Introduction
Tom 109 / 2016
Streszczenie
The First International Conference Arithmetic Methods in Mathematical Physics and Biology was held on August 3–8, 2014, in the Mathematical Research and Conference Center of the Institute of Mathematics of Polish Academy of Sciences, at Będlewo, Poland. We organized this conference with the aim of bringing classical areas of mathematics, such as arithmetic, algebra, algebraic number theory, fractal geometry to the attention of mathematical physicists and biologists to establish some novel tools for research on physical and biological complexity. The state-of-the-art lectures defined complexity as a consequence of interactions within a complex system. Some general principles, such as the Complexity Correspondence Principle, were proposed starting from the formal mathematics underlying the theory of many-body physical systems. This principle suggests that the system can only be designed and controlled by a second system of equal or greater complexity. A number of auspicious arithmetic approaches to a variety of complexity issues were presented. For example, growth and self-organization of cells was \hbox{modeled} by cellular automata. The universal circular fractal model of adenocarcinomas enabled the stratification of prostate carcinomas into the classes of equivalence defined by the cut-off values of the global capacity fractal dimension D0 and, in consequence, the quantitative evaluation of tumor aggressiveness by complexity measures. Both Galois theory and algebraic number theory were shown to be useful for Bethe Ansatz, the quantum mechanics problem with important applications to Quantum Computing. $p$-adic numbers were applied to solve problems in probability theory, dynamical systems, cryptography, and cognitive science. Arithmetic occurred to be very useful in Biometry. The next Conference is scheduled on August 5–11, 2018 at Będlewo.
