Fractal networks: Topology, dimension, and complexity.

Over the past two decades, the study of self-similarity and fractality in discrete structures, particularly complex networks, has gained momentum. This surge of interest is fueled by the theoretical developments within the theory of complex networks and the practical demands of real-world applications. Nonetheless, translating the principles of fractal geometry from the domain of general topology, dealing with continuous or infinite objects, to finite structures in a mathematically rigorous way poses a formidable challenge.

Antigenic cooperation in viral populations: Transformation of functions of intra-host viral variants.

In this paper, we study intra-host viral adaptation by antigenic cooperation - a mechanism of immune escape that serves as an alternative to the standard mechanism of escape by continuous genomic diversification and allows to explain a number of experimental observations associated with the establishment of chronic infections by highly mutable viruses. Within this mechanism, the topology of a cross-immunoreactivity network forces intra-host viral variants to specialize for complementary roles and adapt to the host's immune response as a quasi-social ecosystem. Here we study dynamical changes in immune adaptation caused by evolutionary and epidemiological events.

Local Immunodeficiency: Combining Cross-Immunoreactivity Networks.

This article continues the analysis of the recently observed phenomenon of local immunodeficiency (LI), which arises as a result of antigenic cooperation among intrahost viruses organized into a network of cross-immunoreactivity (CR). We study here what happens as the result of combining (connecting) the simplest CR networks, which have a stable state of LI. It turned out that many possibilities occur, particularly resulting in a change of roles of some viruses in the CR network.

Elliptic Flowers: New Types of Dynamics to Study Classical and Quantum Chaos.

We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong to a new class of dynamical systems called elliptic flowers billiards. Such systems demonstrate a variety of new behaviors which have never been observed or predicted to exist.

Detecting intrinsic global geometry of an obstacle via layered scattering.

Given a closed k-dimensional submanifold K, encapsulated in a compact domain M ⊂ , k ≤ n - 2, we consider the problem of determining the intrinsic geometry of the obstacle K (such as volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular ϵ-neighborhood T ( K , ϵ ) of K in M. The geodesics that participate in this scattering emanate from the boundary ∂ M and terminate there after a few reflections from the boundary ∂ T ( K , ϵ ). However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of K by entering some trap there so that this ray will have infinitely many reflections from ∂ T ( K , ϵ ).