In this paper we continue our study of the Frattini p-subalgebra of a Lie />-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which X. is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal als...

4. Proof of Theorem 2. Harish-Chandra [l] and others have proved that every Lie algebra over a field of characteristic zero has a faithful representation. Consequently by Lemma 4, 8 has a faithful representation x—*Qx whose matrices have elements in an algebraic extension $ of g such that t(QxQv) =0 for all x of 31 and all y of 8. We now apply another form of Cartan's criterion for solvability ...

Introduction. In this note, we will announce a characterization of a connected, simply connected Type I solvable Lie group, G, and present a complete description of the set of all unitary equivalence classes of irreducible unitary representations of G together with a construction of an irreducible representation in each equivalence class. This result subsumes the results previously obtained on ...

The q-state chiral Potts model can describe systems that exhibit rich phase diagrams (see, for instance, ref. [l]) and interesting physical properties like commensurate-incommensurate transitions [2]. The chiral Potts model has been extensively studied in the last few years from the point of view of exactly solvable models. Au-Yang et al. discovered integrable cases for the q-state chiral Potts...

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ=(G). In this paper the problem of determinig the value of equitable chromatic number for multic...

A vertex k-ranking of a graph G is a function c : V (G) → {1, . . . , k} such that if c(u) = c(v), u, v ∈ V (G) then each path connecting vertices u and v contains a vertex w with c(w) > c(u). If each vertex v has a list of integers L(v) and for a vertex ranking c it holds c(v) ∈ L(v) for each v ∈ V (G) then c is called L-list k-ranking, where L = {L(v) : v ∈ V (G)}. In this paper we investigat...

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In ...

Let 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation f(x) = y, where x, y belong to l and f has continuous bounded gradient in an inverseclosed subalgebra of B(l), the Banach algebra of all bounded linear operators on the Hilbert space l. We introduce strict monotonicity property for functions f on Banach spaces l so that the above nonlinear functional equation is s...

The variance of the particle number (equivalently the total charge) in a domain of length L of a one-component plasma (OCP) on a cylinder of circumference W at the reciprocal temperature ;=2, is shown to remain bounded as L . This exactly solvable system with average density \ has a density profile which is periodic with period (\W ) along the axis of the infinitely long cylinder. This illustra...