Deformations of smooth toric surfaces

For a complete, smooth toric variety Y, we describe the graded vector space ${T_Y^1}$ . Furthermore, we show that smooth toric surfaces are unobstructed and that a smooth toric surface is rigid if and only if it is Fano. For a given toric surface we then construct homogeneous deformations by means of Minkowski decompositions of polyhedral subdivisions, compute their images under the Kodaira-Spencer map, and show that they span ${T_Y^1}$ .     

Web servers and services for electrostatics calculations with APBS and PDB2PQR

APBS and PDB2PQR are widely utilized free software packages for biomolecular electrostatics calculations. Using the Opal toolkit, we have developed a Web services framework for these software packages that enables the use of APBS and PDB2PQR by users who do not have local access to the necessary amount of computational capabilities. This not only increases accessibility of the software to a wider range of scientists, educators, and students but also increases the availability of electrostatics calculations on portable computing platforms. Users can access this new functionality in two ways. First, an Opal-enabled version of APBS is provided in current distributions, available freely on the web. Second, we have extended the PDB2PQR web server to provide an interface for the setup, execution, and visualization of electrostatic potentials as calculated by APBS. This web interface also uses the Opal framework which ensures the scalability needed to support the large APBS user community. Both of these resources are available from the APBS/PDB2PQR website: http://www.poissonboltzmann.org/.     

Synthesis of unsymmetrical 3,4-diaryl-3-pyrrolin-2-ones utilizing pyrrole Weinreb amides

A regiocontrolled synthesis of unsymmetrical 3,4-diaryl-3-pyrrolin-2-ones has been achieved in three steps from 1,2-diaryl-1-nitroethenes with pyrrole-2-carboxamides (pyrrole Weinreb amides) serving as the key linchpin intermediates. Two different methods for the preparation of the requisite nitroalkenes were investigated: (1) modified Henry reaction between arylnitromethanes and arylimines; and (2) Suzuki-Miyaura cross-coupling reaction of 2-aryl-1-bromo-1-nitroethenes with arylboronic acids. Some difficulty was encountered in the preparation of arylnitromethanes, thus leading to the exploration of a cross-coupling strategy that proved more useful. A Barton-Zard pyrrole cyclocondensation reaction between 1,2-diaryl-1-nitroethenes and N-methoxy-N-methyl-2-isocyanoacetamide gave the corresponding pyrrole Weinreb amides, which were then converted into the desired 3-pyrrolin-2-ones in two steps. Overall, this method allowed for the construction of 3,4-diaryl-3-pyrrolin-2-ones with complete regiocontrol of the substituents with respect to the lactam carbonyl. The utility of this synthetic methodology was demonstrated by the preparation of eight unsymmetrical and symmetrical 3,4-diaryl-3-pyrrolin-2-ones including the N-H lactam analogue of the selective COX-II inhibitor, rofecoxib.     

A systematic investigation of quaternary ammonium ions as asymmetric phase-transfer catalysts. Synthesis of catalyst libraries and evaluation of catalyst activity

Despite over three decades of research into asymmetric phase-transfer catalysis (APTC), a fundamental understanding of the factors that affect the rate and stereoselectivity of this important process are still obscure. This paper describes the initial stages of a long-term program aimed at elucidating the physical organic foundations of APTC employing a chemoinformatic analysis of the alkylation of a protected glycine imine with libraries of enantiomerically enriched quaternary ammonium ions. The synthesis of the quaternary ammonium ions follows a diversity-oriented approach wherein the tandem inter[4 + 2]/intra[3 + 2] cycloaddition of nitroalkenes serves as the key transformation. A two-part synthetic strategy comprised of (1) preparation of enantioenriched scaffolds and (2) development of parallel synthesis procedures is described. The strategy allows for the facile introduction of four variable groups in the vicinity of a stereogenic quaternary ammonium ion. The quaternary ammonium ions exhibited a wide range of activity and to a lesser degree enantioselectivity. Catalyst activity and selectivity are rationalized in a qualitative way on the basis of the effective positive potential of the ammonium ion.     

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC–WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319–338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC–WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.     

Pressure broadening and collisional shift of the Rb D2 absorption line by CH4, C2H6, C3H8, n-C4H10, and He

The pressure broadening and shift rates of the rubidium D2 absorption line 5²S_1/2→5²P_3/2 (780.24 nm) with CH₄, C₂H₆, C₃H₈, n-C₄H₁₀, and He were measured for pressures ≤80 Torr using high-resolution laser spectroscopy. The broadening rates γ_B for CH₄, C₂H₆, C₃H₈, n-C₄H₁₀, and He are 28.0, 28.1, 30.5, 31.3, and 20.3 (MHz/Torr), respectively. The corresponding shift rates γ_S are −8.4, −8.8, −9.7, −10.0, and 0.39 (MHz/Torr), respectively. The measured rates of Rb for the hydrocarbon buffer gas series of this study are also compared to the theoretically calculated rates of a purely attractive van der Waals difference potential. Good agreement is found to exist between measured and theoretical rates.     

Evaluation of the performance of inexact GMRES

The inexact GMRES algorithm is a variant of the GMRES algorithm where matrix-vector products are performed inexactly, either out of necessity or deliberately, as part of a trading of accuracy for speed. Recent studies have shown that relaxing matrix-vector products in this way can be justified theoretically and experimentally. Research, so far, has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is liable to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up being greater than that of the exact GMRES if there were not enough savings in the matrix-vector products. In this paper, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that provide instructive insights into results affected by the build-up of the inexactness. Such information is of vital importance to practitioners who need to decide whether switching their workflow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments that gauge the effectiveness of the inexact scheme and its suitability for use in addressing certain problems, depending on how much inexactness is allowed in the matrix-vector products.     

Noncrossing partitions and the shard intersection order

We define a new lattice structure \((W,\preceq)\) on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC?(W) as a sublattice. The new construction of NC?(W) yields a new proof that NC?(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of \((W,\preceq)\), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC?(W). There is a natural dimension-preserving bijection between simplices in the order complex of \((W,\preceq)\) (i.e. chains in \((W,\preceq)\)) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC?(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron.The lattice \((W,\preceq)\) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.     

Generic transient memory formation in disordered systems with noise

Out-of-equilibrium disordered systems may form memories of external driving in a remarkable fashion. The system "remembers" multiple values from a series of training inputs yet "forgets" nearly all of them at long times despite the inputs being continually repeated. Here, learning and forgetting are inseparable aspects of a single process. The memory loss may be prevented by the addition of noise. We identify a class of systems with this behavior, giving as an example a model of non-Brownian suspensions under cyclic shear.