Serre duality for noncommutative projective schemes

We prove the Serre duality theorem for the noncommutative projective scheme when is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.

     

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, normal, and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small H cells. Our results are of the formN=AL ^D _SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD _SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.     

Rigid complexes via DG algebras

Let be a commutative ring, a commutative -algebra and a complex of -modules. We begin by constructing the square , which is also a complex of -modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism , then the pair is called a rigid complex over relative to (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes.

We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks (resp. ) along a finite (resp. essentially smooth) ring homomorphism .

In the subsequent paper, Rigid Dualizing Complexes over Commutative Rings, we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in our paper Rigid Dualizing Complexes and Perverse Sheaves on Schemes, where we give a comprehensive version of Grothendieck duality for schemes.

The idea of rigid complexes originates in noncommutative algebraic geometry, and is due to Van den Bergh (1997).

     

Fast enzyme dynamics at the active site of formate dehydrogenase

The role of femtosecond-picosecond structural dynamics of proteins in enzyme-catalyzed reactions is a hotly debated topic. We report infrared photon echo measurement of the formate dehydrogenase-NAD+-azide ternary complex. In contrast to earlier studies of protein dynamics, the data show complete spectral diffusion on the femtosecond-picosecond time scale with no static heterogeneity. This result indicates that this transition-state analogue complex completely samples the distribution of structures that determine the distribution of azide vibrational frequencies within a few picoseconds and that there are no slower motions that perturb the H-bond network at the active site.     

Nucleus Independent Chemical Shift (NICS) at Small Distances from the Molecular Plane: The Effect of Electron Density

When close to the molecular plane, the behavior of nucleus independent chemical shift (NICS) as a function of the distance from the molecular plane deviates from its behavior at larger distances. By using a dense grid of NICS-probes (BQs) it is shown that, when close to the molecular plane, maximal (absolute) NICS values are obtained above the atoms. These maxima move towards the center as the grid is elevated until the (absolute) maximum NICS is obtained at the center and stay there when the grid is further elevated. It is shown that this behavior is a result of the current density, which is influenced by the electron density, according to the Biot-Savart law, which, in turn, causes the induced magnetic field measured by the NICS. It is thus concluded that if magnetic aromaticity is studied, the NICS calculations should be carried out at a large enough distance so that only the π-ring current affects the NICS. At distances ≥2 Å, NICS(r)_π,zz=A+B*C^r. Using non-linear correlation for obtaining A, B and C and extrapolate to NICS(1)_π,zz and NICS(1.7)_π,zz is recommended as measures for aromaticity.     

Double-stranded cycles: toward C84's belt region

The reactivity of the double-stranded hydrocarbon cycle with two ether bridges (1) toward iodotrimethylsilane (TMSI) was investigated in some detail. The carbon skeleton of cycle 1 resembles the belt region of a C84 fullerene which makes it a potential precursor to the long sought after fully aromatic derivative. Upon exposure to TMSI, cycle 1 undergoes a cascade of reactions which involve different states of iodination/reduction which ultimately lead to the hydrogenated cycle 5a, whose structure was proven by single-crystal X-ray analysis. A deeper insight into mechanistic aspects of this sequence of conversions was gained by performing the reaction under dry and wet conditions, whereby the latter involved both normal and deuterated water. With the help of detailed NMR correlation studies and DFT computations, all important aspects were clarified including an unexpected selective H/D exchange at the naphthalenic moieties.     

Can Substituted Cyclopentadiene Become Aromatic or Antiaromatic?

Cyclopentadiene derivatives with electronegative (F, Cl) or electropositive (H(3)Si, Me(3)Si) bis-5,5-substituents were studied at the B3LYP/6-311G* level of theory. It was found that there is no special stabilization or destabilization for any of the derivatives; the energetic effects that were previously attributed to aromatic stabilization or antiaromatic destabilization are the result of interactions in the reference systems. A nucleus-independent chemical shift (NICS) scan study at the HF-GIAO/6-311+G* theoretical level of these and similar derivatives suggest that they all show different magnitudes of diamagnetic ring current. None of the derivatives shows a paramagnetic ring current. Thus, cyclopentadienes are neither aromatic nor antiaromatic. It is also concluded that a diamagnetic ring current is perhaps necessary but certainly not a sufficient condition for aromaticity. The NICS scan procedure describes the type of ring current in the system, whereas a single isotropic NICS value (i.e., NICS(1)) may wrongly assign the type of ring current. It is shown that neither NICS(1) nor the NICS scan procedure can be used as a single aromaticity criterion.     

Electrically pumped two-dimensional Bragg grating lasers

We demonstrate electrically pumped InGaAsP two-dimensional Bragg grating (2DBG) lasers with two line defects. The 2DBG structure uses a weak 2D index perturbation surface grating to control the optical modes in the plane of the wafer. Measurements of the 2DBG lasers show that modal control in both the longitudinal and transverse directions is due to the gratings and defects. The 2DBG lasers are promising candidates for single-mode, high power, and high efficiency large-area lasers.