Manin conjecture for statistical pre-Frobenius manifolds, hypercube relations and motivic Galois group in coding
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This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of Eilenberg–Moore algebras over a Giry monad, the collection Cap_n of all probability distributions on the measurable space (Ω_n, 𝒜) (where Ω is discrete with n issues) and it turns out that there exists an embedding relation of Segre type among the product of Cap_n's. We unravel hidden symmetries of these type of embeddings and show that there exists a hypercubic relation. Secondly, we show that the Manin conjecture – initially defined concerning the diophantine geometry of Fano varieties – is true in the case of exponential statistical manifolds, defined over a discrete sample space. Thirdly, we introduce a modified version of the parenthesised braids (𝐦𝐏𝐚𝐁), which forms a key tool in code-correction. This modified version 𝐦𝐏𝐚𝐁 presents all types of mistakes that could occur during a transmission process. We show that the standard parenthesised braids 𝐏𝐚𝐁 form a full subcategory of 𝐦𝐏𝐚𝐁. We discuss the role of the Grothendieck–Teichmüller group in relation to the modified parenthesised braids. Finally, we prove that the motivic Galois group is contained in the automorphism Aut(𝐦𝐏𝐚𝐁). We conclude by presenting an open question concerning rational points, Commutative Moufang Loops and information geometry.
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