Creation and Growth of Components in a Random Hypergraph Process

Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend

Microscopic magnetic modeling for the SS=1/2 alternating chain compounds Na3_3Cu2_2SbO6_6 and Na2_2Cu2_2TeO6_6

The spin-1/2 alternating Heisenberg chain system Na$_3$Cu$_2$SbO$_6$ features two relevant exchange couplings: $J_{1a}$ within the structural Cu$_2$O$_6$ dimers and $J_{1b}$ between the dimers. Motivated by the controversially discussed nature of $J_{1a}$, we perform extensive density-functional-theory (DFT) calculations, including DFT+$U$ and hybrid functionals. Fits to the experimental magnetic susceptibility using high-temperature series expansions and quantum Monte Carlo simulations yield the optimal parameters $J_{1a}$ = $-$217 K and $J_{1b}$ = 174 K with the alternation ratio $\alpha = J_{1a}/J_{1b} \simeq$ $-$1.25. For the closely related system Na$_2$Cu$_2$TeO$_6$, DFT yields substantially enhanced $J_{1b}$, but weaker $J_{1a}$. The comparative analysis renders the buckling of the chains as the key parameter altering the magnetic coupling regime. Numerical simulation of the dispersion relations of the alternating chain model clarify why both antiferromagnetic and ferrromagnetic $J_{1a}$ can reproduce the experimental magnetic susceptibility data.Comment: published version: 11 pages, 8 figures, 5 tables + Supplemental materia

Magnetism of CuX2 frustrated chains (X = F, Cl, Br): the role of covalency

Periodic and cluster density-functional theory (DFT) calculations, including DFT+U and hybrid functionals, are applied to study magnetostructural correlations in spin-1/2 frustrated chain compounds CuX2: CuCl2, CuBr2, and a fictitious chain structure of CuF2. The nearest-neighbor and second-neighbor exchange integrals, J1 and J2, are evaluated as a function of the Cu-X-Cu bridging angle, theta, in the physically relevant range 80-110deg. In the ionic CuF2, J1 is ferromagnetic for theta smaller 100deg. For larger angles, the antiferromagnetic superexchange contribution becomes dominant, in accord with the Goodenough-Kanamori-Anderson rules. However, both CuCl2 and CuBr2 feature ferromagnetic J1 in the whole angular range studied. This surprising behavior is ascribed to the increased covalency in the Cl and Br compounds, which amplifies the contribution from Hund's exchange on the ligand atoms and renders J1 ferromagnetic. At the same time, the larger spatial extent of X orbitals enhances the antiferromagnetic J2, which is realized via the long-range Cu-X-X-Cu paths. Both, periodic and cluster approaches supply a consistent description of the magnetic behavior which is in good agreement with the experimental data for CuCl2 and CuBr2. Thus, owing to their simplicity, cluster calculations have excellent potential to study magnetic correlations in more involved spin lattices and facilitate application of quantum-chemical methods

Terrestrial planets across space and time

The study of cosmology, galaxy formation and exoplanets has now advanced to a stage where a cosmic inventory of terrestrial planets may be attempted. By coupling semi-analytic models of galaxy formation to a recipe that relates the occurrence of planets to the mass and metallicity of their host stars, we trace the population of terrestrial planets around both solar-mass (FGK type) and lower-mass (M dwarf) stars throughout all of cosmic history. We find that the mean age of terrestrial planets in the local Universe is $7\pm{}1$ Gyr for FGK hosts and $8\pm{}1$ Gyr for M dwarfs. We estimate that hot Jupiters have depleted the population of terrestrial planets around FGK stars by no more than $\approx 10\%$, and that only $\approx 10\%$ of the terrestrial planets at the current epoch are orbiting stars in a metallicity range for which such planets have yet to be confirmed. The typical terrestrial planet in the local Universe is located in a spheroid-dominated galaxy with a total stellar mass comparable to that of the Milky Way. When looking at the inventory of planets throughout the whole observable Universe, we argue for a total of $\approx 1\times 10^{19}$ and $\approx 5\times 10^{20}$ terrestrial planets around FGK and M stars, respectively. Due to light travel time effects, the terrestrial planets on our past light cone exhibit a mean age of just $1.7\pm 0.2$ Gyr. These results are discussed in the context of cosmic habitability, the Copernican principle and searches for extraterrestrial intelligence at cosmological distances.Comment: 11 pages, 8 figures. v.2: Accepted for publication in ApJ. Some changes in quantitative results compared to v.1, mainly due to differences in IMF assumption

Large quantum fluctuations in the strongly coupled spin-1/2 chains of green dioptase: a hidden message from birds and trees

