Controlled nanoparticle targeting and nanoparticle-driven nematic structural transition

We study experimentally and theoretically controlled targeting of specific nanoparticles (NPs) to different regions within nematic liquid crystal. Using a simple mesoscopic Landau-de Gennes-type model in terms of a tensor nematic order parameter, we demonstrate a general mechanism which could be exploited for controlled targeting of NPs within a spatially nonhomogeneous nematic texture. Furthermore, we experimentally demonstrate using polarising microscopy that even a relatively low concentration of localised appropriate NPs could trigger a nematic structural transition. A simple estimate is derived to account for the observed transition

A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)

Let Δ be the line graph of PG(n –1,2), Alt(n,2) be the graph of the n-dimensional alternating forms over GF(2), n ≥ 4. Let Γ be a connected locally Δ graph such that 1. the number of common neighbours of any pair of vertices at distance two is the same as in Alt(n,2). 2. the valency of the subgraph induced on the second neighbourhood of any vertex is the same as in Alt(n,2). It is shown that Γ is covered either by Alt(n,2) or by the graph of (n – l)-dimensional GF(2)-quadratic forms Quad(n – 1,2).Published versio

The automorphism group and the convex subgraphs of the quadratic forms graph in characteristic 2

We determine the automorphism group and the convex subgraphs of the quadratic forms graph Quad(n,q),q even.Accepted versio

Extended F4-buildings and the baby monster

Let Θ be the Baby Monster graph which is the graph on the set of {3, 4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Θ is locally the commuting graph of central (root) involutions in 2E6(2). The graph Θ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ε(B) with diagram for t = 4 and the action of B on ε(B) is flag-transitive. We show that ε(B) contains subgeome¬tries ε(2E6(2)) and ε(Fi22) with diagrams c.F4(2) and c.F4(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of 2E6(2) : 2 and Fi22 : 2, respectively. The geometries ε(B), ε(2E6(2)) and ε(Fi22) possess the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F4(t)-geometries satisfying (a) and (b). We construct three further examples for t = 2 with flag-transitive au¬tomorphism groups isomorphic to 3•2E2(2) : 2, E6(2) : 2 and 226.F4(2) and one for t = 1 with flag-transitive automorphism group 3 • Fi22 : 2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F4(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ε(B) is the only c.F4(4)-geometry, satisfying (a) and (b).Accepted versio

Liquid crystals: viscous and elastic properties

Covering numerous practical applications as yet not covered in any single source of information, this monograph discusses the importance of viscous and elastic properties for applications in both display and non-display technologies. The very well-known authors are major players in this field of research and pay special attention here to the use of liquid crystals in fiber optic devices as applied in telecommunication circuits

Non−abelian representations of some sporadic geometries

For a point-line incidence system =(P, L) with three points per line we define the universal representation group of as[formula]We prove that if is the 2-local parabolic geometry of the sporadic simple groupF1(the Monster) orF2(the Baby Monster) thenR( )F1or 2·F2, respectively

Non-abelian representations of some sporadic geometries

For a point-line incidence system S =(P, L) with three points per line we define the universal representation group of S asR(S)= 〈zp, p∈P|zp^2=1 for p∈P,zp zq zr = 1 for {p,q,r} ∈ L〉We prove that if G is the 2-local parabolic geometry of the sporadic simple group F1(the Monster) or F2(the Baby Monster) thenR(G)≅F1or 2·F2, respectively.Accepted versio

Majorana representations of the symmetric group of degree 4

AbstractThe Monster group M acts on a real vector space VM of dimension 196,884 which is the sum of a trivial 1-dimensional module and a minimal faithful M-module. There is an M-invariant scalar product (,) on VM, an M-invariant bilinear commutative non-associative algebra product ⋅ on VM (commonly known as the Conway–Griess–Norton algebra), and a subset A of VM∖{0} indexed by the 2A-involutions in M. Certain properties of the quintetM=(M,VM,A,(,),⋅) have been axiomatized in Chapter 8 of Ivanov (2009) [Iv09] under the name of Majorana representation of M. The axiomatization enables one to study Majorana representations of an arbitrary group G (generated by its involutions). A representation might or might not exist, but it always exists whenever G is a subgroup in M generated by the 2A-involutions contained in G. We say that thus obtained representation is based on an embedding of G in the Monster. The essential motivation for introducing the Majorana terminology was the most remarkable result by S. Sakuma (2007) [Sak07] which gave a classification of the Majorana representations of the dihedral groups. There are nine such representations and every single one is based on an embedding in the Monster of the relevant dihedral group. It is a fundamental property of the Monster that its 2A-involutions form a class of 6-transpositions and that there are precisely nine M-orbits on the pairs of 2A-involutions (and also on the set of 2A-generated dihedral subgroups in M). In the present paper we are making a further step in building up the Majorana theory by classifying the Majorana representations of the symmetric group S4 of degree 4. We prove that S4 possesses precisely four Majorana representations. The Monster is known to contain four classes of 2A-generated S4-subgroups, so each of the four representations is based on an embedding of S4 in the Monster. The classification of 2A-generated S4-subgroups in the Monster relies on calculations with the character table of the Monster. Our elementary treatment shows that there are (at most) four isomorphism types of subalgebras in the Conway–Griess-Norton algebra of the Monster generated by six Majorana axial vectors canonically indexed by the transpositions of S4. Two of these subalgebras are 13-dimensional, the other two have dimensions 9 and 6. These dimensions, not to mention the isomorphism type of the subalgebras, were not known before