We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear functional of a linear least...

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least s...

the author studies the $bf r$$g$-module $a$ such that $bf r$ is an associative ring‎, ‎a group $g$ has infinite section $p$-rank (or infinite 0-rank)‎, ‎$c_{g}(a)=1$‎, ‎and for every‎ ‎proper subgroup $h$ of infinite section $p$-rank (or infinite 0-rank respectively) the quotient module $a/c_{a}(h)$ is‎ ‎a finite $bf r$-module‎. ‎it is proved that if the group $g$ under‎ ‎consideration is local...

Let S = K[x1, . . . , xn] be a polynomial ring over a field K, and E = ∧ 〈y1, . . . , yn〉 an exterior algebra. The linearity defect ldE(N) of a finitely generated graded E-module N measures how far N departs from “componentwise linear”. It is known that ldE(N) < ∞ for all N . But the value can be arbitrary large, while the similar invariant ldS(M) for an S-module M is always at most n. We will ...

We are grateful that Hastie points out the connection to degrees of freedom for LARS which leads to another—and often better—definition of degrees of freedom for boosting in generalized linear models. As Hastie writes and as we said in the paper, our formula for degrees of freedom is only an approximation: the cost of searching, for example, for the best variable in componentwise linear least s...

We present a novel method to compute componentwise transient bounds, componentwise ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an in...

If I is a monomial ideal with linear quotients, then it has componentwise quotients. However, the converse of this statement an open question. In paper, we provide two classes ideals for which holds. First class polymatroidal in K[x, y] and second one strong exchange property.