The green mineral dioptase Cu6Si6O18(H2O)6 has been known since centuries and plays an important role in esoteric doctrines. In particular, the green dioptase is supposed to grant the skill to speak with trees and to understand the language of birds. Armed with natural samples of dioptase, we were able to unravel the magnetic nature of the mineral (presumably with hidden support from birds and trees) and show that strong quantum fluctuations can be realized in an essentially framework-type spin lattice of coupled chains, thus neither frustration nor low-dimensionality are prerequisites. We present a microscopic magnetic model for the green dioptase. Based on full-potential DFT calculations, we find two relevant couplings in this system: an antiferromagnetic coupling J_c, forming spiral chains along the hexagonal c axis, and an inter-chain ferromagnetic coupling J_d within structural Cu2O6 dimers. To refine the J_c and J_d values and to confirm the proposed spin model, we perform quantum Monte-Carlo simulations for the dioptase spin lattice. The derived magnetic susceptibility, the magnetic ground state, and the sublattice magnetization are in remarkably good agreement with the experimental data. The refined model parameters are J_c = 78 K and J_d = -37 K with J_d/J_c ~ -0.5. Despite the apparent three-dimensional features of the spin lattice and the lack of frustration, strong quantum fluctuations in the system are evidenced by a broad maximum in the magnetic susceptibility, a reduced value of the Neel temperature T_N ~ 15 K >> J_c, and a low value of the sublattice magnetization m = 0.55 Bohr magneton. All these features should be ascribed to the low coordination number of 3 that outbalances the three-dimensional nature of the spin lattice.Comment: Dedicated to Stefan-Ludwig Drechsler on the occasion of his 60th birthday (9 pages, 6 figures

On the identification of Ti-Ta-Nb-oxides in “wiikites” from Karelia

Purpose. With respect to the nature of Ti-Ta-Nb-oxides, which are included in the mineral associations that have historically gone by the now outdated name of “wiikites”, there is no unanimity of opinion. The main difficulty in identifying them is associated with the complexity of their chemical composition, their metamict structure and the substantial changes in their initial mineral form due to the effects of natural weathering. The aim of this work is the development of methodology to identify Ti-Ta-Nb-oxides corresponding to the mineral formulas AB2O6 and A2B2O7. Methods. The methodology developed in the work includes two experimental approaches: 1) electron probe microanalysis of phases revealed through use of the SEM-BSE method. Based on the resulting data, a hypothesis about the nature of the Ti-Ta-Nb-oxides in a number of wiikite samples is proposed. 2) XRD analysis of those phases that are formed in the course of thermal annealing of samples from 200 to 1000°C. Based on the resulting data here, the previous hypotheses are either accepted or rejected for each sample. Findings. Using this methodology, three “wiikite” samples were analyzed (the experimental codes were Wk-2, Wk-3 and Wk-7). The Ti-Ta-Nb-oxides in two of these wiikites (Wk-2 and Wk-3) were preliminarily determined to be hydroxyyttropyrochlore and hydroxycalciobetafite, respectively. In the third sample (Wk-7), the Ti-Ta-Nb-oxide was classified as polycrase. The results of XRD phase analysis of the annealed samples were compatible with the primary hypotheses that hydroxycalciobetafite and polycrase were the initial minerals in “wiikites” 3 and 7, respectively. Originality. The paper considers the issue of the chemical conditions necessary for the recovery of the original Ti-Ta-Nb-oxide during recrystallization of the metamict structures. This is a problem that has not been substantively addressed in the literature. Also for the first time experimentally proved the existence of the polycrase in “wiikites”. Practical implications. Completed work expands our knowledge of the mineral composition of the granitic pegmatites of the Baltic shield and the possibility of their use both for research purposes and for needs of national economy.Мета. Розробка методології ідентифікації Ti-Ta-Nb-оксидів складного хімічного складу з метаміктною структурою, що відповідають мінеральним формулами АВ2О6 та А2В2О7. Методика. Для розробки методології ідентифікації у роботі використано два експериментальних підходи: 1) рентген-спектральний мікроаналіз фаз, виявлених методом скануючої електронної мікроскопії зразків у назад відображених електронах. На основі отриманих результатів висунута гіпотеза щодо природи Ti-Ta-Nb-оксидів. 2) рентген-дифракційний аналіз фаз, що утворюються в процесі термічного відпалу зразків від 200 до 1000°С. На основі отриманих результатів для кожного зразка первинна гіпотеза приймалася або відкидалася. Результати. З використанням даної методики були вивчені три зразки “віікітів” (шифри зразків: Wk-2, Wk-3 і Wk-7). Наявність Ti-Ta-Nb-оксидів попередньо визначено e двох зразках (Wk-2 і Wk-3) відповідно, як гідроксііттропірохлор та гідроксікальціобетафіт. У третьому зразку (Wk-7) Ti-Ta-Nb діагностований як полікраз. Результати рентген-дифракційного аналізу відпалених зразків підтвердили гіпотези про те, що початковими мінералами у Wk-3 і Wk-7 були гідроксікальціобетафіт і полікраз відповідно. Наукова новизна. Визначено хімічні умови, необхідні для утворення початкового Тi-Ta-Nb під час рекристалізації метаміктної структури, що раніше не розглядалось у відповідній тематичній науковій літературі. Вперше експериментально виявлено присутність у “віікітах” полікраза. Практична значимість. Отримані результати розширюють й доповнюють знання щодо мінерального складу гранітних пегматитів Балтійського щита та надають можливість їх використання в дослідницьких цілях і потребах народного господарства.Цель. Разработка методологии идентификации Ti-Ta-Nb-оксидов сложного химического состава с метамиктной структурой, соответствующих минеральным формулам АВ2О6 и А2В2О7. Методика. Для разработки методологии идентификации в работе использовано два экспериментальных подхода: 1) рентген-спектральный микроанализ фаз, выявленных методом сканирующей электронной микроскопии образцов в обратно отраженных электронах. На основе полученных результатов выдвинута гипотеза о природе Ti-Ta-Nb-оксидов. 2) рентген-дифракционный анализ фаз, образующихся в процессе термического отжига образцов от 200 до 1000°С. На основе полученных результатов для каждого образца первоначальная гипотеза принималась или отвергалась. Результаты. С использованием данной методики были изучены три образца “виикитов” (шифры образцов: Wk-2, Wk-3 и Wk-7). Наличие Ti-Ta-Nb-оксидов предварительно определено в двух образцах (Wk-2 и Wk-3) соответственно, как гидроксииттропирохлор и гидроксикальциобетафит. В третьем образце (Wk-7) Ti-Ta-Nb диагностирован как поликраз. Результаты рентген-дифракционного анализа отожженных образцов подтвердили гипотезы о том, что первоначальными минералами в Wk-3 и Wk-7 были гидроксикальциобетафит и поликраз соответственно. Научная новизна. Определены химические условия, необходимые для образования первоначального Тi-Ta-Nb во время рекристаллизации метамиктной структуры, что ранее не рассматривалось в соответствующих тематических научных литературе. Впервые экспериментально выявлено присутствие в “виикитах” поликраза. Практическая значимость. Полученные результаты расширяют и дополняют знания о минеральном составе гранитных пегматитов Балтийского щита и дают возможности их использования в исследовательских целях и нуждах народного хозяйства.The study was conducted using equipment of the Center of X-ray Diffraction Studies and the Center for Microscopy and Microanalysis of the Research Park at Saint Petersburg State University, as well as the analytical laboratory of OJSC MMC Norilsk Nickel in Saint Petersburg

