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 ).

Collision of a hard ball with singular points of the boundary.

Recently, physical billiards were introduced where a moving particle is a hard sphere rather than a point as in standard mathematical billiards. It has been shown that in the same billiard tables, the physical billiards may have totally different dynamics than mathematical billiards. This difference appears if the boundary of a billiard table has visible singularities (internal corners if the billiard table is two-dimensional); i.

Using earth mover's distance for viral outbreak investigations.

Background: RNA viruses mutate at extremely high rates, forming an intra-host viral population of closely related variants, which allows them to evade the host's immune system and makes them particularly dangerous. Viral outbreaks pose a significant threat for public health, and, in order to deal with it, it is critical to infer transmission clusters, i.e.

Graph fractal dimension and the structure of fractal networks.

Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks.

Local Immunodeficiency: Role of Neutral Viruses.

This paper analyzes the role of neutral viruses in the phenomenon of local immunodeficiency. We show that, even in the absence of altruistic viruses, neutral viruses can support the existence of persistent viruses and thus local immunodeficiency. However, in all such cases neutral viruses can maintain only bounded (relatively small) concentration of persistent viruses.

Early-Time Exponential Instabilities in Nonchaotic Quantum Systems.

The majority of classical dynamical systems are chaotic and exhibit the butterfly effect: a minute change in initial conditions has exponentially large effects later on. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts.

Physical versus mathematical billiards: From regular dynamics to chaos and back.

In standard (mathematical) billiards, a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite-size hard sphere moves at the same billiard table, virtually anything may happen. Namely, a nonchaotic billiard may become chaotic and vice versa.

Local immunodeficiency: Minimal networks and stability.

Some basic aspects of the recently discovered phenomenon of local immunodeficiency (Skums et al. [1]) generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated. We prove that local immunodeficiency (LI) that is stable under perturbations already occurs in very small networks and under general conditions on their parameters.

QUENTIN: reconstruction of disease transmissions from viral quasispecies genomic data.

Motivation: Genomic analysis has become one of the major tools for disease outbreak investigations. However, existing computational frameworks for inference of transmission history from viral genomic data often do not consider intra-host diversity of pathogens and heavily rely on additional epidemiological data, such as sampling times and exposure intervals. This impedes genomic analysis of outbreaks of highly mutable viruses associated with chronic infections, such as human immunodeficiency virus and hepatitis C virus, whose transmissions are often carried out through minor intra-host variants, while the additional epidemiological information often is either unavailable or has a limited use.

Some new surprises in chaos.

"Chaos is found in greatest abundance wherever order is being sought.It always defeats order, because it is better organized"Terry PratchettA brief review is presented of some recent findings in the theory of chaotic dynamics. We also prove a statement that could be naturally considered as a dual one to the Poincaré theorem on recurrences.

Antigenic cooperation among intrahost HCV variants organized into a complex network of cross-immunoreactivity.

Hepatitis C virus (HCV) has the propensity to cause chronic infection. Continuous immune escape has been proposed as a mechanism of intrahost viral evolution contributing to HCV persistence. Although the pronounced genetic diversity of intrahost HCV populations supports this hypothesis, recent observations of long-term persistence of individual HCV variants, negative selection increase, and complex dynamics of viral subpopulations during infection as well as broad cross-immunoreactivity (CR) among variants are inconsistent with the immune-escape hypothesis.

At grammatical faculty of language, flies outsmart men.

Using a symbolic dynamics and a surrogate data approach, we show that the language exhibited by common fruit flies Drosophila ('D.') during courtship is as grammatically complex as the most complex human-spoken modern languages. This finding emerges from the study of fifty high-speed courtship videos (generally of several minutes duration) that were visually frame-by-frame dissected into 37 fundamental behavioral elements.

Isospectral compression and other useful isospectral transformations of dynamical networks.

It is common knowledge that a key dynamical characteristic of a network is its spectrum (the collection of all eigenvalues of the network's weighted adjacency matrix). We demonstrated that it is possible to reduce a network, considered as a graph, to a smaller network with fewer vertices and edges while preserving the spectrum (or spectral information) of the original network [L. A.

Many faces of stickiness in Hamiltonian systems.

We discuss the phenomenon of stickiness in Hamiltonian systems. By visual examples of billiards, it is demonstrated that one must make a difference between internal (within chaotic sea(s)) and external (in vicinity of KAM tori) stickiness. Besides, there exist two types of KAM-islands, elliptic and parabolic ones, which demonstrate different abilities of stickiness.

Introduction to Focus Issue: statistical mechanics and billiard-type dynamical systems.

Dynamical systems of the billiard type are of fundamental importance for the description of numerous phenomena observed in many different fields of research, including statistical mechanics, Hamiltonian dynamics, nonlinear physics, and many others. This Focus Issue presents the recent progress in this area with contributions from the mathematical as well as physical stand point.

Path lengths in protein-protein interaction networks and biological complexity.

We investigated the biological significance of path lengths in 12 protein-protein interaction (PPI) networks. We put forward three predictions, based on the idea that biological complexity influences path lengths. First, at the network level, path lengths are generally longer in PPIs than in random networks.

Suppressing Fermi acceleration in a driven elliptical billiard.

We study dynamical properties of an ensemble of noninteracting particles in a time-dependent elliptical-like billiard. It was recently shown [Phys. Rev.

Suppressing Fermi acceleration in two-dimensional driven billiards.

We consider a dissipative oval-like shaped billiard with a periodically moving boundary. The dissipation considered is proportional to a power of the velocity V of the particle. The three specific types of power laws used are: (i) F∝-V ; (ii) F∝-V(2) and (iii) F∝-V(δ) with 1<δ<2 .

One-particle and few-particle billiards.

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics.

Open circular billiards and the Riemann hypothesis.

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container.