Asymmetric Electrophilic alpha-Amidoalkylation, VII1): Generation, Crystal Structure, and Trapping Reactions of a Chiral 6,7-Dimethoxy-1,2,3,4-tetrahydroisoquinoline Derived N-Acyliminium Ion

The camphanic acid amide 4 has efficiently been oxidized with triphenylcarbenium tetrafluoroborate (3) to yield the chiral N-acyliminium ion 1. Trapping reactions of 1 with the silyl nucleophiles 7a-c and 10a-f proceeded with stereoselective bond formation, affording the diastereomers (R)-8/(S)-9a-c and (R)-11/(S)-12a-f, respectively, with diastereoselectivities of up to 93.9/6.1. The amido ketones (R)-8/(S)-9a-c were employed in the synthesis of the secondary amines (R)-16a-c, (S)-16a and for the preparation of (-)-homolaudanosine (R)-18. By X-ray crystallography the conformation of 1 in the crystal lattice was established and the preferred conformation of 1 in solution was elucidated by NOE experiments. Finally, the addition reaction of 7a to the iminium ion 21 derived from menthyl carbamate 20 was investigated, which reaction, however, proceeded only with insignificant asymmetric induction

Comparison of two notions of subharmonicity on non-archimedean curves

We show that a continuous function on the analytification of a smooth proper algebraic curve over a non-archimedean field is subharmonic in the sense of Thuillier if and only if it is psh, i.e. subharmonic in the sense of Chambert-Loir and Ducros. This equivalence implies that the property psh for continuous functions is stable under pullback with respect to morphisms of curves. Furthermore, we prove an analogue of the monotone regularization theorem on the analytification of the projective line and Mumford curves using this equivalence.Comment: v3: To appear in Mathematische Zeitschrift. 32 page

A Note on Diphthongization

published or submitted for publicationis peer reviewe

Topology-guided sampling of nonhomogeneous random processes

Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only through experimental observation or numerical simulations. In both cases, the data on which the topological measurements are based are derived via some form of finite sampling or discretization. In this paper, we present a probabilistic approach to quantifying the number of components of generalized nodal domains of nonhomogeneous random processes on the real line via finite discretizations, that is, we consider excursion sets of a random process relative to a nonconstant deterministic threshold function. Our results furnish explicit probabilistic a priori bounds for the suitability of certain discretization sizes and also provide information for the choice of location of the sampling points in order to minimize the error probability. We illustrate our results for a variety of random processes, demonstrate how they can be used to sample the classical nodal domains of deterministic functions perturbed by additive noise and discuss their relation to the density of zeros.Comment: Published in at http://dx.doi.org/10.1214/09-AAP652 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer

Probabilistic validation of homology computations for nodal domains

Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper, we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random fields in one and two space dimensions, which furnishes explicit probabilistic a priori bounds for the suitability of certain discretization sizes. We illustrate our results for the special cases of random periodic fields and random trigonometric polynomials.Comment: Published at http://dx.doi.org/10.1214/105051607000000050 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analytical description of interference between two misaligned and mismatched complete Gaussian beams

A typical application for laser interferometers is a precision measurement of length changes that result in interferometric phase shifts. Such phase changes are typically predicted numerically, due to the com- plexity of the overlap integral that needs to be solved. In this paper we will derive analytical representations of the interferometric phase and contrast (aka. fringe visibility) for two beam interferometers, both homodyne and heterodyne. The fundamental Gaussian beams can be arbitrarily misaligned and mismatched to each other. A limitation of the analytical result is that both beams must be detected completely, which can experimentally be realized by a sufficiently large single-element photodetector.Comment: 8 pages, 2 figure

Chromosome Centromeres: Structural and Analytical Investigations with High Resolution Scanning Electron Microscopy in Combination with Focused Ion Beam Milling

Whole mount mitotic metaphase chromosomes of different plants and animals were investigated with high resolution field emission scanning electron microscopy (FESEM) to study the ultrastructural organization of centromeres, including metacentric, acrocentric, telocentric, and holocentric chromosome variants. It could be shown that, in general, primary constrictions have distinctive ultrastructural features characterized by parallel matrix fibrils and fewer smaller chromomeres. Exposure of these structures depends on cell cycle synchronization prior to chromosome isolation, chromosome size, and chromosome isolation technique. Chromosomes without primary constrictions, small chromosomes, and holocentric chromosomes do not exhibit distinct ultrastructural elements that could be directly correlated to centromere function. Putative spindle structures, although rarely observed, spread over the primary constriction to the bordering pericentric regions. Analytical FESEM techniques, including specific DNA staining with Pt blue, staining of protein as a substance class with silver-colloid, and artificial loosening of fixed chromosomes with proteinase K, were applied, showing that centromere variants and ultrastructural elements in the centromere differ in DNA and protein distribution. Immunogold localization allowed high-resolution comparison between chromosomes with different centromere orientations of the distribution of centromere-related histone variants, phosphorylated histone H3 (ser10), and CENH3. A novel application of FESEM combined with focused ion beam milling (FIB) provided new insights into the spatial distribution of these histone variants in barley chromosomes. Copyright (C) 2009 S. Karger AG, Base