The collection of k-closed subsets of X forms a topology, which contains the original topology of X (i.e. closed subsets are always k-closed). Notation 1.2. Let kX denote the space whose underlying set is that of X, but equipped with the topology of k-closed subsets of X. Because the k-topology contains the original topology on X, the identity function id : kX → X is continuous. Definition 1.3....

In this work we present an “out-of-the-box” application of Machine Learning (ML) optimizers for industrial optimization problem. We introduce a piecewise polynomial model (spline) fitting $$\mathcal {C}^k$$ -continuous functions, which can be deployed in cam approximation setting. then use the gradient descent context provided by machine learning framework TensorFlow to optimize parameters with...

It is shown that for each k > 1, if f is a Baire one function and f is the product of k bounded Darboux (quasi–continuous) functions, then f is the product of k bounded Darboux (quasi–continuous) Baire one functions as well.

In this paper, we derived recurrence relations for joint moment generating functions of nonadjacent generalized order statistics (GOS) of random samples drawn from doubly truncated class of continuous distributions. Recurrence relations for joint moments of nonadjacent GOS (ordinary order statistics (OOS) and k-upper records (k-RVs) as special cases) are obtained. Single and product moment gene...