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In addition to the LS closed form solution, two well-established algorithms were considered in this study.
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In general Im ( L ) = L. Clearly, Im ( L ) is closed in Z and dim Ker ( L ) = codim Im ( L ) = 1, thus L is a Fredholm operator of index zero.
An SP-class of L-algebras is a class of L-algebras closed under subalgebras and direct products.
Furthermore, one easily shows that (operatorname{Im}(L)) is closed in Z and begin{aligned} operatorname{dim} operatorname{ker}(L =n+m=operatorname{codim} operatorname{Im}(L). end{aligned} Therefore, L is a Fredholm mapping of index zero.
Let L be closed and connected.
So L is closed.
KerL has a finite dimension; Im L is closed and has a finite codimension.
A language L is open if it can be extended to a new language L′ without changes in the meanings of other expressions; otherwise L is closed.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com