410 research outputs found
Perturbations of roots under linear transformations of polynomials
Let \cP_n be the complex vector space of all polynomials of degree at most
. We give several characterizations of the linear operators T\in\cL(\cP_n)
for which there exists a constant such that for all nonconstant
p\in\cP_n there exist a root of and a root of with
. We prove that such perturbations leave the degree unchanged and,
for a suitable pairing of the roots of and , the roots are never
displaced by more than a uniform constant independent on . We show that such
``good'' operators are exactly the invertible elements of the commutative
algebra generated by the differentiation operator. We provide upper bounds in
terms of for the relevant constants.Comment: 23 page
Homology and Robustness of Level and Interlevel Sets
Given a function f: \Xspace \to \Rspace on a topological space, we consider
the preimages of intervals and their homology groups and show how to read the
ranks of these groups from the extended persistence diagram of . In
addition, we quantify the robustness of the homology classes under
perturbations of using well groups, and we show how to read the ranks of
these groups from the same extended persistence diagram. The special case
\Xspace = \Rspace^3 has ramifications in the fields of medical imaging and
scientific visualization
Exponential bounds for the support convergence in the Single Ring Theorem
We consider an by matrix of the form , with some
independent Haar-distributed unitary matrices and a deterministic matrix.
We prove that for and
, as tends to infinity, we have
This gives a simple proof (with slightly weakened hypothesis) of the
convergence of the support in the Single Ring Theorem, improves the available
error bound for this convergence from to and
proves that the rate of this convergence is at most .Comment: 15 pages, 1 figure. Minor typos corrected, references added. To
appear in J. Funct. Ana
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