We briefly survey results about Invertible Algebras (algebras having bases that consist entirely of units) and other related notions. In addition, we consider the existence of an augmentation map as a possible way in which results about group rings, the archetypical invertible algebras, may be extended to more general settings. 1 Invertible Algebras and Some Related Notions Section 1 of this pa...

We study the equivariant genera of strongly invertible and periodic knots. Our techniques include some new concordance group invariants, Donaldson's theorem, g-signature. find many examples where 4-genus is larger than 4-genus.

In multi-agent (multi-robot) environment each agent tries to achieve its own goal and it implies that in most cases the agent goals conflict. However, there exists a group of problems where it is possible to find a way to satisfy all goals even in conflict situations. Such problems can be modelled as a STRIPS system (for instance Block World environment). If STRIPS planning problem is invertibl...

Let M(R) denote the measure algebra on the additive group of the reals. R. G. Douglas recently pointed out to us the importance of the following question in the study of Wiener-Hopf integral equations: if fxÇ:M(R) is invertible, then under what conditions does jit = exp(j>) for some vGM(R)? The relevance of the above question in integral equations stems from the fact that if JJLÇÏM(R) is invert...

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈G, a 〉 \G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates ag = g−1ag of a by elements g ∈ G generate a semigroup denoted 〈ag | g ∈ G〉. We classify the finite permutation groups G on a finite set...

Various digital watermarking techniques have been proposed in recent years as the methods to protect the copyright of multimedia data. However, as pointed out by the IBM research group, the rightful ownership problem has not been properly solved. Currently, there are two proposals to solve the ownership problem. Unfortunately, one proposal lacks a formal proof and the other can be easily defeat...

The issue of testing invertibility of cellular automata has been often discussed. Constructing invertible automata is very useful for simulating invertible dynamical systems, based on local rules. The computation universality of cellular automata has long been positively resolved, and by showing that any cellular automaton could be simulated by an invertible one having a superior dimension, Tof...