The union of minimal hitting sets: Parameterized combinatorial bounds and counting

A k-hitting set in a hypergraph is a set of at most k vertices that intersects all hyperedges. We study the union of all inclusion-minimal k-hitting sets in hypergraphs of rank r (where the rank is the maximum size of hyperedges). We show that this union is relevant for certain combinatorial inference problems and give worst-case bounds on its size, depending on r and k. For r=2 our result is tight, and for each r3 we have an asymptotically optimal bound and make progress regarding the constant factor. The exact worst-case size for r3 remains an open problem. We also propose an algorithm for counting all k-hitting sets in hypergraphs of rank r. Its asymptotic runtime matches the best one known for the much more special problem of finding one k-hitting set. The results are used for efficient counting of k-hitting sets that contain any particular vertex.     

Nonempirical calculations of NMR indirect spin-spin coupling constants. Part 14: azomethines of the alpha, beta-unsaturated aldehydes

Configurational assignment of seven azomethines obtained from the alpha-functionally substituted and nonsubstituted alpha,beta-unsaturated aldehydes has been performed on the basis of experimental measurements and the high-level ab initio calculations of their 1J(C,C) and 1J(C,H), involving the alpha-imino carbon that demonstrated the marked stereochemical dependence of both coupling constants upon the orientation of the nitrogen lone pair in the diverse isomers of the title azomethines.     

A priori resolution of the intermediate spectra in the bacteriorhodopsin photocycle: the time evolution of the L spectrum revealed

Resolution of the spectra of the intermediates in the photocycle of wild-type bacteriorhodopsin (BR) was achieved by singular value decomposition with exponential-fit-assisted self-modeling (SVD-EFASM) treatment of multichannel difference spectra measured at 5 degrees C during the course of the photocycle. New is the finding that two spectrally distinct L intermediates, L(1) and L(2), form sequentially. Our conclusion is that the photocycle is more complex than most published schemes. The dissection of the spectrally different L forms eliminates stoichiometric discrepancies usually appearing as systematically varying total intermediate concentrations before the onset of BR recovery. In addition, our analysis reveals that the red tails in the spectra of K and L(1) are more substantial than those of L(2) and BR. We suggest that these subtle differences in the shapes of the spectra reflect torsional and/or environmental differences in the retinyl chromophore.     

Synthesis and structures of silicon-, germanium- and tin-containing tungsten imido alkyl complexes (ArN)2W(CH2EMe3)2 (E = Si, Ge, Sn)

The novel silicon-, germanium- and tin-containing imido alkyl complexes of tungsten of the type (ArN)₂W(CH₂EMe₃)₂ (; E = Si (1), Ge (2), Sn (3)) have been prepared by the reactions of (ArN)₂WCl₂(dme) (dme = 1,2-dimethoxyethane) with heteroelement-containing alkyllithium or Grignard reagents Me₃ECH₂Li (E = Si, Ge), Me₃ECH₂MgCl (E = Ge, Sn). The title compounds were isolated in high yields as crystalline solids and characterized by elemental analysis, IR, ¹H, ¹³C, ²⁹Si and ¹¹⁹Sn NMR spectroscopy and X-ray diffraction studies. The geometry of the W atoms in the compounds can be described as a distorted tetrahedron.     

Solid-state laser spectral narrowing using a volumetric photothermal refractive Bragg grating cavity mirror

Dramatic spectral narrowing of two normally broadband lasers, Ti:sapphire and Cr:LiSAF, was achieved by simply replacing the output mirror with a reflective, volumetric Bragg grating recorded in photothermal refractive glass. The output power of each laser was unchanged from that obtained using dielectric coated output mirrors with the same output coupling as the Bragg grating while spectral brightness increased by 3 orders of magnitude.     

Effect of photonic crystal structure on the nonlinear optical anisotropy of birefringent porous silicon

Anisotropic photonic crystal structures consisting of birefringent porous silicon layers with alternating porosity were fabricated. The in-plane birefringence formed as a result of anisotropic etching in Si(110) results in unique multilayered structures with two distinct photonic bandgaps for orthogonal light polarizations. Nonlinear optical studies based on the third-harmonic generation from these structures demonstrate variation in the symmetry of the nonlinear optical response.     

A New Fluorescent Squaraine Probe for the Measurement of Membrane Polarity

The present study was undertaken to evaluate the sensitivity of newly synthesized squaraine dye 1 to the changes in lipid bilayer physical properties and compared it with the well-known dye 2. Partitioning of the dye 1 into lipid bilayer was found to be followed by significant increase of its fluorescence intensity and red-shift of emission maximum, while intensity of the dye 2 fluorescence increased only slightly on going from aqueous to lipidic environment. This suggests that dye 1 is more sensitive to the changes in membrane properties as compared to dye 2. Partition coefficients of the dye 1 have been determined for the model membranes composed of zwitterionic phospholipid phosphatidylcholine (PC) and its mixtures with positively charged detergent cetyltrimethylammonium bromide (CTAB), anionic phospholipid cardiolipin (CL), and sterol (Chol). The spectral responses of the dye 1 in different liposome media proved to correlate with the increase of bilayer polarity induced by Chol and CL or its decrease caused by CTAB. It was concluded that dye 1 can be used as fluorescent probe for examining membrane-related processes.     

One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.     

Homogenization of Unidirectional and Arbitrarily Oriented Fiber-Reinforced Materials by the Method of Conditional Moments

In this paper the method of conditional moments is developed for the case of a two–component matrix composite with randomly distributed unidirectional and arbitrarily oriented ellipsoidal inclusions. The algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters and orientation of inclusions is discussed. It is assumed that the components of the composite show orthotropic symmetry of thermoelastic properties. As a numerical example arbolite (straw particle inclusions in a cement matrix) is considered. The dependencies of Young's moduli, Poisson's ratios and shear moduli from the concentration of inclusions and for certain orientations of the inclusions are predicted and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transition wave in a supported heavy beam

We consider a heavy, uniform, elastic beam rested on periodically distributed supports as a simplified model of a bridge. The supports are subjected to a partial destruction propagating as a failure wave along the beam. Three related models are examined and compared: (a) a uniform elastic beam on a distributed elastic foundation, (b) an elastic beam in which the mass is concentrated at a discrete set of points corresponding to the discrete set of the elastic supports and (c) a uniform elastic beam on a set of discrete elastic supports. Stiffness of the support is assumed to drop when the stress reaches a critical value. In the formulation, it is also assumed that, at the moment of the support damage, the value of the ‘added mass’, which reflects the dynamic response of the support, is dropped too. Strong similarities in the behavior of the continuous and discrete-continuous models are detected. Three speed regimes, subsonic, intersonic and supersonic, where the failure wave is or is not accompanied by elastic waves excited by the moving jump in the support stiffness, are considered and related characteristic speeds are determined. With respect to these continuous and discrete-continuous models, the conditions are found for the failure wave to exist, to propagate uniformly or to accelerate. It is also found that such beam-related transition wave can propagate steadily only at the intersonic speeds. It is remarkable that the steady-state speed appears to decrease as the jump of the stiffness increases.