Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

     

Combining Swaps and Node Weights in an Adaptive Greedy Approach for the Maximum Clique Problem

In this work, the NP-hard maximum clique problem on graphs is considered. Starting from basic greedy heuristics, modifications and improvements are proposed and combined in a two-phase heuristic procedure. In the first phase an improved greedy procedure is applied starting from each node of the graph; on the basis of the results of this phase a reduced subset of nodes is selected and an adaptive greedy algorithm is repeatedly started to build cliques around such nodes. In each restart the selection of nodes is biased by the maximal clique generated in the previous execution. Computational results are reported on the DIMACS benchmarks suite. Remarkably, the two-phase procedure successfully solves the difficult Brockington-Culberson instances, and is generally competitive with state-of-the-art much more complex heuristics.     

Projective varieties with projectively equivalent singular 0-dimensional sections

Fix integers m, n such that 1 ≤ m ≤ n ? 3. Let X ? Pⁿ be an integral non-degenerate m-dimensional variety. Assume either char(K) = 0 or char(K) > deg(X). Here we prove that all general 0-dimensional sections of X containing a tangent vector to a smooth point of X are protectively equivalent if and only if n ? m + 1 ≤ deg(X) ≤ n ? m + 2.     

Laser spectroscopic investigation of a new precursor-type polyparaphenylenevinylene

Optical, structural and morphological properties of thin films of polyparaphenylenevinylene (PPV) formed by an alkyl sulfinyl precursor route have been studied. Thin films were fabricated on an optical glass and on quartz glass either by spin-coating of the precursor solution or by layer-by-layer deposition using Langmuir–Blodgett technique. PPV precursor films were also spin-coated on gold-coated glass in order to study thin-film optical parameters by surface plasmon spectroscopy. We have been successful in forming about 40 precursor mono layers on quartz glass by Langmuir–Blodgett technique using optimized surface pressure and dipping conditions. After thermal conversion of the precursor layers good quality fluorescent PPV films of yellow colour have been obtained. Optical characterization of the films was carried out by linear absorption and emission spectroscopy, ellipsometry, and surface plasmon spectroscopy. Structural and morphological studies on the thin films were carried out by using X-ray scattering and atomic force microscopy. Wave-guided travelling-wave laser action has been achieved in a PPV film on quartz glass. The sample was transversally pumped with picosecond laser pulses (wavelength 347.15 nm, duration 35 ps). Laser emission occurred at 550 nm for pump pulse energy densities above .     

The non-isotropic character of electric and dielectric properties of ammonium zinc chloride crystal

The dielectric constant, ε, and the d.c. conductivity, σ, were measured along the a-, b- and c-axes of (NH₄)₂ZnCl₄ (AZC) crystal in the 300-450 K temperature range. Crystals of AZC grown from aqueous solutions containing excess of ZnCl₂ were used. The value of the dielectric permittivity of AZC is extremely small compared to other ferroelectric crystals. Pronounced broad or step-like peaks at the phase transition temperatures were detected along the a- and b-axes, while ε along the c-axis is temperature independent up to the end of the measuring range. Reciprocal of the dielectric permittivity in the range of the commensurate to incommensurate phase transition obeys a relation similar to the Curie-Weiss law that is valid for second order ferroelectric/paraelectric phase transitions. The constants of the proposed relationship applied to the cooling run are given. The J-E characteristics along the three crystallographic axes were measured in the normal, incommensurate, commensurate and antiferroelectric phases. Hence, the type of conduction mechanism has been estimated. Parameters of Poole-Frenkel and Richardson-Schottky types of conduction mechanism have been determined. The effect of applied electric field on the conductivity measurement was also tested. Conductivity anomalies with different character were observed at the phase transition temperatures. The lnσ−1000/T dependence revealed thermal activation energy of conduction along the a-, b- and c-axes with different values in different phases of AZC.     

The Loewner Equation: Maps and Shapes

An approach called Schramm–Loewner evolution (SLE) provides a new method for dealing with a wide variety of scale-invariant problems in two dimensions. This approach is based upon an older method called Loewner Evolution (LE), which connects analytic and geometrical constructions in the complex plane. In this paper, the bases of LE and SLE are described and some simple applications are discussed in relatively non-technical form. A bibliography of the subject is presented.     

Gravitational Collapse of Self-Similar Perfect Fluid in 2 + 1 Gravity

Perfect fluid with kinematic self-similarity is studied in 2+1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. These include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth and second kinds. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either a black hole or a null singularity. It is also shown that one solution can have two different kinds of kinematic self-similarity.     

Peculiar relics from Primordial Black Holes in the inflationary paradigm

Depending on various assumptions on the energy scale of inflation and assuming a primordial power spectrum of a step‐like structure, we explore new possibilities for Primordial Black Holes (PBH) and Planck relics to contribute substantially to Cold Dark Matter in the Universe. A recently proposed possibility to produce Planck relics in four‐dimensional string gravity is considered in this framework. Possible experimental detection of PBHs through gravitational waves is also explored. We stress that inflation with a low energy scale, and also possibly when Planck relics are produced, leads unavoidably to relics originating from PBHs that are not effectively classical during their formation, rendering the usual formalism inadequate for them.