A sequence in the additive group Z n of integers modulo n is called n-zero-free if it does not contain subsequences with length n and sum zero. The article characterizes the n-zero-free sequences in Z n of length greater than 3n/2−1. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same form to shorter sequence lengths. Consequences of the main result are best possible lower bounds for the maximum multiplicity of a term in an n-zero-free sequence of any given length greater than 3n/2−1 in Z n , and also for the combined multiplicity of the two most repeated terms. Yet another application is finding the values in a certain range of a function related to the classic theorem of Erdős, Ginzburg and Ziv.

Erdős et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119-135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length.

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The necessity of new approaches for structural synthesis of mechanical systems, which directly identify a limited number of structures that carry a potential for solving the technical problems and meet the specific requirements for the design of mechanical systems, mainly for specialised robots, is specified. The identified five types of kinematic chains (primary, parallel, secondary, additional and subsidiary), with different functionality, allow the synthesis of manipulators structures according to the defined goal tasks and specific functional requirements for hybrid systems, in particular specialised industrial robots. Two mechatronic approaches for functional structural synthesis of mechanical systems of industrial robots, where the main manipulation mechanism is a path generator, were introduced. A limited number of structures that meets the set of objectives and technical requirements to design mechanisms, is directly determined. Emphasis is placed on the tasks of passive (mechanical) control of manipulation systems associated with specialized robotics. Two mechatronic approaches for functional structural synthesis of mechanical systems of industrial robots, where the main manipulation mechanism is a motion generator, are introduced. A limited number of structures is directly defined that meets the set of technical objectives and requirements for the designed mechanisms. Emphasis is placed on the tasks of passive control of the manipulation systems associated with the specialized robotics.

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