The Perturbation ϕ2, 1 of the M(p,p + 1) Models of Conformal Field Theory and Related Polynomial-Character Identities

Using q-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p + 1) models of conformal field theory perturbed by the operator _{2, 1}. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime 1⁺ of A _p–1 dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for all Virasoro characters of unitary minimal series.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

Conductive Coatings Based on Polyvinyl Alcohol and Graphene Oxide

Russian Journal of General Chemistry - Thin conductive films have been obtained via dehydration and reduction of composite films based on polyvinyl alcohol and graphene oxide. Composition and...     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

New polynomial analogues of Jacobi''s triple product and Lebesgue''s identities

In a recent paper by the authors, a bounded version of Göllnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of Jacobi and Lebesgue. We also derive a two parameter extension of Jacobi's famous triple product identity.     

Rheological properties of polyacrylonitrile solutions containing highly dispersed carbon nanotubes

Rheological properties of dispersions of carbon nanotubes in dimethylsulfoxide solutions of polyacrylonitrile are studied at different concentrations of the components. The viscosity values of all the studied dispersions are substantially dependent on shear rate. For a number of systems, the energies of activation of viscous flow are determined. The relationship of viscosity and the energy of activation of flow to the compositions and possible structural features of dispersions is discussed. The concentration regime where a thermally reversible mixed network exists with crosslinks formed by physical contacts of the polymer with nano-tubes is characterized.