Power Partial Isometry Index and Ascent of a Finite Matrix

We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$-by-$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power partial isometry index $p(A)$ of a matrix $A$ is the supremum, possibly infinity, of nonnegative integers $j$ such that $I, A, A^2, \ldots, A^j$ are all partial isometries while the ascent $a(A)$ of $A$ is the smallest integer $k\ge 0$ for which $\ker A^k$ equals $\ker A^{k+1}$. It was known before that, for any matrix $A$, either $p(A)\le\min\{a(A), n-1\}$ or $p(A)=\infty$. In this paper, we prove more precisely that there is an $n$-by-$n$ matrix $A$ such that $p(A)=j$ and $a(A)=k$ if and only if one of the following conditions holds: (a) $j=k\le n-1$, (b) $j\le k-1$ and $j+k\le n-1$, and (c) $j\le k-2$ and $j+k=n$. This answers a question we asked in a previous paper.Comment: 11 page

2-irreducible and strongly 2-irreducible ideals of commutative rings

An ideal I of a commutative ring R is said to be irreducible if it cannot be written as the intersection of two larger ideals. A proper ideal I of a ring R is said to be strongly irreducible if for each ideals J, K of R, J\cap K\subseteq I implies that J\subset I or K\subset I. In this paper, we introduce the concepts of 2-irreducible and strongly 2-irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. We say that a proper ideal I of a ring R is 2-irreducible if for each ideals J, K and L of R, I= J\cap K\cap L implies that either I=J\cap K or I=J\cap L or I=K\cap L. A proper ideal I of a ring R is called strongly 2-irreducible if for each ideals J, K and L of R, J\cap K\cap L\subseteq I implies that either J\cap K\subseteq I or J\cap L\subseteq I or K\cap L\subseteq I.Comment: 15 page

Lattice-point generating functions for free sums of convex sets

Let \J and \K be convex sets in $\R^{n}$ whose affine spans intersect at a single rational point in \J \cap \K, and let \J \oplus \K = \conv(\J \cup \K). We give formulas for the generating function {equation*} \sigma_{\cone(\J \oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K) \cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points in all integer dilates of \J \oplus \K in terms of \sigma_{\cone \J} and \sigma_{\cone \K}, under various conditions on \J and \K. This work is motivated by (and recovers) a product formula of B.\ Braun for the Ehrhart series of \P \oplus \Q in the case where $\P$ and \Q are lattice polytopes containing the origin, one of which is reflexive. In particular, we find necessary and sufficient conditions for Braun's formula and its multivariate analogue.Comment: 17 pages, 2 figures, to appear in Journal of Combinatorial Theory Series