The classical Heawood inequality states that if the complete graph K n on n vertices is embeddable in the sphere with g handles, then g (n − 3)(n − 4) 12 . A higherdimensional analogue of the Heawood inequality is the Kühnel conjecture. In a simplified form it states that for every integer k > 0 there is c k > 0 such that if the union of k-faces of n-simplex embeds into the connected sum of g copies of the Cartesian product S k × S k of two k-dimensional spheres, then g c k n k+1 . For k > 1 only linear estimates were known. We present a quadratic estimate g c k n 2 .
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows.Kolmogorov Theorem. There are continuous functions ϕ 1 , . . . , ϕ 5 : [ 0, 1 ] → [ 0, 1 ] such that for any continuous function f :The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
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