Metric and Periodic Lines in Real Inner Product Space Geometries

The lines of euclidean and hyperbolic geometries are characterized as metric lines in the sense of Blumenthal–Menger, and the lines of spherical and elliptic geometry as periodic ones. In euclidean geometries there do not exist periodic lines.     

On the Functional Equation of Periodic Lines

It will be proved that the functional equation of periodic lines has no solution for hyperbolic distances.     

Intact size-selected Au(n) clusters on a TiO2(110)-(1 x 1) surface at room temperature

Experiments in which mass-selected gold clusters were deposited on a surface have found that the catalytic properties depend strongly on cluster size. However, these experiments have not established definitively that the clusters maintain their size after deposition. We report here work in which we deposit low kinetic energy, mass-selected Aun+ (n = 1-8) clusters on a rutile TiO2(1 x 1) surface and use ultrahigh vacuum scanning tunneling microscopy (UHV-STM) to determine their size and shape.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

U6 is unsuitable for normalization of serum miRNA levels in patients with sepsis or liver fibrosis

MicroRNA (miRNA) levels in serum have recently emerged as potential novel biomarkers for various diseases. miRNAs are routinely measured by standard quantitative real-time PCR (qPCR); however, the high sensitivity of qPCR demands appropriate normalization to correct for nonbiological variation. Presently, RNU6B (U6) is used for data normalization of circulating miRNAs in many studies. However, it was suggested that serum levels of U6 themselves might differ between individuals. Therefore, no consensus has been reached on the best normalization strategy in ‘circulating miRNA''. We analyzed U6 levels as well as levels of spiked-in SV40-RNA in sera of 44 healthy volunteers, 203 intensive care unit patients and 64 patients with liver fibrosis. Levels of U6 demonstrated a high variability in sera of healthy donors, patients with critical illness and liver fibrosis. This high variability could also be confirmed in sera of mice after the cecal ligation and puncture procedure. Most importantly, levels of circulating U6 were significantly upregulated in sera of patients with critical illness and sepsis compared with controls and correlated with established markers of inflammation. In patients with liver fibrosis, U6 levels were significantly downregulated. In contrast, levels of spiked-in SV40 displayed a significantly higher stability both in human cohorts (healthy, critical illness, liver fibrosis) and in mice. Thus, we conclude that U6 levels in the serum are dysregulated in a disease-specific manner. Therefore, U6 should not be used for data normalization of circulating miRNAs in inflammatory diseases and previous studies using this approach should be interpreted with caution. Further studies are warranted to identify specific regulatory processes of U6 levels in sepsis and liver fibrosis.     

Convective Temperature Fluctuations in Liquid Gallium in Dependence on Static and Rotating Magnetic Fields

The influence of static axial, static transversal, and rotating magnetic fields on convective temperature fluctuations in liquid gallium (temperature range around 800 °C, Prandtl number 2.5 · 10⁻³) has been investigated. For a Rayleigh number of 6.3 · 10⁻⁵ (ΔT = 10°C, h = 50 mm), convective temperature fluctuations with peak to peak values of 3 °C have been measured. Depending on the strength, the frequency, and the configuration of the field, they could be eliminated to ΔT < 0.01 °C (static axial field of 182 mT), damped to a nearly periodic state with 0.1 °C amplitudes (static transversal field of 45 mT), or reduced to 0.03 °C oscillations by applying a rotating field.     

On a general principle in geometry that leads to functional equations

Summary Suppose that an invariant (or an invariant notion) of some geometry is given, like the distance between two points, the cross ratio of four points, the tangential distance between two spheres (or like the notion of orthogonality, of order, of a circle). One may ask what are the functions preserving (or preserving partially) that invariant (invariant notion). Originating from this principle some functional equation problems are formulated, namely the functional equations of distance, of area, of angle preservance.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.