Proof of a conjecture of S. Chowla

Let θ(k, p) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ … + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0(}\text{mod}\text{p)}$ has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: $\text{θ(k) = O(k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>+?</mtext>}}\text{)}$.     

On the usage of qualitative reasoning as an approach towards enterprise modelling

This paper analyzes the computational requirements of enterprise modelling, presents a reasoning framework, and reports about a prototype implementation. Enterprise modelling studies complex systems which are normally only partially known, and focuses on studying the qualitative properties of the underlying system, and thus differs from traditional quantitative approaches. Various modelling approaches from computer science and management science are discussed with respect to their suitability for enterprise modelling. Qualitative reasoning is identified as a promising basis to develop a computational framework for enterprise modelling. A rules-constraints-method of reasoning with imprecise knowledge, which is essentially an interval-based version of difference equations, is proposed as a powerful and versatile tool for enterprise modelling. As an illustrative example, a model of a business organization, derived from the concept of a financial statement, is developed and simulated.     

UV-B exposure causes DNA damage and changes in protein expression in northern pike (Esox lucius) posthatched embryos

The ongoing anthropogenically caused ozone depletion and climate change has increased the amount of biologically harmful UV-B radiation, which is detrimental to fish in embryonal stages. The effects of UV-B radiation on the levels and locations of DNA damage manifested as cyclobutane pyrimidine dimers (CPDs), heat shock protein 70 (HSP70) and p53 protein in newly hatched embryos of pike were examined. Pike larvae were exposed in the laboratory to current and enhanced doses of UV-B radiation. UV-B exposure caused the formation of CPDs in a fluence rate-dependent manner, and the CPDs were found deeper in the tissues with increasing fluence rates. UV-B radiation induced HSP70 in epidermis, and caused plausible p53 activation in the brain and epidermis of some individuals. Also at a fluence rate occurring in nature, the DNA damage in the brain and eyes of pike and changes in protein expression were followed by severe behavioral disorders, suggesting that neural molecular changes were associated with functional consequences.     

Channeling‐enhanced EDX for polarity resolved crystal orientation determination

Using channeling‐enhanced energy‐dispersive X‐ray spectroscopy (EDX), we demonstrate polarity sensitive orientation determination of a non‐centrosymmetric crystal in the scanning electron microscope. The authors observe a characteristic asymmetry in the channeling‐enhanced, angle‐dependent EDX data of a GaAs sample, which is in good agreement to simulations using the dynamical theory of diffraction for the incident electron beam. This allows us to assign the orientation of the GaAs crystal according to the non‐centrosymmetric point group. The method shown here overcomes the limitation of a reduced point‐group sensitivity of electron backscatter diffraction patterns and electron channeling patterns for crystalline phases that contain atoms of approximately equal electron scattering cross sections.     

Site-specific recoil diffraction of backscattered electrons in crystals

A novel diffraction effect in high-energy electron backscattering is demonstrated: the formation of element-specific diffraction patterns via nuclear recoil. For sapphire (Al(2)O(3)), the difference in recoil energy allows us to determine if an electron scattered from aluminum or from oxygen. The angular electron distribution obtained in such measurements is a strong function of the recoiling lattice site. These element-specific recoil diffraction features are explained using the dynamical theory of electron diffraction. Our observations open up new possibilities for local, element-resolved crystallographic analysis using quasielastically backscattered electrons in scanning electron microscopy.     

Perpendicular magnetic anisotropy induced by tetragonal distortion of FeCo alloy films grown on Pd(001)

We grew tetragonally distorted FexCo1-x alloy films on Pd(001). Theoretical first-principles calculations for such films predicted a high saturation magnetization and a high uniaxial magnetic anisotropy energy for specific values of the lattice distortion c/a and the alloy composition x. The magnetic anisotropy was investigated using the magneto-optical Kerr effect. An out-of-plane easy axis of magnetization was observed for Fe0.5Co0.5 films in the thickness range of 4 to 14 monolayers. The magnetic anisotropy energy induced by the tetragonal distortion is estimated to be almost 2 orders of magnitude larger than the value for bulk FeCo alloys. Using LEED Kikuchi patterns, a change of the easy axis of magnetization can be related to a decrease of the tetragonal distortion with thickness.     

Growth conditions for entire functions with only bounded Fatou components

Let f be a transcendental entire function of order less than 1/2. Denote the maximum and minimum modulus of f by M(r, f) = max{|f(z)|: |z| = r} and m(r, f) = min{|f(z)|: |z| = r}. We obtain a minimum modulus condition satisfied by many f of order zero that implies all Fatou components are bounded. A special case of our result is that if

Limit Functions for Convergence Groups and Uniformly Quasiregular Maps

We investigate the limit functions of iterates of a functionbelonging to a convergence group or of a uniformly quasiregularmapping. We show that it is not possible for a subsequence ofiterates to tend to a non-constant limit function, and for anothersubsequence of iterates to tend to a constant limit function.It follows that the closure of the stabiliser of a Siegel domainfor a uniformly quasiregular mapping is a compact abelian Liegroup, which we further conjecture to be infinite. This resultconcerning possible limits of convergent subsequences of iteratesfor holomorphic rational functions on the Riemann sphere isknown, and the only known method of proof involves universalcovering surfaces and Möbius groups. Hence, our methodyields a new and perhaps more elementary proof also in thatcase.     

Experimental observation of the strong influence of crystal orientation on Electron Rutherford Backscattering Spectra

In Electron Rutherford Backscattering Spectroscopy (ERBS) energetic electrons (in our case up to 40 keV) impinge on a target and one measures the energy of elastically scattered electrons. This energy depends on the mass of the scattering atom, due to the recoil effect. This technique thus provides information about the sample composition. For single crystals the interaction of the projectile electron with the crystal potential modifies the angular intensity distribution of the scattered electrons. This leads, for example, to the well-known Kikuchi patterns. Here we investigate if such modified angular distribution has any influence on the intensity ratio of the observed elastic peaks in ERBS. Dramatic effects are found. Implications of these observations for quantitative surface analysis using energetic electrons are discussed.