One of the graphs associated with any ring R is its distant graph G(R, Δ) with points of the projective line P(R) over R as vertices. We prove that the distant graph of any commutative, Artinian ring is a Cayley graph. The main result is the fact that G(Z, Δ) is a Cayley graph of a nonartinian commutative ring. We indicate two non-isomorphic subgroups of P SL2(Z) corresponding to this graph.Mathematics Subject Classification. Primary 05C25, Secondary 51C05.
We introduce the concept of an extreme relation for a topological flow as an analogue of the extreme measurable partition for a measure-preserving transformation considered by Rokhlin and Sinai, and we show that every topological flow has such a relation for any invariant measure. From this result, it follows, among other things, that any deterministic flow has zero topological entropy and any flow which is a K-system with respect to an invariant measure with full support is a topological K-flow.
We show that solution to the Hermite-Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.
There is presented an infinite class of subgroups of the modular group PSL(2, Z) that serve as Cayley representations of the distant graph of the projective line of integers. They are infinite countable free products of subgroups of PSL(2, Z) isomorphic with Z2, Z3 and Z subject to the restriction that the number of copies of Z is 0 or 2. The proof technique is based on a 1-1 correspondence between some involutions ι of Z that fulfill the equation ι(ι(n) − δn) = ι(n + 1) + δn+1, δn = ± 1, δ ι(n) = δn, and groups from this class.
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