arXiv subjects

Serguei Samborski

Publications and source records attributed to Serguei Samborski.

3 recordsLinked to original sources

arXiv2005

Set-valued differentiation as an operator

We introduce real vector spaces composed of set-valued maps on an open set. They are also complete metric spaces, lattices, commutative rings. The set of differentiable functions is a dense subset of these spaces and the classical gradient may be extended in these spaces as a closed operator. If a function f belongs to the domain of such extension, then f is locally lipschitzian and the values of extended gradient coincide with the values of Clarke's gradient. However, unlike Clarke's gradient, our generalized gradient is a linear operator.

math.OC↗

arXiv2004

On Linearity of Nonclassical Differentiation

We introduce a real vector space composed of set-valued maps on an open set X and note it by S. It is a complete metric space and a complete lattice. The set of continuous functions on X is dense in S as in a metric space and as in a lattice. Thus the constructed space plays the same role for the space of continuous functions with uniform convergence as the field of reals plays for the field of rationals. The classical gradient may be extended in the space S as a close operator. If a function f belongs to its domain then f is locally lipschitzian and the values of our gradient coincide with the values of Clarke's gradient. However, unlike Clarke's gradient, our gradient is a linear operator.

math.OC↗

arXiv2004

On Metric Completness and Completness in sense of Lattices

We give an answer to the following question: for which metric in an abstract lattice the completion as a metric space coincides with the completion as a lattice. We obtain the answer for inductive limits of lattices which are complete in both senses. As an application we construct such a metric in the lattice of continuous functions endowed with the uniform convergence.

math.FA↗