arXiv subjects
Publications and source records attributed to Vania Mascioni.
Let $T=α_0 I + α_1 D + ...+α_n D^n$, where $D$ is the differentiation operator and $α_0\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-α_1/α_0$.
Let $\cP_n$ be the complex vector space of all polynomials of degree at most $n$. We give several characterizations of the linear operators $T\in\cL(\cP_n)$ for which there exists a constant $C > 0$ such that for all nonconstant $p\in\cP_n$ there exist a root $u$ of $p$ and a root $v$ of $Tp$ with $|u-v|\leq C$. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of $p$ and $Tp$, the roots are never displaced by more than a uniform constant independent on $p$. We show that such ``good'' operators $T$ are exactly the invertible elements of the commutative algebra generated by the differentiation operator. We provide upper bounds in terms of $T$ for the relevant constants.
We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots. In fact, endowing the space of monic polynomials of a fixed degree $n$ and the space of $n$ roots with suitable topologies, we are able to formulate the classical theorems in the form of a homeomorphism. Related topological facts are also considered.
The aim of this paper is to characterize those linear maps from a von Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital *-antihomomorphism.
We give a structural characterisation of linear operators from one $C^\ast$% -algebra into another whose adjoints map extreme points of the dual ball onto extreme points. We show that up to a $\ast$-isomorphism, such a map admits of a decomposition into a degenerate and a non-degenerate part, the non-degenerate part of which appears as a Jordan $\ast$-morphism followed by a ``rotation'' and then a reduction. In the case of maps whose adjoints preserve pure states, the degenerate part does not appear, and the ``rotation'' is but the identity. In this context the results concerning such pure state preserving maps depend on and complof Størmer [Stø2; 5.6 \& 5.7]. In conclusion we consider the action of maps with ``extreme point preserving'' adjoints on some specific $C^\ast$-algebras.
It is proved that every function of finite Baire index on a separable metric space $K$ is a $D$-function, i.e., a difference of bounded semi-continuous functions on $K$. In fact it is a strong $D$-function, meaning it can be approximated arbitrarily closely in $D$-norm, by simple $D$-functions. It is shown that if the $n^{th}$ derived set of $K$ is non-empty for all finite $n$, there exist $D$-functions on $K$ which are not strong $D$-functions. Further structural results for the classes of finite index functions and strong $D$-functions are also given.