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
During a survey of the coccidian parasites of reptiles, caryosporan oocysts were found in the faeces of wild and captive European viperid snakes Vipera berus (L.) and V. ammodytes (L.). Thirty two of 37 examined V. herus (86%) and 9 of 17 examined V. ammodytes (53%) specimens were found to be passing caryosporan oocysts. Morphological characters of all caryosporan isolates were identical and fitted well with the description of Caryospora simplex Léger, 1904. Experimental inoculation of severe combined immunodeficient (SC1D) mice with seven isolates of C. simplex from V. berus or V. ammodytes confirmed the heteroxenous life cycle pattern, for the first time for isolates of evidently European origin. Caryosporan developmental stages were observed in the connective tissues of the nose, cheeks, ear and scrotum in all inoculated SCID mice. V. berus and V. ammodytes represent new hosts for C. simplex. The present paper represents the first widely based report on coccidian parasites of the genus Caryospora Léger in European viperids. Our findings indicate a wide distribution of C. simplex throughout the range of distribution of snakes of the genus Vipera.