In this article we introduce some impulsive models of tumor growth based on classical models as inhibition model, Piantadosi model, and autostimulation model. The basic goal is to describe the medical interventions during the treatment of the cancer process.The used technique is based on the theory of impulsive differential equations.
Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P = {p 1 , . . . , p n } ⊂ Z 2 a maximal integral point set over Z 2 if all pairwise distances are integral and every additional point p n+1 destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases, we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.
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