Nowadays symmetrical patterns are widely used in various industries, such as jewelry design, carpet design, patterns on wallpaper, and textile design. During the design stage, designers perform most of the work manually. Therefore, the development of methods for symmetrical pattern generation is beneficial. In this paper, we are going to present some methods for generating symmetrical patterns using the discrete dynamical system. For this, the discrete dynamical system is considered as a standard iterative method, and then the general algorithm applied for Polynomiography is used for this to generate patterns. Through phase portrait, we analyze the conditions of the existence of some conventional symmetries. Several non-standard iterative methods can be employed to create a variety of visually appealing patterns. These methods include Mann iteration, Ishikawa iteration, and S-iteration which we use them. Through numerous examples, it is demonstrated that by manipulating the parameters and coefficients, it is possible to generate beautiful symmetrical patterns that have potential artistic applications.

Our contribution consists of studying numerical methods based on finite element space and finite difference schema in time of the linear one-dimensional thermoelastic Bresse system with second sound. We establish some a priori error estimates, and present some numerical analysis results of discrete energy under different decay rate profiles. Moreover, we study the behaviors of discrete energy with respect to the system parameters and the initial data. Some numerical simulations will be given in order to validate the theoretical results.

In this paper, a new image watermarking method in Discrete Fourier Transform (DCT) domain based on Fuzzy logic and human vision system is proposed. In first stage, input image is divided into 8*8 blocks and then the Fuzzy system allocates a coefficient for each block based on texture features and illumination which results in more robustness and clarification in watermark process. Next, by calculation of Discrete Fourier Transform (DFT) for each block, the resulting magnitude and phase are separated from each other and watermark is hidden in magnitude. The process of watermark extraction is blind which corresponds to higher security. The shortcoming of the general watermark system DCT-based is that the human vision system is neglected and therefore in order to prevent degradation of image quality, less robustness is resulted. The experimental results show that the proposed method provides more illumination and more robustness against cutting attacks, brightness and darkness of image, additive noise, image filtering and compression in comparison with popular DCT-based methods.