On Gibbs measures of P-adic Potts model on the cayley tree

We consider a nearest-neighbor p-adic Potts (with q ≥ 2 spin values and coupling constant J ? _p) model on the Cayley tree of order k ≥ 1. It is proved that a phase transition occurs at k = 2, q ? p and p ≥ 3 (resp. q ? 2², p = 2). It is established that for p-adic Potts model at k ≥ 3 a phase transition may occur only at q ? p if p ≥ 3 and q ? 2² if p = 2.     

Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

Abstract We prove that there are non-recursive r.e. sets A and C with A < _T C such that for every set . Both authors are supported by “863” and the National Science Foundation of China     

Cayley color graphs of inverse semigroups and groupoids

The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.     

Cayley graphs of the group ℤ4 and limits for minimal vertex-primitive graphs of HA-type

We point out a countable set of pairwise nonisomorphic Cayley graphs of the group ℤ⁴ that are limit for finite minimal vertex-primitive graphs admitting a vertex-primitive automorphism group containing a regular Abelian normal subgroup. Supported by RFBR grant No. 06-01-00378. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 203–214, March–April, 2008.     

Minimal cutwidth linear arrangements of abelian Cayley graphs

We find the minimal cutwidth and bisection width values for abelian Cayley graphs with up to 4 generators and present an algorithm for finding the corresponding optimal ordering. We also find minimal cuts of each order.     

Nowhere-zero 3-flows in Cayley graphs and Sylow 2-subgroups

Tutte’s 3-Flow Conjecture suggests that every bridgeless graph with no 3-edge-cut can have its edges directed and labelled by the numbers 1 or 2 in such a way that at each vertex the sum of incoming values equals the sum of outgoing values. In this paper we show that Tutte’s 3-Flow Conjecture is true for Cayley graphs of groups whose Sylow 2-subgroup is a direct factor of the group; in particular, it is true for Cayley graphs of nilpotent groups. This improves a recent result of Potočnik et al. (Discrete Math. 297:119–127, 2005) concerning nowhere-zero 3-flows in abelian Cayley graphs.     

On automorphism groups of quasiprimitive 2-arc transitive graphs

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs. This work forms part of an ARC grant project and is supported by a QEII Fellowship.     

Improved Sobolev inequalities on groups of Iwasawa type in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. We prove a refined version of the Sobolev inequality on N in presence of symmetry.     

George Boole and the origins of invariant theory

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.