A short and simple proof of necessity in the McCullough-Quiggin characterization of positive semi-definite kernels with the complete Pick property is presented.
We outline a simple proof of Hulanickis theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilberts 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $varphi_1,ldots,varphi_5 : [,0, 1,]to [,0,1,]$ such that for any continuous function $f: [,0,1,]^2tomathbb R$ there is a continuous function $h: [,0,3,]tomathbb R$ such that for any $x,yin [,0, 1,]$ we have $$f(x,y)=sumlimits_{k=1}^5 hleft(varphi_k(x)+sqrt{2},varphi_k(y)right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given in terms of Hankel determinants of the large and small Schroder numbers.
We give a simple proof of the exponential de Finetti theorem due to Renner. Like Renners proof, ours combines the post-selection de Finetti theorem, the Gentle Measurement lemma, and the Chernoff bound, but avoids virtually all calculations, including any use of the theory of types.
Schoenberg showed that a function $f:(-1,1)rightarrow mathbb{R}$ such that $C=[c_{ij}]_{i,j}$ positive semi-definite implies that $f(C)=[f(c_{ij})]_{i,j}$ is also positive semi-definite must be analytic and have Taylor series coefficients nonnegative at the origin. The Schoenberg theorem is essentially a theorem about the functional calculus arising from the Schur product, the entrywise product of matrices. Two important properties of the Schur product are that the product of two rank one matrices is rank one, and the product of two positive semi-definite matrices is positive semi-definite. We classify all products which satisfy these two properties and show that these generalized Schur products satisfy a Schoenberg type theorem.