Fractional Calculus as Modelling Tool

Cancer is a complex disease, responsible for a significant portion of global deaths. The increasing prioritisation of know-why over know-how approaches in biological research has favoured the rising use of both white- and black-box mathematical techniques for cancer modelling, seeking to better grasp the multi-scale mechanistic workings of its complex phenomena (such as tumour-immune interactions, drug resistance, tumour growth and diffusion, etc.). In light of this wide-ranging use of mathematics in cancer modelling, the unique memory and non-local properties of Fractional Calculus (FC) have been sought after in the last decade to replace ordinary differentiation in the hypothesising of FC’s superior modelling of complex oncological phenomena, which has been shown to possess an accumulated knowledge of its past states. As such, this review aims to present a thorough and structured survey about the main guiding trends and modelling categories in cancer research, emphasising in the field of oncology FC’s increasing employment in mathematical modelling as a whole. The most pivotal research questions, challenges and future perspectives are also outlined.publishersversionpublishe

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Individual-Based Modeling and Data Analysis of Ecological Systems Using Machine Learning Techniques

Artificial life (Alife) studies the logic of living systems in an artificial environment in order to gain a deeper insight of the complex processes and governing rules in such systems. EcoSim, an Alife simulation for ecological modeling, is an individual-based predator-prey ecosystem simulation and a generic platform designed to investigate several broad ecological questions, as well as long-term evolutionary patterns and processes in biology and ecology. Speciation and extinction of species are two essential phenomena in evolutionary biology. Many factors are involved in the emergence and disappearance of species. Due to the complexity of the interactions between different factors, such as interaction of individuals with their environment, and the long time required for the observation, studying such phenomena is not easy in the real world. Using data sets obtained from EcoSim and machine learning techniques, we predicted speciation and extinction of species based on numerous factors. Experimental results showed that factors, such as demographics, genetics, and environment are important in the occurrence of these two events in EcoSim.We identified the best species-area relationship (SAR) models, using EcoSim, along with investigating how sampling approaches and sampling scales affect SARs. Further, we proposed a machine learning approach, based on extraction of rules that provide an interpretation of SAR coefficients, to find plausible relationships between the models\u27 coefficients and the spatial information that likely affect SARs. We found the power function family to be a reasonable choice for SAR. Furthermore, the simple power function was the best ranked model in nested sampling amongst models with two coefficients. For some of the SAR model coefficients, we obtained clear correlations with spatial information, thereby providing an interpretation of these coefficients. Rule extraction is a method to discover the rules explaining a predictive model of a specific phenomenon. A procedure for rule extraction from Random Forest (RF) is proposed. The proposed methods are evaluated on eighteen UCI machine learning repository and four microarray data sets. Our experimental results show that the proposed methods outperform one of the state-of-the art methods in terms of scalability and comprehensibility while preserving the same level of accuracy

Three Risky Decades: A Time for Econophysics?

Our Special Issue we publish at a turning point, which we have not dealt with since World War II. The interconnected long-term global shocks such as the coronavirus pandemic, the war in Ukraine, and catastrophic climate change have imposed significant humanitary, socio-economic, political, and environmental restrictions on the globalization process and all aspects of economic and social life including the existence of individual people. The planet is trapped—the current situation seems to be the prelude to an apocalypse whose long-term effects we will have for decades. Therefore, it urgently requires a concept of the planet's survival to be built—only on this basis can the conditions for its development be created. The Special Issue gives evidence of the state of econophysics before the current situation. Therefore, it can provide excellent econophysics or an inter-and cross-disciplinary starting point of a rational approach to a new era

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal