of fuzzy systems for function approximation. By Fourier analysis, the uniform approximation rates are estimated for the fuzzy systems with various shapes of if‐part fuzzy sets in the case of single‐input and single‐output. Based on the approximation rates, the relationships between the approximating capabilities and the shapes of fuzzy sets are developed and compared.

The since functions as the input membership functions in fuzzy systems are proved to have the almost best approximation property in a particular class of membership functions.

From the viewpoint of function approximation, the input membership functions are not necessarily positive in fuzzy systems.

For engineers, the sinc‐shaped membership function is a good choice to improve their fuzzy systems in real applications.

The uniform approximation rates of fuzzy systems for function approximation are estimated. Mathematically, the relationships between the approximating capabilities and the shapes of fuzzy sets are analyzed for fuzzy systems.

The investigation of the efficiency of optimisation technique based on approximation of objective function by multiquadric (MQ) function, used for induction heating devices was the aim of the paper.

The optimisation package based on Matlab language and using Flux2D commercial program for calculation of electromagnetic and thermal fields was built. It allows the use of different optimisation techniques for induction heating devices, e.g. based on MQ function approximation. In the paper two algorithms of approximated points generating have been tested.

The efficiency of MQ optimisation method strongly depends on the applied algorithm of approximated point generating. To ensure high efficiency of MQ optimisation method, the stochastic element of the algorithm of approximated point generating should have a significant role.

The efficiency of elaborated algorithms of MQ function approximated point generating should be proved in other applications.

The efficient optimisation technique of induction heating devices has been proposed.

The two new algorithms for generation of MQ function approximated points have been proposed. The paper could be useful for designers of induction heating devices.

In the paper piecewise continuous analogues of Liapunov's functions have been introduced for systems of differential equations with impulses. They serve to study the problems for uniqueness, continuity and boundedness of the solutions of nonlinear systems with impulses. The asymptotic behaviour of the bounded solutions has also been considered.