Synthese von Dibenzo[a,g]chinolizinen — Ein neuer Zugang zu Protoberberinen

Summary A novel synthesis to prepare dibenzo-8-quinolizinones8 is reported using the intramolecular cyclisation of the benzylated Reissert compounds6 for the key step. Methylation of8 by methyllithium gives partial deoxygenated coralynes10 which in turn can be reduced by NaBH₄ yielding the 8-methyltetrahydro-protoberberines11. The structures are assigned by NMR-spectroscopy.

     

A Discrete Element Approach for Wave Propagation in Thin Rods

Wave propagation and evoked side effects, such as material failure, are important parts of analysis of dynamically exposed structures like buildings or machinery. One of the numerical analysis tools for wave propagation is the well-known Finite Element Method (FEM) with its impressive performance but also with the drawback of not being able to model easily material failure, discontinuities and contacts. In contrast, the Discrete Element Method (DEM) is capable to describe these effects on a meso-scale more easily. The here selected concept is based on a DEM particle which is considered as deformable, and can establish and remove lasting bonds with other particles. The simulation example used is a thin rod that has been an extensive numerical and experimental research subject since the last century. A longitudinal wave is excited within the thin rod by simulating an impact on one end of the rod in a free-free configuration. It is found, that the simulation data, the velocity profile and the resulting displacement at the end of the rod, are in good agreement with experimental obtained data. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.     

Comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule

A comparison principle for viscosity sub- and super-solutions of second order elliptic partial differential equations is derived using the “fuzzy sum rule” of non-smooth calculus. This method allows us to weaken the assumptions made on the function F when the equation F(x,u,Du,D²u)=0 is under consideration.     

Continuous convolution semigroups integrating a submultiplicative function

Let be a submultiplicative function on a locally compact group G and let S be a convolution semigroup on G with Lévy measure . It is shown that the measures of S integrate if and only if integrates outside some neighbourhood of the identity of G (Theorem 1). Moreover if (X(t);t0) is the G-valued process with independent increments associated with the semigroup S it is shown that the measures of S integrate if and only if the random variable sup { (X(t)):0t1} is integrable (Theorem 2).     

Automatic differentiation of numerical integration algorithms

Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

     

An ALE finite element method for a coupled Stefan problem and Navier–Stokes equations with free capillary surface

In this article, an ALE finite element method to simulate the partial melting of a workpiece of metal is presented. The model includes the heat transport in both the solid and liquid part, fluid flow in the liquid phase by the Navier–Stokes equations, tracking of the melt interface solid/liquid by the Stefan condition, treatment of the capillary boundary accounting for surface tension effects and a radiative boundary condition. We show that an accurate treatment of the moving boundaries is crucial to resolve their respective influences on the flow field and thus on the overall energy transport correctly. This is achieved by a mesh‐moving method, which explicitly tracks the phase boundary and makes it possible to use a sharp interface model without singularities in the boundary conditions at the triple junction. A numerical example describing the welding of a thin‐steel wire end by a laser, where all aforementioned effects have to be taken into account, proves the effectiveness of the approach.Copyright © 2012 John Wiley & Sons, Ltd.     

A high-order full-discretization method using Hermite interpolation for periodic time-delayed differential equations

A high-order full-discretization method(FDM)using Hermite interpolation(HFDM) is proposed and implemented for periodic systems with time delay. Both Lagrange interpolation and Hermite interpolation are used to approximate state values and delayed state values in each discretization step. The transition matrix over a single period is determined and used for stability analysis. The proposed method increases the approximation order of the semidiscretization method and the FDM without increasing the computational time. The convergence, precision, and efficiency of the proposed method are investigated using several Mathieu equations and a complex turning model as examples. Comparison shows that the proposed HFDM converges faster and uses less computational time than existing methods.