Mathematical Researches

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Zobov’s Theorem is one of the theorems which indicate the conditions for the stability of a nonlinear system with specific attraction region. We have applied neural networks to approximate some functions mentioned in Zobov’s theorem in order to find the controller of a nonlinear controlled system whose law in a mathematical manner is difficult to make. Finally, the effectiveness and the applicability of the proposed method are demonstrated through using numerical examples.

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Due to the dynamic structure and nonlinear fluctuations of the stock market, it is difficult to accurately predict the trend of this market using the old methods. In this study, in order to improve the accuracy of predicting the index trend in different industries, we propose a new algorithm that combines algorithms fractal interpolation and support vector machine regression, abbreviated as fracsion algorithm. . For this purpose, after recognizing the fractal structure of industries using the Hurst exponent of each industry, we consider the value of the index in each fractal industry as the primary data to predict the trend of the index. Then, by modifying the fractal interpolation algorithm, we will generate new data, and finally, by calling the support vector regression algorithm on the obtained data, we will predict the index trend. The results of the implementation of the Hybrid fracsion algorithm and its comparison with two conventional methods, namely artificial neural network and support vector machine regression, indicate the superiority of the predictive accuracy of the proposed algorithm.

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Introduction
As the Forex market becomes increasingly complex, accurate trend forecasting has gained critical importance for traders and researchers. Unlike most studies that focus on price prediction, this paper introduces a novel bi-timeframe framework (1-hour and 4-hour) that integrates the Ichimoku Kinko Hyo strategy with deep learning models to predict directional movements in currency pairs. 

Materials and Methods
The approach employs convolutional neural networks (CNNs) and hybrid architectures (CNN-LSTM, CNN-GRU), with hyperparameters optimized using the Particle Swarm Optimization Algorithm (PSO). Models are trained on historical EURUSD data (2019--2024) from MetaTrader5 and evaluated on eight highly correlated ($pm$80%) currency pairs. Due to the limitations of regression metrics (MAE, MSE, MAPE) in trading contexts, regression outputs are used solely for 4-hour trend classification, with Accuracy and F1-score as primary performance measures. 

Results and Discussion
Results show that PSO-optimized models, particularly Ichimoku-CNN-GRU-PSO (ICGP), consistently outperform standard variants, achieving the highest Accuracy (up to 80.23% on USDSGD) and F1-score across most pairs. 

Conclusion
The findings confirm that Ichimoku-based features, combined with hybrid deep learning and metaheuristic optimization, significantly enhances trend forecasting reliability and generalization in volatile financial markets.

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In this paper, a new numerical method based on the neural network and the discrete Hahn wavelets is presented for solving the Bratu-type equations. First, the discrete Hahn wavelets and some of their important properties are expressed to construct the Hahn wavelets neural network. The presented neural network consists of three layers: input layer, hidden layer, output layer. In this method, the Hahn wavelets and sinh functions are used as the activation functions of the hidden layer and output layer, respectively. Then, an analytical optimization method is employed to adjust the neural network weights so that the approximated solution satisfies the given problem. Using this method, the original problem is converted into a system of algebraic equations, which is solved by Newton’s iterative method. Finally, the accuracy and efficiency of the proposed method are examined through several numerical examples.