Numbers with fixed sum of digits in linear recurrent number systems

We study the set of integers with a given sum of digits with respect to a linear recurrent digit system. An asymptotic formula for the number of integers ≤N with given sum of digits is determined, and the distribution in residue classes is investigated, thus generalizing results due to Mauduit and Sárközy. It turns out that numbers with fixed sum of digits are uniformly distributed in residue classes under some very general conditions. Namely, the underlying linear recurring sequence must have the property that there is no prime factor P of the modulus such that all but finitely many members of the sequence leave the same residue modulo P. The key step in the proof is an estimate for exponential sums using known theorems from Diophantine approximation.     

Heavy Elements and Electromagnetic Transients from Neutron Star Mergers

Compact binary mergers involving neutron stars can eject a fraction of their mass to space. Being extremely neutron rich, this material undergoes rapid neutron capture nucleosynthesis, and the resulting radioactivity powers fast, short-lived electromagnetic transients known as kilonova or macronova. Such transients are exciting probes of the most extreme physical conditions and their observation signals the enrichment of the Universe with heavy elements. Here the current understanding of the mass ejection mechanisms, the properties of the ejecta, and the resulting radioactive transients are reviewed. The first well-observed event in the aftermath of GW170817 delivered a wealth of insights, but much of today's picture of such events is still based on a patchwork of theoretical studies. Apart from summarizing the current understanding, questions where no consensus has been reached yet are also pointed out, and possible directions for the future research are sketched. In an appendix, a publicly available heating rate library based on the WinNet nuclear reaction network is described, and a simple fit formula to alleviate the implementation in hydrodynamic simulations is provided.     

On the preservation of periodic monotonicity

A 2-periodic continuous real functionf is said to beperiodically monotone if it has the following property: there exist numbert ₁t ₂t ₃t ₁+2 such thatf is nonincreasing fort ₁t ₂ and nondecreasing int ₂tt ₃. For any 2-periodic, integrable real functiong with ₀ ² |g(t|dt) we define

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV _n (z), wheren is the degree of the polynomial. The polynomial approximationsV _n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV _λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV _n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV _λ *f→f locally uniformly as λ→∞, and in fact when λ₂ > λ₁. We also consider Hadamard products of theV _λ with complex-valued harmonic mappings of the disk. This work was supported by the Volkswagen Stiftung (RiP-program at Oberwolfach). S. R. received partial support also from INTAS (Project 99-00089) and the German-Israeli Foundation (grant G-643-117.6/1999).     

Solid state 19F NMR parameters of fluorine-labeled amino acids. Part II: aliphatic substituents

A representative set of amino acids with aliphatic 19F-labels has been characterized here, following up our previous compilation of NMR parameters for single 19F-substituents on aromatic side chains. Their isotropic chemical shifts, chemical shift tensor parameters, intra-molecular 19F dipole-dipole couplings and temperature-dependent T1 and T2 relaxation times were determined by solid state NMR on twelve polycrystalline amino acid samples, and the corresponding isotropic 19F chemical shifts and scalar couplings were obtained in solution. Of particular interest are amino acids carrying a trifluoromethyl-group, because not only the 19F chemical shift but also the intra-CF3 homonuclear dipolar coupling can be used for structural studies of 19F-labeled peptides and proteins. The CF3-groups are further compared with CH2F-, CD2F-, and CD3-groups, using both 19F and 2H NMR to describe their motional behavior and to examine the respective linebroadening effects of the protonated and deuterated neighbors. We have also characterized two unnatural amino acids in which a CF3-label is rigidly connected to the backbone by a phenyl or bicyclopentyl moiety, and which are particularly well suited for structure analysis of membrane-bound polypeptides. The 19F NMR parameters of the polycrystalline amino acids are compared with data from the correspondingly labeled side chains in synthetic peptides.     

Solid state NMR analysis of the dipolar couplings within and between distant CF3-groups in a membrane-bound peptide

Dipolar couplings contain information on internuclear distances as well as orientational constraints. To characterize the structure of the antimicrobial peptide gramicidin S when bound to model membranes, two rigid 4-CF3-phenylglycine labels were attached to the cyclic backbone such that they reflect the behavior of the entire peptide. By solid state 19F NMR we measured the homonuclear dipolar couplings of the two trifluoromethyl-groups in oriented membrane samples. Using the CPMG experiment, both the strong couplings within each CF3-group as well as the weak coupling between the two CF3-groups could be detected. An intra-CF3-group dipolar coupling of 86 Hz and a weak inter-group coupling of 20 Hz were obtained by lineshape simulation of the complex dipolar spectrum. It is thus possible to explore the large distance range provided by 19F-labels and to resolve weak dipolar couplings even in the presence of strong intra-CF3 couplings. We applied this approach to distinguish and assign two epimers of the labeled gramicidin S peptide on the basis of their distinct 19F dipolar coupling patterns.     

Isotope (δ13C, δ15N, δ2H) diet–tissue discrimination in African grey parrot Psittacus erithacus: implications for forensic studies

Diet–tissue isotopic relationships established under controlled conditions are informative for determining the dietary sources and geographic provenance of organisms. We analysed δ¹³C, δ¹⁵N, and non-exchangeable δ²H values of captive African grey parrot Psittacus erithacus feathers grown on a fixed mixed-diet and borehole water. Diet–feather Δ¹³C and Δ¹⁵N discrimination values were +3.8?±?0.3?‰ and +6.3?±?0.7?‰ respectively; significantly greater than expected. Non-exchangeable δ²H feather values (?62.4?±?6.4?‰) were more negative than water (?26.1?±?2.5?‰) offered during feather growth. There was no positive relationship between the δ¹³C and δ¹⁵N values of the samples along each feather with the associated samples of food offered, or the feather non-exchangeable hydrogen isotope values with δ²H values of water, emphasising the complex processes involved in carbohydrate, protein, and income water routing to feather growth. Understanding the isotopic relationship between diet and feathers may provide greater clarity in the use of stable isotopes in feathers as a tool in determining origins of captive and wild-caught African grey parrots, a species that is widespread in aviculture and faces significant threats to wild populations. We suggest that these isotopic results, determined even in controlled laboratory conditions, be used with caution.     

Single Molecule Fluorescence of Native and Refolded Peridinin–Chlorophyll–Protein Complexes

Single molecule spectroscopy was applied to study the optical properties of native and refolded peridinin-chlorophyll-protein (PCP) complexes. The native system is a trimer with six chlorophyll a (Chl a) molecules, while the refolded one contains two Chl a and resembles structurally and spectroscopically the PCP monomer. The fluorescence emission of single PCP complexes strongly broadens with increasing excitation power. Simultaneously, the distribution of fluorescence maximum frequencies is also broadened. These spectral changes are attributed to photoinduced conformational changes of the protein that influence the fluorescence of embedded chromophores. Comparison of fluorescence intensities measured for PCP complexes in two different solvents indicates that the native PCP trimers are preserved in EDTA Tris buffer, while in PVA polymer matrix only monomers are stable.     

Impurity clustering and ferromagnetic interactions that are not carrier induced in dilute magnetic semiconductors: the case of Cu2O:Co

Current models for ferromagnetism in diluted magnetic semiconductors, such as "p-d exchange" or "double-exchange", rely on the presence of partially filled gap states. We point out a new mechanism, not requiring partially filled states, in which ferromagnetic coupling arises from the occupation of previously unoccupied levels when two transition metal impurities form a close pair. We find from first-principles calculations that this mechanism explains strong ferromagnetic coupling between Co impurities in Cu2O, and at the same time gives rise to Co clustering.