Analytic calculation of scaling dimensions: Tricritical hard squares and critical hard hexagons

The finite-size corrections, central chargesc, and scaling dimensionsx of tricritical hard squares and critical hard hexagons are calculated analytically. This is achieved by solving the special functional equation or inversion identity satisfied by the commuting row transfer matrices of these lattice models at criticality. The results are expressed in terms of Rogers dilogarithms. For tricritical hard squares we obtainc=7/10,x=3/40, 1/5, 7/8, 6/5 and for hard hexagons we obtainc=4/5,x=2/15, 4/5, 17/15, 4/3, 9/5, in accord with the predictions of conformal and modular invariance.     

Theory of helicenic hydrocarbons. Part 1: Invariants and symmetry

Helicenes form a subclass of polyhexes and correspond to hydrocarbons of considerable chemical interest. This paper is the first part of a general graph-theoretical treatment of helicenes. The invariants are studied: the relations between them, their possible values, and their upper and lower bounds in helicenes. Extremal helicenes and circular helicenes are useful definitions of subclasses of the systems under consideration. Finally an account of symmetry of helicenes is given.On leave from: Department of Mathematics, Xinjiang University, Wulumuqi Xinjiang 830046, People's Republic of China.     

On the Twisted Cubic of PG(3, q)

In this paper we classify the lines of PG(3, q) whose points belong to imaginary chords of the twisted cubic of PG(3, q). Relying on this classification result, we obtain a complete classification of semiclassical spreads of the generalized hexagon H(q).     

Slim Near Polygons

In this paper we initiate the study of hybrid slim near hexagons. These are near hexagons which are not dense and not a generalized hexagon in which each line is incident with exactly three points. In the present paper, we will emphasize slim near hexagons with at least one W(2)-quad or Q(5, 2)-quad. Such near hexagons are finite if there are no special points, i.e. points which lie at distance at most 2 from any other point. We will determine upper bounds for the number of lines through a fixed point. We will also look at the special case where the near hexagon has an order. We will determine all slim near hexagons with an order which contain at least one (necessarily big) Q(5,2)-quad, or at least one big W(2)-quad.AMS Classification 05B25, 51E12Communicated by: M.J. de ResminiPostdoctoral Fellow of the Research Foundation–Flanders.     

Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon S, and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group.     

Uniform Mesoporous Silica Hexagon and Its Two‐Dimensional Colloidal Crystal

Well‐ordered mesoporous silica nanoparticles with uniform hexagonal disk shapes are synthesized under dilute alkaline conditions with a two‐step process, separating the nucleation and growth process. The resulting monodisperse hexagons can be arranged in a 2‐dimensional (2D) ordered periodical super‐structure. The hexagonal symmetry is similar in both scales. A statistical mechanical cell model is applied to analyze consequences of the interesting packing structure, including osmotic bulk modulus, phase separation and defects.     

Geometry and Algebra of Prime Fano 3-folds of Genus 12

The connection between these Fano 3-folds and plane quartic curves is explained.     

On Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces

The flag geometry =( ) of a finite projective plane of order s is the generalized hexagon of order (s, 1) obtained from by putting equal to the set of all flags of , by putting equal to the set of all points and lines of and where I is the natural incidence relation (inverse containment), i.e., is the dual of the double of in the sense of Van Maldeghem Mal:98. Then we say that is fully and weakly embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s = q, if the set of points of generates PG(d, q), and if the set of points of not opposite any given point of does not generate PG(d, q). Preparing the classification of all such embeddings, we construct in this paper the classical examples, prove some generalities and show that the dimension d of the projective space belongs to {6,7,8}.     

Effect of AlN doping on the growth morphology of SiC

AlN doped SiC films were deposited on on‐axis Si‐face 4H‐SiC (0001) substrates by the physical vapor transport (PVT) method. Thick film in the range of 20 μm range was grown and morphology was characterized. Films were grown by physical vapor deposition (PVD) in a vertical geometry in the nitrogen atmosphere. We observed that nucleation occurred in the form of discs and growth occurred in hexagonal geometry. The X‐ray studies showed (001) orientation and full width of half maxima (FWHM) was less than 0.1° indicating good crystallinity. We also observed that film deposited on the carbon crucible had long needles with anisotropic growth very similar to that of pure AlN. Some of the needles grew up to sizes of 200 μm in length and 40 to 50 μm in width. It is clear that annealing of SiC‐AlN powder or high temperature physical vapor deposition produces similar crystal structure for producing AlN‐SiC solid solution. SEM studies indicated that facetted hexagons grew on the top of each other and coarsened and merged to form cm size grains on the substrate. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)