The interdisciplinary workshops, focused on methodological and philosophical aspects, have helped - and still undoubtedly are helping - in forging links among different members of the academic community or research teams, which are today described as "intellectual networks" or "invisible colleges". Their focus of interest is known to transcend the boundaries of the traditionally divided scientific disciplines or research areas. That is also why the network that was instrumental in shaping the genesis of cybernetics includes, to this day, the names of C.Shannon, the pioneer of the mathematical theory of information, J.von Neumann, the founder of the theory of games and decision-making, linguist R.Jakobson, the above-mentioned specialists in the medicine- and biology-oriented branches, and many other scholars. Similar conceptions and aspirations connected therewith also proved to be conducive to the emergence and expansion of the works and studies devoted to the role of the sign, its creation, significance and function in communication, i.e. semantics and semiotics. Efforts were made to uncover more profound links and subsequently to outline paths leading to a unification of different scientific domains, particularly by integrating the language of science. There was a mounting interest in methodological and epistemological problems and - generally speaking - in finding ways and means of attaining more profound and thorough learning. All these and similar tendencies had and still have one common trait: they kept enhancing respect, weight and significance attached to mathematics, to mathematical methods of expressing and depicting problems, and to mathematical thinking in general.
The entropy region is a fundamental object of study in mathematics, statistics, and information theory. On the one hand, it involves pure group theory, governing inequalities satisfied by subgroup indices, whereas on the other hand, computing network coding capacities amounts to a convex optimization over this region. In the case of four random variables, the points in the region that satisfy the Ingleton inequality (corresponding to abelian groups and to linear network codes) form a well-understood polyhedron, and so attention has turned to Ingleton-violating points in the region. How far these points extend is measured by their Ingleton score, where points with positive score are Ingleton-violating. The Four-Atom Conjecture stated that the Ingleton score cannot exceed 0.089373, but this was disproved by Matúš and Csirmaz. In this paper we employ two methods to investigate Ingleton-violating points and thereby produce the currently largest known Ingleton scores. First, we obtain many Ingleton-violating examples from non-abelian groups. Factorizability appears in many of those and is used to propose a systematic way to produce more. Second, we rephrase the problem of maximizing Ingleton score as an optimization question and introduce a new Ingleton score function, which is a limit of Ingleton scores with maximum unchanged. We use group theory to exploit symmetry in these new Ingleton score functions and the relations between them. Our approach yields some large Ingleton scores and, using this methodology, we find that there are entropic points with score 0.09250007770, currently the largest known score., Nigel Boston and Ting-Ting Nan., and Obsahuje bibliografické odkazy