No abstract

Abstract-The paper is devoted to discussion of the minimal cycles of the so called Kaprekar's transformations and some of its generalizations. The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions. In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovsky's theorem and with the Erdős-Szekeres theorem concerning the monotonic subsequences. Because of the size the study is divided into two parts. Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekar's transformations. We find here all the minimal orbits of Kaprekar's transformations Tn, for n = 3, ..., 7. Moreover, we define many different generalizations of the Kaprekar's transformations and we discuss their minimal orbits for the selected cases. In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations. For example, we notice there that each fixed point, familiar to us, of any Kaprekar's transformation generates an infinite sequence of fixed points of the other Kaprekar's transformations. The observed facts concern also several generalizations of the Kaprekar's transformations defined in Part I.

Recovering energy from waste is a positive element in the operation of a waste incineration plant. Hazardous waste is a very diverse group, including in terms of its fuel properties. Carrying out the thermal process in this case is associated with the difficulty in maintaining stable conditions. This may translate into the efficiency of energy recovery from waste. The article presents a tool supporting the work of hazardous waste incineration plant operators, the aim of which is to select waste for a batch of input material in a manner that ensures process stability and efficient energy recovery. The tool is an application in which the bee algorithm is implemented. It selects the optimal solution to the problem, in accordance with the assumed parameters. The application tests in laboratory conditions were satisfactory and indicated compliance with the assumptions and stability of the solution.

Computer tomography has a wide field of applicability; however, most of its applications assume that the data, obtained from the scans of the examined object, satisfy the expectations regarding their amount and quality. Unfortunately, sometimes such expected data cannot be achieved. Then we deal with the incomplete set of data. In the paper we consider an unusual case of such situation, which may occur when the access to the examined object is difficult. The previous research, conducted by the author, showed that the CT algorithms can be used successfully in this case as well, but the time of reconstruction is problematic. One of possibilities to reduce the time of reconstruction consists in executing the parallel calculations. In the analyzed approach the system of linear equations is divided into blocks, such that each block is operated by a different thread. Such investigations were performed only theoretically till now. In the current paper the usefulness of the parallel-block approach, proposed by the author, is examined. The conducted research has shown that also for an incomplete data set in the analyzed algorithm it is possible to select optimal values of the reconstruction parameters. We can also obtain (for a given number of pixels) a reconstruction with a given maximum error. The paper indicates the differences between the classical and the examined problem of CT. The obtained results confirm that the real implementation of the parallel algorithm is also convergent, which means it is useful.