Invertible point transformations,Painlevé test,and the second Painlevé transcendent

The techniques of invertible point transformations and the Painlevé analysis can be used to construct integrable ordinary differential equations. We compare both techniques for the second Painlevé transcendent.     

Chaotic Maps, Control Parameter, and Liapunov Exponent

We describe a one-dimensional chaotic map wherethe Liapunov exponent is a smooth function of a controlparameter.     

Lie-symmetry vector fields for linear and nonlinear wave equations

We derive the Lie symmetry vector fields for the linear wave equation u=0 and nonlinear wave equation u=u ³. The conformal vector fields for the underlying metric tensor fieldg are also given. We construct the conservation laws and derive similarity solutions. Furthermore, we perform a Painlevé test for the nonlinear wave equation and discuss whether Lie-Bäcklund vector fields exist.     

X-ray emission spectroscopy with EPMA applied to high Tc Superconductors using new spectrometer crystals

The High Temperature Superconductors are characterized by a very anisotropical structure and by typical arrangements of Cu- and O-atoms within CuO₂-planes. In studying the electronic structure of these compounds there is a strong demand for high resolution valence band spectroscopy. Using an electron probe microanalyzer we study the X-ray emission of O-K_ga and Cu-L emerging from polycrystalline YBa₂Cu₃O₇ and Tl₂Ba₂Ca₂Cu₃O₁₀. The O-K emission band is analyzed using a chlinochlore crystal in (001) orientation (2d = 28.4 Å). For the Cu-L emission band we use a beryl crystal in (10¯10) orientation (2d = 15.9 Å). Furthermore, orientation dependent X-ray emission spectroscopy of single-crystalline YBa₂Cu₃O₇ and Tl₂Ba₂Ca₁Cu₂O₈ specimens was performed. We present the partial O-K spectra which represent the O-2p ( = x, y, z) electronic densities of states of the valence band and compare them with calculated data.     

Non-linear autonomous systems of differential equations and Carleman linearization procedure

The non-linear autonomous of differential equations

$\text{x}\text{?}{}_{\mathrm{i}}\text{=}\text{∑}\text{j}\text{a}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{j}}\text{+}\text{∑}\text{j,k}\text{b}{}_{\mathrm{ijk}}\text{x}{}_{\mathrm{j}}\text{x}{}_{\mathrm{k}}\text{(}\text{x}\text{?}{}_{\mathrm{i}}\text{=dx}{}_{\mathrm{i}}\text{/dt, i, j, k= 1,2,…n)}$

which plays an important role in chemical kinetics and other fields of physics (turbulence and plasma physics) is investigated using the Carleman linearization procedure.     

A doubly degenerate Hubbard model in the strongly correlated limit

Using a doubly degenerate Hubbard model we derive an effective Hamiltonian in the limiting case where the Coulomb repulsion is much greater than the hopping constant. A brief discussion of this Hamiltonian is given.     

Fully Entangled Quantum States in C^{N^2 } and Bell Measurement

Entangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding, and quantum teleportation. We describe how to generate fully entangled states in the Hilbert space C ^N C ^N starting from a unitary matrix and show that they form an orthonormal basis in this space.     

Soft X-ray emission and the local densities of states of the Cu x Rh1−x alloy system

Soft X-ray emission experiments on Cu _x Rh_1–x alloys for the compositionsx=0.0, 0.05, 0.10, 0.80 and 1.0, have been performed. In this paper we present the spectra of Cu-and Rh-L _III core level-conduction band transitions. Using the data of recent self-consistent KKR-CPA electronic structure calculations we evaluate the corresponding SXS-cross sections and compare these theoretical results to our measurements.     

Relaxed loading conditions for higher order homogenisation approaches

The present paper deals with the formulation of minimal loading conditions for the application of numerical homogenisation techniques, namely the FE² methodology. Based on the set of volume averaging rules connecting the heterogeneous micro and the homogeneous macro scale, the minimal constraints on the deformation of a micro volume are derived for a classical Cauchy continuum as well as for a micromorphic continuum theory. For both cases, numerical studies are included highlighting the main aspects of the proposed procedure within the context of small deformations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal loading conditions for higher-order numerical homogenisation schemes

The present study deals with the formulation of minimal loading conditions for microscale applications in numerical two-scale modelling (FE²) approaches. From the homogenisation concept, a set of volume average rules constrains the microscale PDE to be solved. They are considered to be the minimal set of loading conditions and can be specified by additional polynomial or periodic assumptions, for example, on the microscale displacement field. Whereas the resulting volume integrals can be transformed into surface integrals for so-called first-order homogenisation schemes, this is not possible for a second-order homogenisation of second gradient or micromorphic effective media substituting a heterogeneous microcontinuum represented by a volume element on the microscale. Several numerical examples compare the minimal loading condition concept with standard techniques discussed in literature.