Periods implying almost all periods, trees with snowflakes, and zero entropy maps

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$. We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a cycle of period $n$, then $f$ has cycles of all periods greater than $2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime number greater than $End(X)$ and $f$ has cycles of all periods from 1 to $2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture of Misiurewicz for tree maps). Together with the spectral decomposition theorem for graph maps it implies that $h(f)>0$ iff there exists $n$ such that $f$ has a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd integer with prime divisors less than $End(X)+1$

Thermal radiation of conducting nanoparticles

The thermal radiation of small conducting particles was investigated in the region where the Stephan-Boltzmann law is not valid and strongly overestimates radiation losses. The new criterion for the particle size, at which black body radiation law fails, was formulated. The approach is based on the magnetic particle polarization, which is valid until very small sizes (cluster size) where due to drop of particle conductivity the electric polarization prevails over the magnetic one. It was also shown that the radiation power of clusters, estimated on the basis of the experimental data, is lower than that given by the Stephan-Boltzmann law.Comment: 12 pages, 5 figures, 1 tabl

Typical orbits of quadratic polynomials with a neutral fixed point: Brjuno type

We describe the topological behavior of typical orbits of complex quadratic polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type. Here we prove that for such Brjuno values of alpha the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then combining with Part I of this work, we show that the limit set of the orbit of a typical point in the Julia set is equal to the closure of the critical orbit.Comment: 38 pages, 5 figures; fixed the issues with processing the figure

The multipliers of periodic points in one-dimensional dynamics

It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle

Complex bounds for multimodal maps: bounded combinatorics

We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to extend the renormalization theory of unimodal maps to multimodal maps.Comment: 20 pages, 3 figure

Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets

Composition law of cardinal order permutations

In this paper the theorems that determine composition laws for both cardinal ordering permutations and their inverses are proven. So, the relative positions of points in a hs-periodic orbit become completely known as well as in which order those points are visited. No matter how a hs-periodic orbit emerges, be it through a period doubling cascade (s=2^n) of the h-periodic orbit, or as a primary window (like the saddle-node bifurcation cascade with h=2^n), or as a secondary window (the birth of a $s-$periodic window inside the h-periodic one). Certainly, period doubling cascade orbits are particular cases with h=2 and s=2^n. Both composition laws are also shown in algorithmic way for their easy use

Energy Spectra, Altitude Profiles and Charge Ratios of Atmospheric Muons

We present a new measurement of air shower muons made during atmospheric ascent of the High Energy Antimatter Telescope balloon experiment. The muon charge ratio mu+ / mu- is presented as a function of atmospheric depth in the momentum interval 0.3-0.9 GeV/c. The differential mu- momentum spectra are presented between 0.3 and about 50 GeV/c at atmospheric depths between 13 and 960 g/cm^2. We compare our measurements with other recent data and with Monte Carlo calculations of the same type as those used in predicting atmospheric neutrino fluxes. We find that our measured mu- fluxes are smaller than the predictions by as much as 70% at shallow atmospheric depths, by about 20% at the depth of shower maximum, and are in good agreement with the predictions at greater depths. We explore the consequences of this on the question of atmospheric neutrino production.Comment: 11 pages, 8 figures, to appear in Phys. Rev. D (2000

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure

Tuberculosis outcomes related to the Mycobacterium tuberculosis genotype

Mycobacterium tuberculosis strains of different phylogenetic lineages and genetic families differ in biological properties that determine, to some extent, epidemiological features and clinical manifestation in tuberculosis (TB) patients.The aim of the study was to assess the risk of an adverse outcome of the disease in TB patients caused by various M. tuberculosis genotypes.Materials and methods. A total of 425 patients with respiratory TB were enrolled in this study. They were registered at phthisiatric facilities in the Omsk region from March 2015 to June 2017 period and included: males — 73.1%, mean age 39.9 years, females — 26.9%, mean age 42.0 years. M. tuberculosis culture and drug susceptibility testing and DNA extraction were performed in accordance with standard methods. Strains were assigned to the M. tuberculosis Beijing genotype and its epidemiologically relevant clusters B0/W148 and 94-32 by PCR based detection of specific markers. Non-Beijing strains were subjected to spoligotyping.Results. We found that 66.5% isolates belonged to the Beijing genotype, 12.8% — to LAM, 10.1% — to T, and 4.7% — to the Ural genotype. Multi-drug resistance (MDR) to anti-TB drugs was observed in 195 M. tuberculosis strains (45.9%). Moreover, Beijing genotype was more often isolated from patients with MDR-TB infection (PR = 2.09 (95% CI 1.6–2.74) and TB infection associated with HIV infection (PR = 1.14 (95% CI 1.01–1.31). Lethal outcome was double higher in patients infected with Beijing vs. non-Beijing strains, 28.6% vs. 14.0% (PR = 2.03; 95% CI 1.3–3.17). The risk factors were identified as follows: young age 18–44 years (RR = 1.7; 95% CI 1.18–2.7), co-morbidity with HIV (RR = 5.0; 95% CI 3.39–7.45), multiple (RR = 1.7; 95% CI 1.14–2.55) and extensive drug resistance (RR = 2.57; 95% CI 1.35–4.92), and association with the Beijing genotype (RR = 2.0, 95% CI 1.3–3.17).Conclusion. M. tuberculosis spread in the Omsk region is characterised by significant prevalence of the Beijing genotype, associated with multiple and extensive drug resistance. A significant association of adverse clinical outcomes and various factors, including association with the Beijing genotype, requires development of new approaches in the fight against tuberculosis