Physical Mathematics

Counting independent sets is a fundamental problem in combinatorics with applications in various fields such as computer science, graph theory and statistical physics. In this article, we delve into the counting rules for computing the number of independent sets in graphs. We explore different techniques and algorithms employed to efficiently determine the count of independent sets, discussing their applications and implications in diverse domains.

Metaheuristic algorithms have emerged as powerful tools for solving optimization problems across various domains. These algorithms offer innovative approaches to finding high-quality solutions, often outperforming traditional optimization techniques. In this article, we delve into the realm of metaheuristic algorithms, exploring their principles, applications and comparative advantages. We discuss several prominent metaheuristic algorithms, including genetic algorithms, simulated annealing, particle swarm optimization and ant colony optimization. By understanding these algorithms' underlying mechanisms and characteristics, practitioners can effectively apply them to tackle complex optimization challenges.

Nanofluids, colloidal suspensions of nanoparticles in base fluids, exhibit fascinating thermophysical properties that have garnered significant attention in various fields, particularly in thermal engineering and nanotechnology. Accurate prediction of these properties is crucial for their effective utilization in applications such as heat transfer enhancement, cooling systems and advanced manufacturing processes. Traditional methods for predicting nanofluids properties often face challenges due to the complex interactions between nanoparticles and base fluids. In recent years, artificial intelligence (AI) techniques have emerged as promising tools for predicting the thermophysical properties of nanofluids. This article provides a comprehensive review of the application of AI methods, including machine learning and deep learning, in predicting the thermophysical properties of nanofluids. The review explores various AI algorithms, data sources and modelling approaches employed in this domain, highlighting their advantages, limitations and future prospects.

The Monge–Ampère equation, stemming from classical geometry, has found significant applications across various fields, including differential geometry, optimal transportation and even image processing. The solvability criterion for systems arising from Monge–Ampère equations plays a pivotal role in understanding the existence and uniqueness of solutions to these equations. In this article, we delve into the mathematical intricacies of this criterion, exploring its theoretical foundations and practical implications. We begin by providing an overview of the Monge–Ampère equation and its significance. Subsequently, we delve into the formulation of systems derived from this equation and elucidate the solvability criterion, discussing its mathematical underpinnings and implications in diverse contexts. Through concrete examples and applications, we illustrate the relevance and utility of this criterion in various mathematical and applied domains.