The M -triangle of a ranked locally finite poset P is the generating function P u,w∈P μ(u,w)x rk uyrk w, where μ(., .) is the Möbius function of P . We compute the M triangle of Armstrong’s poset of m-divisible non-crossing partitions for the root systems of type E7 and E8. For the other types exceptDn this had been accomplished in the earlier paper “The F -triangle of the generalised cluster c...

We study the quotient of a free product of groups by the normal closure of a word that is contained in a certain type of 2-generator subgroup, and hence can be expressed in the form of a pushout involving a generalized triangle group. Using the theory of generalized triangle groups, we extend the range of conditions under which a number of results for one-relator products of groups, such as the...

We initiate a triangle geometry in the projective metrical setting, based on the purely algebraic approach of universal geometry, and yielding in particular a new form of hyperbolic triangle geometry. There are three main strands: the Orthocenter, Incenter and Circumcenter hierarchies, with the last two dual. Formulas using ortholinear coordinates are a main objective. Prominent are five partic...

This paper presents a simple but fully functioning and complete artificial visual system. The triangle is the simplest object of perception and therefore the simplest visual system is one which sees only triangles. The system presented is complete in the sense that it will see any triangle presented to it as visual input in bitmap form, even triangles with illusory contours that can only be det...

In this paper we discuss our work on plan management in the Autominder cognitive orthotic system. Autominder is being designed as part of an initiative on the development of robotic assistants for the elderly. Autominder stores and updates user plans, tracks their execution via input from robot sensors, and provides carefully chosen and timed reminders of the activities to be performed. It will...

This paper describes a novel encoding for sequential numeric planning into the problem of determining the satisfiability of a logical theory T . We introduce a novel technique, orthogonal to existing work aiming at producing more succinct encodings that enables the theory solver to roll up an unbounded yet finite number of instances of an action into a single plan step, greatly reducing the hor...

We introduce the new notion of sydpoints into projective triangle geometry with respect to a general bilinear form. These are analogs of midpoints, and allow us to extend hyperbolic triangle geometry to non-classical triangles with points inside and outside of the null conic. Surprising analogs of circumcircles may be defined, involving the appearance of pairs of twin circles, yielding in gener...