Cross-bifix-free sets are sets of words such that no prefix of any word is a suffix of any other word. In this paper, we introduce a general constructive method for the sets of crossbifix-free binary words of fixed length. It enables us to determine a cross-bifix-free words subset which has the property to be non-expandable.

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We define a variable-length code having the property that no (nonempty) prefix of each its codeword is a suffix of any other one, and vice versa. This kind of code can be seen as an extension of two well-known codes in literature, called respectively fix-free code and non-overlapping code. In this paper, some constructive algorithms for such codes are presented as well as numerical results about their cardinality.

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We define a set of binary matrices where any two of them can not be placed one on the other in a way such that the corresponding entries coincide. The rows of the matrices are obtained by means of Dyck words. The cardinality of the set of such matrices involves Catalan numbers.

In recent years, there is an increasing attention on air quality derived services for the final users. A dense grid of measures is needed to implement services such as conditional routing, alerting on data values for personal usage, data heatmaps for Dashboards in control room for the operators, and for web and mobile applications for the city users. Therefore, the challenge consists of providing high density data and services starting from scattered data and regardless of the number of sensors and their position to a large number of users. To this aim, this paper is focused on providing an integrated solution addressing at the same time multiple aspects: To create and optimize algorithms for data interpolation (creating regular data from scattered), making it possible to cope with the scalability and providing support for on demand services to provide air quality data in any point of the city with dense data. To this end, the accuracy of different interpolation algorithms has been evaluated comparing the results with respect to real values. In addition, the trends of heatmaps interpolation errors have been exploited to detected devices’ dysfunctions. Such anomalies may often be useful to request a maintenance action. The solution proposed has been integrated as a Micro Services providing data analytics in a data flow real time process based on Node.JS Node-RED, called in the paper IoT Applications. The specific case presented in this paper refers to the data and the solution of Snap4City for Helsinki. Snap4City, which has been developed as a part of Select4Cities PCP of the European Commission, and it is presently used in a number of cities and areas in Europe.

A cross-bifix-free set of words is a set in which no prefix of any length of any word is the suffix of any other word in the set. A construction of cross-bifix-free sets has recently been proposed by Chee et al. in 2013 within a constant factor of optimality. We propose a trace partitioned Gray code for these cross-bifix-free sets and a CAT algorithm generating it.

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