On the moments of Kloosterman sums and fibre products of Kloosterman curves

Let q=p^r with p=3 and r2. We give a recursion formula for the moments of a Kloosterman sum over the finite field , which utilizes known weight formulae for the ternary Melas code M of length q−1. The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As an application for the formulae, and for their analogues obtained earlier in case p=2, we get the exact number of rational points on fibre products of certain Kloosterman curves. As a corollary we obtain identities between Ramanujan's tau-function, Kronecker class numbers, and Dickson polynomials.     

Pore structure of pyrolyzed scrap tires

Internal structure of carbon black produced by pyrolysis (CBp) of rubber samples from the top and bottom parts of sidewall and tread of a passenger car tire was investigated in nitrogen flow at different temperatures. The pore structure (specific surface area, pore size distribution, and porosity) of CBp and commercial CB, was compared. The development of pore structure and the increase of the specific surface area were most intensive during the thermal decomposition at temperatures ranging from 300°C to 500°C. This is caused by the intensive release of volatiles during the pyrolysis. After the pyrolysis was finished, at temperatures above 500°C, further decomposition of solid matter was associated with a slight increase of the specific surface area. Presented at the 34th International Conference of the Slovak Society of Chemical Engineering, Tatranské Matliare, 21–25 May 2007.     

The trapezoidal cylinder phase: a new mode of self-assembly in liquid-crystalline soft matter

A new rectangular columnar liquid crystalline phase with p2gg lattice is reported, which represents a polygonal cylinder array composed of cylinders with trapezoidal cross section. In these polygonal cylinders, one of the sides has a different length and is composed of a different material than the others. This tiling pattern was obtained in two series of T-shaped facial amphiphilic triblock molecules in which a rigid rod-like p-terphenyl core is substituted laterally by a polar and flexible oligoethylene glycol chain, terminated either by a hydrogen-bonding COOH group or by a Li carboxylate group, and having identical or different alkyl groups in the terminal positions. The trapezoidal cylinder phase provides an improved packing for relatively long and rigid alkyl chains at lower temperature and more space inside the polygonal cylinders than triangular cylinders. This combination of conformational and space-filling effects leads to different phase sequences. The trapezoidal cylinder phases pave the way to a new level of complexity in LC engineering and show the huge potential of the general concept of polyphilic tectons for the design of new complex soft matter structures.     

Directional states of symmetric-top molecules produced by combined static and radiative electric fields

We show that combined electrostatic and radiative fields can greatly amplify the directional properties, such as axis orientation and alignment, of symmetric top molecules. In our computational study, we consider all four symmetry combinations of the prolate and oblate inertia and polarizability tensors, as well as the collinear and perpendicular (or tilted) geometries of the two fields. In, respectively, the collinear or perpendicular fields, the oblate or prolate polarizability interaction due to the radiative field forces the permanent dipole into alignment with the static field. Two mechanisms are found to be responsible for the amplification of the molecules' orientation, which ensues once the static field is turned on: (a) permanent-dipole coupling of the opposite-parity tunneling doublets created by the oblate polarizability interaction in collinear static and radiative fields and (b) hybridization of the opposite parity states via the polarizability interaction and their coupling by the permanent dipole interaction to the collinear or perpendicular static field. In perpendicular fields, the oblate polarizability interaction, along with the loss of cylindrical symmetry, is found to preclude the wrong-way orientation, causing all states to become high-field seeking with respect to the static field. The adiabatic labels of the states in the tilted fields depend on the adiabatic path taken through the parameter space comprised of the permanent and induced-dipole interaction parameters and the tilt angle between the two field vectors.     

Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3

A benchmark set of 28 medium-sized organic molecules is assembled that covers the most important classes of chromophores including polyenes and other unsaturated aliphatic compounds, aromatic hydrocarbons, heterocycles, carbonyl compounds, and nucleobases. Vertical excitation energies and one-electron properties are computed for the valence excited states of these molecules using both multiconfigurational second-order perturbation theory, CASPT2, and a hierarchy of coupled cluster methods, CC2, CCSD, and CC3. The calculations are done at identical geometries (MP26-31G*) and with the same basis set (TZVP). In most cases, the CC3 results are very close to the CASPT2 results, whereas there are larger deviations with CC2 and CCSD, especially in singlet excited states that are not dominated by single excitations. Statistical evaluations of the calculated vertical excitation energies for 223 states are presented and discussed in order to assess the relative merits of the applied methods. CC2 reproduces the CC3 reference data for the singlets better than CCSD. On the basis of the current computational results and an extensive survey of the literature, we propose best estimates for the energies of 104 singlet and 63 triplet excited states.     

Development of Polar Order in the Liquid Crystal Phases of a 4‐Cyanoresorcinol‐Based Bent‐Core Mesogen with Fluorinated Azobenzene Wings

A bent‐core mesogen consisting of a 4‐cyanoresorcinol unit as the central core and laterally fluorinated azobenzene wings forms four different smectic LC phase structures in the sequence SmA–SmC_s–SmC_sP_AR–M, all involving polar SmC_sP_S domains with growing coherence length of tilt and polar order on decreasing temperature. The SmA phase is a cluster‐type de Vries phase with randomized tilt and polar direction; in the paraelectric SmC_s phase the tilt becomes uniform, although polar order is still short‐range. Increasing polar correlation leads to a new tilted and randomized polar smectic phase with antipolar correlation between the domains (SmC_sP_AR) which then transforms into a viscous polar mesophase M. As another interesting feature, spontaneous symmetry breaking by formation of a conglomerate of chiral domains is observed in the non‐polar paraelectric SmC_s phase.     

Non-Stationary Abstract Friedrichs Systems

Inspired by the results of Ern et al. (Commun Partial Differ Equ 32:317–341, 2007) on the abstract theory for Friedrichs symmetric positive systems, we give the existence and uniqueness result for the initial- (boundary) value problem for the non-stationary abstract Friedrichs system. Despite the absence of the well-posedness result for such systems, there were already attempts for their numerical treatment by Burman et al. (SIAM J Numer Anal 48:2019–2042, 2010) and Bui-Thanh et al. (SIAM J Numer Anal 51:1933–1958, 2013). We use the semigroup theory approach and prove that the operator involved satisfies the conditions of the Hille–Yosida generation theorem. We also address the semilinear problem and apply the new results to a number of examples, such as the symmetric hyperbolic system, the unsteady div–grad problem, and the wave equation. Special attention was paid to the (generalised) unsteady Maxwell system.     

Extensions of the Fill–Fishkind formula and the infimum – parallel sum relation

Recently, in (Arias ML, Corach G, Maestripieri A. Range additivity, shorted operator and the Sherman–Morrison–Woodburry formula, Linear Algebra Appl. 2015;467), authors gave a generalization of a formula by Fill and Fishkind, regarding the Moore–Penrose inverse of the sum of two matrices, in the setting of arbitrary Hilbert spaces. We consider this formula under some weaker assumptions and derive certain conclusions generalizing the mentioned result. We also extend a formula connecting the infimum of two orthogonal projections and their parallel sum to a formula connecting the star-infimum and the parallel sum of operators which need not be positive, using the concept of parallel sum that was introduced in (Antezana J, Corach G, Stojanoff D. Bilateral shorted operators and parallel sums. Linear Algebra Appl. 2006;414).     

Alpha-substituted organic peroxides: synthetic strategies for a biologically important class of gem-dihydroperoxide and perketal derivatives

In this paper we review the recent developments in the synthesis of alpha-substituted hydroperoxides. Alpha-substituted hydroperoxides are interesting compounds due to their chemistry and bioactivity and as intermediates for the synthesis of other peroxides, of which cyclic peroxides are of major importance. Although the emphasis of this report will be on the derivatives of gem-dihydroperoxides, perketals, as well as the less studied nitrogen and sulfur derivatives, will also be covered.