SOME systems of collineation groups which arise in connection with the theory of elliptic functions have been investigated by Klein | and HurwitzJ. One of them is a system in n variables each group of which contains an invariant subgroup of order n. For n SL prime the quotient group with respect to this invariant subgroup is (1, 1) isomorphic with the modular group on two indices of order n(n —...

A short proof of the characterization of idempotent subreducts of semimodules over commutative semirings is presented. It says that an idempotent algebra embeds into a semimodule over a commutative semiring, if and only if it belongs to the variety of Szendrei modes.

This paper presents a query evaluation technique for positive relational algebra queries with aggregates on a representation system for probabilistic data based on the algebraic structures of semiring and semimodule. The core of our evaluation technique is a procedure that compiles semimodule and semiring expressions into so-called decomposition trees, for which the computation of the probabili...

count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss’s continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Han...

We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning’s [9] ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugat...

Equational theories of some linear equations are studied. As a consequence, semimodule representations of the corresponding algebras are obtained. Examples are shown on medial and paramedial equations and some of their general-

Perfect ternary arrays are closely related to difference sets and group invariant weighing matrices. A previous paper by one of the authors demonstrated strong restrictions on difference sets with parameters (120, 35, 10). A perfect ternary array with energy 25 and order 60 – PTA(60,25) – would obey an equation analogous to the difference set equation for (120, 35, 10). A perfect ternary array ...

We provide an expression for the conductor \(c(\varDelta )\) of a semimodule \(\varDelta \) numerical semigroup \(\varGamma with two generators in terms syzygy module and semigroup. In particular, we deduce that difference between is element \), as well formula dual \).

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices nk of the k-iterated line digraph L (G), for k ≥ 0, where L(G) = G. We obtain this result by using the minimal polynomial of a quotient digraph π(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our ...