Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

A commuting graph is a graph denoted by C ( G , X ) where G is any group and X, a subset of a group G, is a set of vertices for C ( G , X ) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C ( G , X ) when G is a symmetric group of degree n, Sym ( n ) , and X is a conjugacy class of elements of order three in G.

The finite embeddability property for some noncommutative knotted extensions of FL

We consider the knotted structural rule x<sup>m</sup>≤x<sup>n</sup> for n different than m and m greater or equal than 1. Previously van Alten proved that commutative residuated lattices that satisfy the knotted rule have the finite embeddability property (FEP). Namely, every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. In our work we replace the commutativity property by some slightly weaker conditions. Particularly, we prove the FEP for the variety of residuated lattices that satisfy the equation xyx=x<sup>2</sup>y and the knotted rule. Furthermore, we investigate some generalizations of this noncommutative property by working with equations that allow us to move variables. We also note that the FEP implies the finite model property. Hence the logics modeled by these residuated lattices are decidable.

On dynamically consistent Jacobian inverse for non-holonomic robotic systems

This paper presents the dynamically consistent Jacobian inverse for non-holonomic robotic system, and its application to solving the motion planning problem. The system’s kinematics are represented by a driftless control system, and defined in terms of its input-output map in accordance with the endogenous configuration space approach. The dynamically consistent Jacobian inverse (DCJI) has been introduced by means of a Riemannian metric in the endogenous configuration space, exploiting the reduced inertia matrix of the system’s dynamics. The consistency condition is formulated as the commutativity property of a diagram of maps. Singular configurations of DCJI are studied, and shown to coincide with the kinematic singularities. A parametric form of DCJI is derived, and used for solving example motion planning problems for the trident snake mobile robot. Some advantages in performance of DCJI in comparison to the Jacobian pseudoinverse are discovered.

IMAGING ALL THE PEOPLE

It is well known that aggregating the degree-of-belief functions of different subjects by linear pooling or averaging is subject to a commutativity dilemma: other than in trivial cases, conditionalizing the individual degree-of-belief functions on a piece of evidence E followed by linearly aggregating them does not yield the same result as first aggregating them linearly and then conditionalizing the resulting social degree-of-belief function on E. In the present paper we suggest a novel way out of this dilemma: adapting the method of update or learning such that linear pooling commutes with it. As it turns out, the resulting update scheme – (general) imaging on the evidence – is well-known from areas such as the study of conditionals and causal decision theory, and a formal result from which the required commutativity property is derivable was supplied already by Gärdenfors (1982) in a different context. We end up determining under which conditions imaging would seem to be right method of update, and under which conditions, therefore, group update would not be affected by the commutativity dilemma.

A commutativity property for rings

We provide a partial answer to the following question: Assume that R is a finite ring of order s such that for every two subsets M and N of cardinalities m and n respectively, there exist x ∈ M and y ∈ N such that xy = yx. What relations among s, m, n guarantee that R is commutative? Also, we give criteria for commutativity of such rings in terms of structure and relations between m and n.

A combinatorial commutativity property for rings

We study commutativity in ringsRwith the property that for a fixed positive integern,xS=Sxfor allx∈Rand alln-subsetsSofR.