We propose in this paper a new normal form for dynamical systems or vector fields which improves the classical normal forms in the sense that it is a further reduction of the classical normal forms. We give an algorithm for an effective computation of these normal forms. Our approach is rational in the sense that if the coefficients of the system are in a field K (which, in practice, is Q,R), s...

A sign pattern is a matrix with entries in {+, −, 0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns that are not full, as well as potentiall...

We classify all polynomial maps of the form [Formula: see text] in case when Jacobian matrix is nilpotent and highest degree no more than 1. In addition, we generalize structure to text].

assume that $(n,l)$, is a pair of finite dimensional nilpotent lie algebras, in which $l$ is non-abelian and $n$ is an ideal in $l$ and also $mathcal{m}(n,l)$ is the schur multiplier of the pair $(n,l)$. motivated by characterization of the pairs $(n,l)$ of finite dimensional nilpotent lie algebras by their schur multipliers (arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any matrix. These called ad-nilpotent ideals and there Catalan number such subspaces. Each ideal I meets a unique largest nilpotent orbit OI in algebra all matrices. main result is that an equivalence relation on studied by Mizuno others, classes OI=O for fixed O. We include two appl...

We define the linear capacity of an algebraic cone, give basic properties of the notion and new formulations of certain known results of the Matrix Theory. We derive in an explicit way the formula for the linear capacity of an irreducible component of the zero cone of a quadratic form over an algebraically closed field. We also give a formula for the linear capacity of the cone over the conjuga...

A matrix T ∈ Mn(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a cru...

It is shown that a matrix satisfying a certain spectral condition which has an infinite sequence of accretive powers is unitarily similar to the direct sum of a normal matrix and a nilpotent matrix. If the sequence of exponents is forcing or semiforcing then the spectral condition is automatically satisfied . If, further, the index of 0 as an eigenvalue of A is at most 1 or the first term of th...