On the Magnitude of the Sum of Bases of the Natural Numbers

We construct two bases of the natural numbers B₁ and B₂, eachof order two, such that (B₁ + B₂ (n) <n^{+c/(log n)}. For alower estimate, it is proved that if B₂ and are two bases, eachof order two, then (B₁+B₂)(n) > n. Generalisations to sumsof bases of order h > 2 are also given.     

Comparison of number–average molecular weights calculated by GPC and membrane osmometry

Data are presented which show that when a polymer contains an appreciable amount of low molecular weight species below the diffusion limit of the osmometer membrane, the osmotic molecular weight, M?_n, is generally higher than the M?_n calculated from gel-permeation chromatography (GPC). Experiments were performed on samples of poly(vinyl chloride) (PVC) and high-cis polybutadiene polymers. Osmotic data were obtained in the usual manner, while GPC data were obtained using the universal calibration approach. It was found that when all polymer species below approximately 10,000 molecular weight were excluded from the calculation of M?_n by GPC, agreement in M?_n was obtained between membrane osmometry and GPC. The data obtained suggest that the choice of M?_n as measured by membrane osmometry in the calibration of the GPC should not be done casually, as the measured M?_n may not reflect the “true” value of that sample, especially when the polymer sample contains an appreciable amount of low molecular weight material.     

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions inL ^p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2. This paper is a part of the author’s Ph.D. thesis supervised by V. Bergelson.     

Nonlinear and non-separable multiscale representations based on Lipschitz perturbation

In this Note, we present a new formalism for nonlinear and non-separable multiscale representations. The new formalism we propose brings about similarities between existing nonlinear multiscale representations and also allows us to alleviate the classical hypotheses made to prove the convergence of the multiscale representations.     

On Fuglede's Conjecture and the Existence of Universal Spectra

Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde’s Spectral Set Conjecture in both directions. Namely, in Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a nonspectral tile. In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang [14]. In particular, we prove here that the USC and the "tile → spectral" direction of Fuglede’s conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó [13] for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas [2], and obtain nonspectral tiles in . Fuglede’s conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].     

The Nonsmooth Dynamics of Combined Slip and Spin Motion Under Dry Friction

Journal of Nonlinear Science - The isothermal quasistatic (i.e. acceleration neglected) hardening-free plasticity at large strains is considered, based on the standard multiplicative decomposition...     

Novel tricyclic diamines 2. Synthesis of 1,7-diazaisoadamantane, 1,5-diazaisoadamantane and 1,6-diazahomobrendane

Three novel tricyclic diamines (1,7-diazaisoadamantane, 1,5-diazaisoadamantane and 1,6-diazahomobrendane) were prepared. A flexible synthetic strategy was employed which used flat, aromatic azaindoles as the starting materials. The requisite azaindoles were prepared by a tandem Sonogashira coupling/intramolecular cyclization reaction. Ring saturation, appropriate functionalization and intramolecular alkylation provided the three novel tricyclic diamines cores.     

High-resolution laser ablation inductively coupled plasma mass spectrometry used to study transport of metallic nanoparticles through collagen-rich microstructures in fibroblast multicellular spheroids

We have efficiently produced collagen-rich microstructures in fibroblast multicellular spheroids (MCSs) as a three-dimensional in vitro tissue analog to investigate silver (Ag) nanoparticle (NP) penetration. The MCS production was examined by changing the seeding cell number (500 to 40,000 cells) and the growth period (1 to 10 days). MCSs were incubated with Ag NP suspensions with a concentration of 5 μg mL⁻¹ for 24 h. For this study, laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) was used to visualize Ag NP localization quantitatively. Thin sections of MCSs were analyzed by LA-ICP-MS with a laser spot size of 8 μm to image distributions of ¹⁰⁹Ag, ³¹P, ⁶³Cu, ⁶⁶Zn, and ⁷⁹Br. A calibration using a NP suspension was applied to convert the measured Ag intensity into the number of NPs present. The determined numbers of NPs ranged from 30 to 7200 particles in an outer rim of MCS. The particle distribution was clearly correlated with the presence of ³¹P and ⁶⁶Zn and was localized in the outer rim of proliferating cells with a width that was equal to about twice the diameter of single cells. Moreover, abundant collagens were found in the outer rim of MCSs. For only the highest seeding cell number, NPs were completely captured at the outer rim, in a natural barrier reducing particle transport, whereas Eosin (⁷⁹Br) used as a probe of small molecules penetrated into the core of MCSs already after 1 min of exposure.

Fibroblast MCS could build up the barrier only for nanoparticles