Multivalued Functions in Digital Topology

We study several types of multivalued functions in digital topology

An ordered framework for partial multivalued functors

The category Rel of sets and relations intimately ties the notions of function, partial multivalued function, and direct image under a function through the description of Rel as the Kleisli category of the covariant power set functor on Set. We present a suitable framework to obtain a similar relationship between the concepts of functor, partial multivalued functor, and the direct image under a functor.Comment: Accepted for presentation at the Asia-Pacific World Congress on Computer Science and Engineering 2015, Fij

Universal Nash Equilibrium Strategies for Differential Games

The paper is concerned with a two-player nonzero-sum differential game in the case when players are informed about the current position. We consider the game in control with guide strategies first proposed by Krasovskii and Subbotin. The construction of universal strategies is given both for the case of continuous and discontinuous value functions. The existence of a discontinuous value function is established. The continuous value function does not exist in the general case. In addition, we show the example of smooth value function not being a solution of the system of Hamilton--Jacobi equation.Comment: 23 page

Multivalued function spaces and Atsuji spaces

[EN] In this paper we present two themes. The first one describes a transparent treatment of some of the recent results in graph topologies on multi-valued functions. The study includes Vietoris topology, Fell topology, Fell uniform topology on compacta and uniform topology on compacta. The second theme concerns when continuity is equivalent to proximal continuity or uniform continuityNaimpally, S. (2003). Multivalued function spaces and Atsuji spaces. Applied General Topology. 4(2):201-209. doi:10.4995/agt.2003.2025.SWORD2012094