19,458 research outputs found
Remark on "When Are All the Zeros of a Polynomial Real and Distinct?"
The purpose of this note is to point out that the main result of [M.
Chamberland, When Are All the Zeros of a Polynomial Real and Distinct? Amer.
Math. Monthly. 127 (2020) 449-451] is implicitly contained in the elementary
lore of the theory of orthogonal polynomials on the real line.Comment: This short piece will not be published anywhere els
Von Neumann Regular Cellular Automata
For any group and any set , a cellular automaton (CA) is a
transformation of the configuration space defined via a finite memory set
and a local function. Let be the monoid of all CA over .
In this paper, we investigate a generalisation of the inverse of a CA from the
semigroup-theoretic perspective. An element is von
Neumann regular (or simply regular) if there exists
such that and , where is the composition of functions. Such an
element is called a generalised inverse of . The monoid
itself is regular if all its elements are regular. We
establish that is regular if and only if
or , and we characterise all regular elements in
when and are both finite. Furthermore, we study
regular linear CA when is a vector space over a field ; in
particular, we show that every regular linear CA is invertible when is
torsion-free elementary amenable (e.g. when ) and , and that every linear CA is regular when
is finite-dimensional and is locally finite with for all .Comment: 10 pages. Theorem 5 corrected from previous versions, in A.
Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata
and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer,
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Coral symbiodinium community composition across the Belize Mesoamerican barrier reef system is influenced by host species and thermal variability
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