In this paper we settle a weak version of a conjecture (i.e. Conjecture 6) by Mills, Robbins and Rumsey in the paper “Self-complementary totally symmetric plane partitions” J. Combin. Theory Ser. A 42, 277–292. In other words we show that the number of shifted plane partitions invariant under the involution γ is equal to the number of alternating sign matrices invariant under the vertical flip....

Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q) numerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed ...

Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q) numerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed ...

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quant...

This brief deals with the asymptotic stability of a class of linear time-invariant systems subject to point constant uncommensurate delays. Results are obtained dependent on and independent of the delays which may be tested be performing a finite set of 1-D Routh–Hurwitz tests on a corresponding set of auxiliary delay-free linear time-invariant systems plus some supplementary conditions related...

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on “transfer principles...

In the Conley index theory, transition matrices are used to detect bifurcations of codimension one connecting orbits in a Morse decomposition of an isolated invariant set. Here we shall give a new axiomatic definition of the transition matrix in order to treat several existing formulations of transition matrices in a unified manner.

Finding the best weights of the state variables and the control variables in the objective function of a linear-quadratic control problem is considered. The weights of these variables are considered as two diagonal matrices with appropriate size and so the objective function of the control problem becomes a function of the diagonal elements of these matrices. The optimization problem which is d...

Given the set of matrix pairs M ⊂ Mm,n(C) × Mn(C) keeping a subspace S ⊂ C invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of M. It generalizes the known results when S is a supplementary subspace of the unobservable one. Keywords-Conditioned invariant subspaces, Miniversal deformation, Stratified manifold, Vertical pairs of matrices.

The componentwise stability of a linear system is a special type of asymptotic stability induced by the existence of exponentially decreasing rectangular sets that are invariant with respect to the free response. An interval system ( ) ( ) ( ) x t Ax t Bu t = + , [ , ] A A A − + ∈ , [ , ] B B B − + ∈ , is componentwise stabilizable if there exists a constant feedback ( ) ( ) u t Fx t = that ens...