On transfer operators for continued fractions with restricted digits

For any non-empty subset I of the natural numbers, let {Lambda}I denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the corresponding transfer operator Formula. for Formula, where Re (rß) = {theta}I is the abscissa of convergence of the series Formula. When acting on a certain Hilbert space HI, rß, we show that the operator LI, rß is conjugate to an integral operator KI, rß. If furthermore rß is real, then KI, rß is selfadjoint, so that LI, rß : HI, rß -> HI, rß has purely real spectrum. It is proved that LI, rß also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C{omega} [0, 1], and on the Fréchet space C{infty} [0, 1]. The analytic properties of the map rß ↦ LI, rß are investigated. For certain alphabets I of an arithmetic nature (for example, I = primes, I = squares, I an arithmetic progression, I the set of sums of two squares it is shown that rß ↦ LI, rß admits an analytic continuation beyond the half-plane Re rß > {theta}I

On the Ruelle eigenvalue sequence

Abstract. For certain real analytic data, we show that the eigenvalue sequence of the associated transfer operator L is insensitive to the holomorphic function space on which L acts. Explicit bounds on this eigenvalue sequence are established

Buoyancy regulation and aggregate formation in Amoebobacter purpureus from Mahoney lake

Abstract The meromictic Mahoney Lake (British Columbia, Canada) contains an extremely dense layer of purple sulfur bacteria (Amoebobacter purpureus). The buoyant density of Amoebobacter cells grown in pure culture at saturating light intensity was significantly higher (1027–1034 kg m−3) than the density of lake water (1015 kg m−3). When stationary cultures were shifted to the dark, the gas-vesicle content increased by a factor of 9 and buoyant density decreased to 1002 kg m−3 within three days. A novel mechanism of cell aggregation was detected for the Mahoney Lake strain. Dense cell aggregates were formed after depletion of sulfide. Formation of aggregates was correlated with an increase in cell surface hydrophobicity. Cell aggregates could be disintegrated within less than 1 s by addition of sulfide or various thiol compounds. Mercaptanes with a branched structure in the vicinity of the terminal thiol group, compounds with esterified thiol groups (methylmercaptanes), reducing compounds lacking thiol groups and detergents did not influence aggregate stability. Cell aggregates disintegrated upon addition of urea or of proteinase K. Addition of various sugars had no effect on aggregation; this points to the absence of lectins. The results indicate that cell-to-cell adhesion in A, purpureus ML1 is mainly caused by a hydrophobic effect and includes a specific mechanism possibly mediated by a surface protein. Extrapolation of laboratory results to field conditions demonstrated that both regulation of buoyant density and formation of cell aggregates result in passive accumulation of cells at the chemocline and contribute to the narrow stratification of A. purpureus in Mahoney Lake

Algal blooms: how are they harming models used for climate management?

18th International Conference on Harmful Algae (ICHA 2018), 21-26 October 2018, Nantes, France.-- 4 pages, 2 figuresMicroalgae blooms are generally associated with bacterial secondary producers. They produce organic matter (OM), some of which associates with the sea surface microlayer (SML). OM in the SML below the actual surface reduces fluxes of energy, including heat and momentum, and substances, including greenhouse gases, aerosols, algae, bacteria and viruses. In addition to the SML-associated OM, another OM fraction, foam (including whitecaps), often lies above the primary SML when windspeeds exceed about 5 m s-1, trapping gas bubbles. Such foam also dramatically increases albedo, reflecting solar radiation back into space, thus reducing solar heating and penetration of photosynthetically active radiation. Mean coverage of the ocean surface by foam has been measured to range between 1-6%, particularly in zones of Trade Winds. Different types of OM, and particularly their mechanical properties, depend on ambient algal abundance, as well as on taxonomic composition, as do the dynamics of foam formation and decay. Air-sea fluxes may thus be influenced by genomic control through the blooming microalgae and Darwinian-type evolution. Bacteria may also play a role. In addition, foam patches on the ocean’s surface serve as a unique microbial habitat. Such blooms, particularly when their taxonomic composition changes unpredictably, are likely to be harming the usefulness of climate models. Some of this harm might be mitigated by studying the relevant effects of these blooms on fluxes, and incorporating these effects into climate modelsJS is supported by National Nature Science Foundation of China grant (41876134) and the Changjiang Scholar Program of Chinese Ministry of Educatio

Multiple imputation for patient reported outcome measures in randomised controlled trials : advantages and disadvantages of imputing at the item, subscale or composite score level

Background Missing data can introduce bias in the results of randomised controlled trials (RCTs), but are typically unavoidable in pragmatic clinical research, especially when patient reported outcome measures (PROMs) are used. Traditionally applied to the composite PROMs score of multi-item instruments, some recent research suggests that multiple imputation (MI) at the item level may be preferable under certain scenarios. This paper presents practical guidance on the choice of MI models for handling missing PROMs data based on the characteristics of the trial dataset. The comparative performance of complete cases analysis, which is commonly used in the analysis of RCTs, is also considered. Methods Realistic missing at random data were simulated using follow-up data from an RCT considering three different PROMs (Oxford Knee Score (OKS), EuroQoL 5 Dimensions 3 Levels (EQ-5D-3L), 12-item Short Form Survey (SF-12)). Data were multiply imputed at the item (using ordinal logit and predicted mean matching models), sub-scale and score level; unadjusted mean outcomes, as well as treatment effects from linear regression models were obtained for 1000 simulations. Performance was assessed by root mean square errors (RMSE) and mean absolute errors (MAE). Results Convergence problems were observed for MI at the item level. Performance generally improved with increasing sample sizes and lower percentages of missing data. Imputation at the score and subscale level outperformed imputation at the item level in small sample sizes (n ≤ 200). Imputation at the item level is more accurate for high proportions of item-nonresponse. All methods provided similar results for large sample sizes (≥500) in this particular case study. Conclusions Many factors, including the prevalence of missing data in the study, sample size, the number of items within the PROM and numbers of levels within the individual items, and planned analyses need consideration when choosing an imputation model for missing PROMs data