Quicksort with unreliable comparisons: a probabilistic analysis

We provide a probabilistic analysis of the output of Quicksort when comparisons can err.Comment: 29 pages, 3 figure

Discovery of a stellar companion to the nearby solar-analogue HD 104304

Sun-like stars are promising candidates to host exoplanets and are often included in exoplanet surveys by radial velocity (RV) and direct imaging. In this paper we report on the detection of a stellar companion to the nearby solar-analogue star HD 104304, which previously was considered to host a planetary mass or brown dwarf companion. We searched for close stellar and substellar companions around extrasolar planet host stars with high angular resolution imaging to characterize planet formation environments. The detection of the stellar companion was achieved by high angular resolution measurements, using the "Lucky Imaging" technique at the ESO NTT 3.5m with the AstraLux Sur instrument. We combined the results with VLT/NACO archive data, where the companion could also be detected. The results were compared to precise RV measurements of HD 104304, obtained at the Lick and Keck observatories from 2001-2010. We confirmed common proper motion of the binary system. A spectral type of M4V of the companion and a mass of 0.21 M_Sun was derived. Due to comparison of the data with RV measurements of the unconfirmed planet candidate listed in the Extrasolar Planets Encyclopaedia, we suggest that the discovered companion is the origin of the RV trend and that the inclination of the orbit of approximately 35 degrees explains the relatively small RV signal.Comment: 4 pages, 4 PNG figures, use aa.cls, accepted for publication in Astronomy & Astrophysic

Drinking related internal-external locus of control scale. Outcomes measurement tool: attitudes & feelings - attitudes (towards drug and alcohol use).

The DRIE was developed to define an individual’s beliefs about the extent to which life events are under personal control (internal locus of control) or under the influence of chance, fate, or powerful others (external locus of control). The DRIE assesses these beliefs specifically with respect to the individual’s perceptions of control around alcohol, drinking behaviour, and recovery