Combinatorial methods for the spectral p-norm of hypermatrices

The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of $2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of $2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to $r$-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for $2$-matrices. The main result of this paper extends the latter result to $r$-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on combinatorial concepts like $r$-partite $r$-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral $p$-norm in general. Thus, another application shows that the spectral $p$-norm and the $p$-spectral radius of a symmetric nonnegative $r$-matrix are equal whenever $p\geq r$. This result contributes to a classical area of analysis, initiated by Mazur and Orlicz around 1930. Additionally, a number of bounds are given on the $p$-spectral radius and the spectral $p$-norm of $r$-matrices and $r$-graphs.Comment: 29 pages. Credit has been given to Ragnarsson and Van Loan for the symmetrant of a matri

Strong Jumps and Lagrangians of Non-Uniform Hypergraphs

The hypergraph jump problem and the study of Lagrangians of uniform hypergraphs are two classical areas of study in the extremal graph theory. In this paper, we refine the concept of jumps to strong jumps and consider the analogous problems over non-uniform hypergraphs. Strong jumps have rich topological and algebraic structures. The non-strong-jump values are precisely the densities of the hereditary properties, which include the Tur\'an densities of families of hypergraphs as special cases. Our method uses a generalized Lagrangian for non-uniform hypergraphs. We also classify all strong jump values for $\{1,2\}$-hypergraphs.Comment: 19 page