In this paper, we introduce the class of ideals with $(d_1,ldots,d_m)$-linear quotients generalizing the class of ideals with linear quotients. Under suitable conditions we control the numerical invariants of a minimal free resolution of ideals with $(d_1,ldots,d_m)$-linear quotients. In particular we show that their first module of syzygies is a componentwise linear module.

In this paper we devote to generalizing some results of componentwise linear modules over a polynomial ring to the ones over a Koszul algebra. Among other things, we show that the i-linear strand of the minimal free resolution of a componentwise linear module is the minimal free resolution of some module which is described explicitly for any i ∈ Z. In addition we present some theorems about whe...

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...

In the second part of this paper we study condition numbers with respect to componentwise perturbations in the input data for linear systems and for matrix inversion, and the distance to the nearest singular matrix. The structures under investigation are linear structures, namely symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Ha...

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ standard graded $S$-algebra. In terms of Gröbner basis defining ideal $J$ we give condition, called $x$-condition, which implies that all components $A_k$ have linear quotients with additional assumptions are componentwise linear. A typical example such an algebra is Rees $\mathcal{R}(I)$ or symmetric $\textrm{Sym}(M)$ module...