Singularities in derived categories

Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category D^b(A) of finitely generated A-modules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an exact triangle N→M⊕Z→Z→N[1]. In this paper we define and study the type of singularity at every degeneration in the bounded derived categrory.     

Asymmetric Synthesis of Adjacent Tri- and Tetrasubstituted Carbon Stereocenters: Organocatalytic Aldol Reaction of an Hydantoin Surrogate with Azaarene 2-Carbaldehydes

A bifunctional amine/squaramide catalyst promoted direct aldol addition of an hydantoin surrogate to pyridine 2-carbaldehyde N-oxides to afford adducts bearing two vicinal tertiary/quaternary carbons in high diastereo- and enantioselectivity (d.r. up to >20:1; ee up to 98 %) is reported. Acid hydrolysis of adducts followed by reduction of the N-oxide group yields enantiopure carbinol-tethered quaternary hydantoin-azaarene conjugates with densely functionalized skeletons. DFT studies of the potential energy surface (B3LYP/6–31+G(d)+CPCM (dichloromethane)) of the reaction correlate the activity of different catalysts and support an intramolecular hydrogen-bond-assisted activation of the squaramide moiety in the transition state of the catalytic reaction.     

Numerical solution of the 3D time dependent Schrödinger equation in spherical coordinates: Spectral basis and effects of split-operator technique

We study a numerical solution of the multi-dimensional time dependent Schrödinger equation using a split-operator technique for time stepping and a spectral approximation in the spatial coordinates. We are particularly interested in systems with near spherical symmetries. One expects these problems to be most efficiently computed in spherical coordinates as a coarse grain discretization should be sufficient in the angular directions. However, in this coordinate system the standard Fourier basis does not provide a good basis set in the radial direction. Here, we suggest an alternative basis set based on Chebyshev polynomials and a variable transformation.     

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we show that there is always an open adjointorbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k),thus answering positively in this gl_n(k) case to a questionraised independently by Michel Duflo and Dmitri Panyushev. Theproof gives an explicit construction, using -filtered modulesof quasi-hereditary algebras arising from quotients of the doubleof quivers of type A. An example of a seaweed Lie algebra ina simple Lie algebra of type E₈ not admitting an open orbitin its nilpotent radical is given.     

Mössbauer and magnetization studies of iron oxide nanocrystals

Monodisperse iron oxide nanocrystals have been produced following non-hydrolytic, thermal decomposition routes. Spherically shaped particles with diameter of 4 and 12 nm and cubic shaped particles with an edge length of 9 nm have been studied. The particles have been shown to consist of mainly maghemite. A reduction of the saturation magnetic hyperfine field is observed for the 4 nm particles as compared to the corresponding bulk value. The anisotropy energy determined from the temperature variation of the magnetic hyperfine field was strongly enhanced for the 4 nm particles.     

Error derivation in Romberg integration

The error term related to a Romberg type extrapolation scheme based on the use of an arbitrary quadrature formula is derived. A complete discussion, utilizing the known properties of the Bernoulli polynomials and their related periodic functions, is presented in the case of a repeated halving of the integration interval. The general expression for the error term is derived in the case of an arbitrary subdivision of the integration interval.Work performed while the author was working as staff member at CERN, Geneva, Switzerland.     

Romberg integration as a problem in interpolation theory

A complete derivation of Romberg integration for an arbitrary sequence of integration steplenghts, using classical interpolation theory only, is given. An explicit expression for the error is derived using Lagrange interpolation. From the general theory developed, several previous known results may be derived as special cases.     

Resonance effects in the binding of a nucleon to an unbound core

The theory of the single-particle form factor for a nucleon bound to an unbound core presented recently is implemented as a concrete three-particle model for ⁶Li. It is shown that for reasonable assumptions concerning the interactions involved one can reproduce the experimental binding energy and size of this nucleus.