A dual logarithmic barrier method for solving large, sparse semidefinite programs is proposed in this paper. The method avoids any explicit use of the primal variable X and therefore is well-suited to problems with a sparse dual matrix S. It relies on inexact Newton steps in dual space which are computed by the conjugate gradient method applied to the Schur complement of the reduced KKT system....

Extended zigzag Schur algebras are quasi-hereditary which conjecturally Morita equivalent to RoCK blocks of classical algebras. We prove that extended Ringel self-dual.

Partial-rogue waves, i.e., waves that ``come from nowhere but leave with a trace", are analytically predicted and numerically confirmed in the Sasa-Satsuma equation. We show that, among class of rational solutions this equation can be expressed through determinants 3-reduced Schur polynomials, partial-rogue would arise if these certain orders, where associated generalized Okamoto polynomials ha...

Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of g...

D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, tog...

In this paper we investigate some matrix structures on C that have a good behaviour under Schur complementation. The first type of structure is closely related to low displacement rank matrices. Next, we show that for a matrix having a low rank submatrix, also the Schur complement must have a low rank submatrix, which we can explicitly determine. This property holds even if the low rank submatr...

A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of cliques that occur as (topological) r-minors. We observe that this tameness notion from algorithmic graph theory is essentially the earlier stability theoretic notion of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence pr...

The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primal-dual interior-point algorithm for the problem, by applying Newton’s method directly to a system of nonlinear e...

The representation theory of Hecke algebras has been an important part toward understanding (ordinary or modular) representation theories of finite groups of Lie type. Schur algebras, as endomorphism algebras, connects the representation theory of general linear groups and the representations of symmetric groups via Schur-Weyl duality. The quantum version of Schur-Weyl duality is established by...