If the load F produces a linear or angular momentum it cannot be resolved by any framework. Therefore we restrict ourselves to loads with ∑ pi fi = 0 and ∑ pi 〈pi , fi〉 = 0. (The vector pi denotes pi rotated by 90 degrees.) If a framework can resolve any load it is called statically rigid. Static rigidity is an equivalent condition for infinitesimal rigidity (see [3]). In general a statically r...

The goal of this paper is to describe, in as much detail as possible, the structure of the algebra of differential invariants of a Lie pseudo-group. Under the assumption of local freeness of the prolonged pseudo-group action, we develop algorithms for locating a finite generating set of differential invariants, establishing the recurrence relations for the differentiated invariants, and fixing ...

We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T , that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex v ∈ T is pointed, that is, v has an incident angle greater than π. Such a tree can be completed to a minimum pseudo-triangulation. In particular, it follo...

In this paper, for any non-empty subset C of a pseudo BCI-algebra X, the concept of p-closure of C, denoted by C(pc), is introduced and some related properties are investigated. Applying this concept, a characterization of the minimal elements of X is given. It is proved that C(pc) is the least closed pseudo BCI-ideal of X containing C and K(X) for any ideal C of X...

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism φ, we assign an AF -algebra Aφ (an operator algebra). It is proved that the assignment is functorial, i.e. every φ, conjugate to φ, maps to an AF algebra Aφ′ , which is stably isomorphic to Aφ. The new invariants of the conjugacy of the pseudo-Anosov automorphisms are ob...

We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grA. Then grA is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a projection π : grA → A0; let R be the algebra ...

We investigate 3-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of E-subtractive varieties (where E is the set of constants in a variety) recently introduced by the f...