In this paper, we give a new proof of the solution of the rational covariance extension problem, an interpolation problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this problem is to parameterize, in useful systems theoretical terms, all rational, (strictly)...

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F -polynomials ∑ j fj−1((x − 1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev t...

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinator...

The resemblance between the Horn-Thompson theorem and a recent theorem by Dacorogna-Marcellini-Tanteri indicates that Schur convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank-one convexity or quasiconvexity. We give in theorem 6.6 simple necessary and sufficient conditions for an isotropic object...

For a set of positive integers A⊆[n], an r-coloring A is rainbow sum-free if it contains no Schur triple. In this paper we initiate the study Erdős-Rothschild problem in context sets, which asks for subsets [n] with maximum number r-colorings. We show that r=3, interval optimal, while r≥8, [⌊n/2⌋,n] optimal. also prove stability theorem r≥4. The proofs rely on hypergraph container method, and s...

In this paper, we consider the boundedness of the Libera operator on Dirichlet spaces in terms of the Schur test. Moreover, we get its point spectrum and norm.

Recent work of A. Postnikov shows that cylindric skew Schur functions, which are a generalisation of skew Schur functions, have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining this motivation, we study cylindric skew Schur functions from the point of view of...