On the steady-states of a two-species non-local cross-diffusion model

We investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.

Additivity condition for reference kiosk based on page load time

The paper analyzes the condition of additivity of the information and reference kiosk, which allows designing a static kiosk, which is essentially a set of pages organized in order to provide users with reliable information in various fields of knowledge and practice, with additive properties. The condition of additivity is analyzed based on the loading time of the pages of the informational kiosk is implemented on the basis of the concept of a simple website with narrow subject matter by adding the concept of a vertical portal, in connection with which introduced the concept of optimal page load time of the informational kiosk, since the main task of the informational kiosk is the display of the user requested page containing the content within a certain time interval, and a key component of the kiosk GUI. Information and reference kiosk, as a small subscriber system in terms of functionality and volume of content, containing information in text, audio or video format, aimed at providing reference information to the user and giving him a means of interaction, by providing all available system objects and functions implemented in the form of graphical components, is not an exception. The article also discusses the key concepts of the information and reference kiosk, such as additivity, ergodicity, and entropy.