Space-time discontinuous Galerkin method for the solution of fluid-structure interaction

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.     

Bounded type Siegel disks of finite type maps with few singular values

Let U be an open subset of the Riemann sphere \(\hat {\mathbb{C}}\). We give sufficient conditions for which a finite type map f: U → \(\hat {\mathbb{C}}\) with at most three singular values has a Siegel disk compactly contained in U and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery à la Douady-Ghys-Herman-?wi?tek. We also give sufficient conditions for which, instead, Δ has not compact closure in U. The main tool is the Schwarzian derivative and area inequalities à la Graczyk-?wi?tek.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

Strange products of projections

Let H be an infinite-dimensional Hilbert space. We show that there exist three orthogonal projections X ₁,X ₂,X ₃ onto closed subspaces of H such that for every 0 ≠ z ₀ ∈ H there exist k ₁, k ₂, · · · ∈ {1, 2, 3} so that the sequence of iterates defined by z _n = X _kn z _n ?1 does not converge in norm.     

Stereocontrolled Photooxygenations–A Valuable Synthetic Tool§

The stereochemical course of the singlet-oxygen ene reaction with acyclic olefins may be controlled if in the substrate conformational fixation (1,3-allylic strain) an allyl-ic substituent for interaction with the attacking oxygen enophile aligns. Various substrates were chosen to elucidate the features of the olefin that are necessary to control the sense (threo versus erythro) and the extent of the ii-facial preference of the singlet-oxygen attack. Depending on the electronic properties of the double bond and the nature of the allylic substituent, threo or erythro selectivity may be imposed through hydrogen bonding, electrostatic and steric effects and stereoelectronic alignment. Such directing properties, especially that of the hydroxy group, were also confirmed in the other reaction modes of singlet oxygen, namely the [4+2] cycloaddition to chiral naphthylenic alcohols and the [2+2] cycloaddition to an adamantylidene-substituted allylic alcohol. The syntheses of the natural products Merucathin and Iso-dihydromahubanolide B are two examples in which such stereocontrolled photooxygenations have been used as key steps to build up the required chirality diastereose-lectively.     

Preparation of optically active allylic hydroperoxy alcohols and 1, 3-diols by enzyme-catalyzed kinetic resolution and photooxygenation of chiral homoallylic alcohols

All four possible enantiomers of the 3-hydroperoxy-4-penten-1-ols 2a, b and their corresponding 4-pentene-1,3-diols 4a,b have been prepared for the first time in high enantiomeric purity (up to 98% ee) and in preparative amounts according to two distinct ways: First the photooxygenation of the racemic homoallylic alcohols 1 gave the racemic hydroperoxy alcohols 2, which have subsequently been kinetically resolved by horseradish peroxidase (HRP); alternatively, first the lipase-catalyzed resolution afforded the optically active homoallylic alcohols 1 and subsequent photooxygenation led to the enantiomerically enriched hydroperoxy alcohols 2.     

Magnetic circular dichroism study of a dicobalt(II) methionine aminopeptidase/fumagillin complex and dicobalt II-II and II-III model complexes

The dicobalt form of the metallohydrolase methionine aminopeptidase from Escherichia coli (CoCo EcMetAP) has an active site with one 5-coordinate Co (II) and a more weakly bound 6-coordinate Co (II). These metal ions are bridged by two carboxylate amino acid side chains and water or hydroxide, potentially enabling magnetic exchange coupling between the metals. We used variable-temperature, variable-field magnetic circular dichroism to determine whether such coupling occurs. CoCo EcMetAP's MCD spectrum shows distinct d-d transitions at 495 and 567 nm caused by 6- and 5-coordinate Co (II), respectively. The magnetization curves for 5- and 6-coordinate Co (II) are very different, indicating that their electronic ground states vary considerably, ruling out any coupling. When the fungal metabolite fumagillin binds to the CoCoEcMetAP, the qualitative MCD spectrum is unchanged; however, VTVH MCD data show that 5- and 6-coordinate Co (II) ions have similarly shaped magnetization curves, indicating that the Co (II) ions now share the same electronic ground state. Fitting the VTVH MCD data to a model in which dimer wave functions are calculated using a spin Hamiltonian with zero-field splitting showed the Co (II) ions to be weakly ferromagnetically coupled, with J = 2.9 cm (-1). Ferromagnetic coupling is unusual for dinuclear Co (II); therefore, to support the CoCoEcMetAP/fumagillin complex results, we also analyzed VTVH MCD data from a matched pair of dinuclear cobalt complexes, 1 and 2. Complex 1 shares the carboxylate and hydroxide-bridged dicobalt(II) structural motif with the active site of CoCo EcMetAP. Complex 2 contains a nearly isostructural Co (II) ion, but the Co (III) is diamagnetic, so any magnetic coupling is switched off, while the spectral features of the Co (II) ion remain. Magnetization data for 1, fitted to the dimer model, showed that the Co (II) ions were weakly ferromagnetically coupled, with J = 1.7 cm (-1). Magnetization data for Co (II) ions in 2, however, reflect loss of magnetic exchange coupling.     

Natural products as an inspiration in the diversity-oriented synthesis of bioactive compound libraries

The purpose of diversity-oriented synthesis is to drive the discovery of small molecules with previously unknown biological functions. Natural products necessarily populate biologically relevant chemical space, since they bind both their biosynthetic enzymes and their target macromolecules. Natural product families are, therefore, libraries of pre-validated, functionally diverse structures in which individual compounds selectively modulate unrelated macromolecular targets. This review describes examples of diversity-oriented syntheses which have, to some extent, been inspired by the structures of natural products. Particular emphasis is placed on innovations that allow the synthesis of compound libraries that, like natural products, are skeletally diverse. Mimicking the broad structural features of natural products may allow the discovery of compounds that modulate the functions of macromolecules for which ligands are not known. The ability of innovations in diversity-oriented synthesis to deliver such compounds is critically assessed.