Formal semantics of programming languages represents a basic formal method connecting theoretical foundations of computer science with the development of reliable software. It allows for rigorous analysis of the correctness and properties of programs independently of the implementation. When formulating it, it is necessary to respect the principles of the programming paradigm, which determines the basic abstractions of the language. At the same time, there are unifying formalisms that allow for systematic comparison of semantic definitions across paradigms. Category theory represents one of the most important such approaches. We formalize this perspective by introducing a general categorical model of state transformations and show how different programming paradigms can be uniformly represented and related within this framework via structure-preserving mappings.
This study explores the Laplacian spectral characteristics of the Cartesian product graph of K3 square Kg, with particular emphasis on the trace, energy, and the coefficients of its characteristic polynomial. We establish general expressions for the trace of powers of the Laplacian matrix and develop recursive formulas for the Laplacian coefficients using trace-based identities. The findings are validated for various values of g, and generalized to arbitrary cases. Additionally, we derive upper bounds and lower bounds for the Laplacian eigenvalues of the graph K3 square Kg
The presence of outliers can significantly reduce the reliability of conventional estimators, often resulting in biased estimation of the population mean. This challenge is particularly relevant in practical survey data, where irregular observations frequently occur. Although this issue is well-recognized, limited work has been done on population mean estimation under stratified random sampling when outliers are present, especially within a quantile regression framework. To address this gap, the present study introduces a new class of robust estimators based on quantile regression. The proposed estimators make use of non-conventional auxiliary information to improve estimation accuracy under a stratified random sampling scheme. By relying on quantile-based methods, the suggested approach provides greater resistance to the influence of outliers. The statistical properties of the proposed estimators are derived analytically, including measures of bias and mean squared error. In addition, their performance is examined through a real-life dataset. The findings indicate that the proposed estimators offer notable gains in efficiency and robustness compared to the adopted estimators, particularly in datasets affected by outliers.
In this study, we employ two efficient analytical techniques, the simplest equation approach and the tanh-coth method to investigate the (1+2) dimensional nonlinear chiral Schr & ouml;dinger's equation. In condensed matter physics and nonlinear optics, this equation is essential, describing the propagation of light in materials with significant nonlinearities and chiral properties. By employing the simplest equation method, we derive exact solutions that reveal the complex relationship between nonlinearity and chirality. Additionally, the tanh-coth method yields exact soliton solutions, enhancing our understanding of soliton dynamics in chiral media. Our results provide new insights into the behavior of nonlinear waves in (1+2) dimensions and demonstrate the effectiveness of these methods in solving complex nonlinear equations. We present analytical solutions for bright and dark solitons. We also discuss the variables of the system in the following section and conditions enabling various solutions, including singular and periodic solitons.
The need for photorealistic rendering of glittery materials, motivated by applications in gaming, film, virtual reality, and particularly automotive visualization, requires real-time techniques to depict millions of minute flakes in paint coatings. Existing approaches often rely on a single grid for glint mapping, leading to repetitive patterns, aliasing, and noticeable popping during camera movement. Addressing these issues is essential for achieving convincing sparkle under varying lighting and viewpoints.We propose a new real-time method that integrates three procedurally generated microflake grids, each with multiple resolution levels, to ensure smooth transitions between levels of detail and eliminate tiling artifacts. Adaptive scaling dynamically adjusts sparkle size and brightness based on camera distance, while per-flake light-color interactions enhance reflection realism. Our custom shader framework, paired with an artist-friendly material editor, enables real-time adjustments to flake density, size, falloff, and color.Compared to single-grid or fixed-scale methods, our multi-grid approach delivers dense, non-repetitive sparkles with consistent quality across distances, albeit with higher GPU demands in complex scenes. The controllable and diverse output of our system also makes it well-suited for generating synthetic training data for AI models focused on material recognition, lighting estimation, or view-dependent appearance. Future work will explore deferred rendering, 3D noise textures, optimized shader branching, and temporal filtering to enhance performance and visual quality.
