In this paper we introduce a sharpening of the Parikh mapping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix product gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting interconnection between mirror images of words and inverses of .

Vehicular crowdsensing aims to utilize the plethora of onboard sensors and resources on smart vehicles to gather sensing data in a large coverage area. Recruitment algorithms aim to select participants within a crowdsensing network such that the most sensing data is obtained for the lowest possible cost. In this paper, we consider two such existing recruitment problems for vehicular crowdsensing and propose several heuristics. We also show that existing algorithms to solve these problems can be arbitrarily bad in the worst case. We also compare our algorithms with both optimal solutions (returned by mixed integer programs) as well as existing heuristics. Performance evaluations on our algorithms show that our algorithms outperform existing algorithms and obtain near optimal solutions.

We consider the transition complexity of regular languages based on the incomplete deterministic finite automata. We establish tight bounds for the transition complexity of Boolean operations, in the case of union the upper and lower bounds differ by a multiplicative constant two. We show that the transition complexity results for union and complementation are very different from the state complexity results for the same operations. However, for intersection, the transition complexity bounds turn out to be similar to the corresponding bounds for state complexity.

We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition. We show that there exist non-regular unary languages that do not have prime decompositions. We also consider infinite factorizations of unary languages.