Native ultrametricity of sparse random ensebles.
We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. Analyzing in detail the spectral density of ensembles of linear subgraphs, we discuss its ultrametric nature and show that near the spectrum boundary, the tails of the spectral density exhibit a Lifshitz singularity typical for Anderson localization. We pay attention to an intriguing connection of the spectral density to the Dedekind η-function. We conjecture that ultrametricity emerges in rare-event statistics and is inherit to generic complex sparse systems.
We describe an algorithm for the maximum clique problem that is parameterized by the graph’s degeneracy d. The algorithm runs in O(nm+nTd) time, where Td is the time to solve the maximum clique problem in an arbitrary graph on d vertices. The best bound as of now is Td=O(2d/4)by Robson. This shows that the maximum clique problem is solvable in O(nm) time in graphs for which d≤4log2m+O(1). The analysis of the algorithm’s runtime is simple; the algorithm is easy to implement when given a subroutine for solving maximum clique in small graphs; it is easy to parallelize. In the case of Bianconi-Marsili power-law random graphs, it runs in 2O(√n)time with high probability. We extend the approach for a graph invariant based on common neighbors, generating a second algorithm that has a smaller exponent at the cost of a larger polynomial factor.
A new approach to investigate a radial artery pulse signal rhythmic structure is considered based on a simultaneous analysis of a set of oscillatory components determined by parameters of various elements of unit oscillations. Based on a comparative analysis of the spectral density types defined by different parameters of a pulse signal, an essential distinction between them has been revealed. An opportunity of increasing the number of informative features by means of simultaneous analysis of pulse signal rhythmic structure oscillatory component set is shown. The study has been carried out using experimental material obtained during children’s clinical examinations focused on detecting the initial stage arterial hypertension in infancy and adolescence. The informativeness of the pulse signal rhythmic structure parameters was estimated as applied to this task, showing that maximum informativeness is inherent to indicators defined by oscillatory components of the pulse signal dicrotic wave temporal parameter.
The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.
Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.