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So far it is believed that, for every series-parallel planetary gear system (PGS), as a coupled gear, a very harmful phenomenon of power circulation must occur in at least one of its closed circuits. In this paper (Part I) and in the next two (Part II and Part III), it will be shown that it is possible to construct a three-row series-parallel PGS in which this phenomenon can be avoided. For this purpose, in Part I, a detailed analysis of the kinematics and statics of a planetary gear with power circulation inside a closed loop was carried out. The determination of the angular velocities of gears and carriers is carried out using Willis formulas and the graphical-analytical method (for verification), while the torques are determined using free body diagrams. The magnitudes of angular velocities and torques were used to determine the directions of power flows with improved energy balance equations in the reference frame related to the stationary gear body and, additionally, only to verify the energy balance equation in the mobile reference frame related to the carrier h i ( i = 2 , 5 , 8 ). The improvement of the methods was based on the use of the original concept of distinguishing active torque from reactive torque, as well as active power from reactive power, which made it very easy to determine the directions of the power flow. The determined paths of the power flow, including the power circulation in the analysed PGS, are presented graphically. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, solutions of fuzzy systems of equations and systems of inequalities , are considered, where , , stands for max-⁎ product and is an unknown vector. In this paper the families of all solutions are described. During the last thirty years this problem was considered by many authors. Our goal is to achieve the most general assumptions for which sets of solutions' systems of equations and inequalities are the union of lattice intervals. Especially, a method of determination of all minimal solutions in such systems is indicated. As a result, the families of all solutions of fuzzy systems of inequalities and are described. As the main part of these results, the method of determination of the family of all minimal solutions in max-⁎ system of equations with an increasing operation ⁎, continuous on the second argument is presented. [Copyright &y& Elsevier]

Abstract: The aim of this work is to complement recent results about distributive binary operations. Our investigation was motivated by non-distributive logical connectives and their generalizations in fuzzy set theory. In spite of the title we neither solve the concerned functional inequalities nor look for characterization of the solutions of such inequalities. We only lay bare conditions for their fulfilment. For monotonic operations with neutral or zero element we present the sufficient conditions to obtain the subdistributivity or superdistributivity in the case of the lack of distributivity for the considered pairs of operations. In particular we describe families of increasing binary operations subdistributive or superdistributive with respect to idempotent uninorms and nullnorms. [Copyright &y& Elsevier]

We extend the result of Zhang et al. (J Fuzzy Math 14:53, ), who discussed the finite fuzzy relation equations with max–min and max–prod composition. In this article, the $$\text{max-}*$$ composition is used for wide family of operations $$*$$. In particular, families of solutions of two relation equations are compared. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to choose diverse definitions of generalized logical connectives and to present them in a coherent order, from the weakest to the richest. Such a rich list of notions allows us to consider the problem of admissible aggregations in the presented classes of unary and binary operations. This gives a contribution to the discussion of tolerance analysis in soft computing, decision making, approximate reasoning, and fuzzy control. First, we present a short survey of the development of MV-logic''s connectives. Next, we discuss postulates used for generalized logical connectives. Finally, we describe families of weak connectives and indicate the preservation of their properties by some aggregation functions. [Copyright &y& Elsevier]

Abstract: Recently, the property of distributivity of binary operations on is of great interest both for practical as well as fundamental reasons. In this paper we summarize some facts connected with distributivity and non-distributivity of binary operations on . Motivated by non-distributive logical connectives in fuzzy set theory we show some new results concerning the above property. A new direction of investigations mainly concerns the distributivity inequalities for triangular norms and conorms and their diverse generalizations. In particular, we look for examples of binary operations fulfilling the distributivity inequalities. [Copyright &y& Elsevier]

Abstract: Recently the distributivity equation was investigated in families of certain operations (e.g. triangular norms, conorms, uninorms and nullnorms). Generally, the assumption of associativity is not necessary in consideration of distributivity equation. Moreover, if we omit commutativity assumption, consideration of the left and right distributivity conditions is reasonable. Previous results about distributivity between uninorms and nullnorms can be obtained as simple corollaries. [Copyright &y& Elsevier]

In this paper MN -convex and MN -concave functions are examined and compared for particular cases of M and N . Characterization of pairs M , N for the family of MN -convex ( MN -concave) functions consisting of all functions and partial characterization of pairs M , N for the family of MN -convex functions consisting of all constant functions will be presented. The condition of MN -convexity was proposed by Aumann in 1933. It was convexity with respect to arbitrary binary means M and N (abbreviated to MN -convexity). Recently many authors have considered this notion with suitable pairs of means. Here aggregation functions M , N will be used for this type of convexity and concavity. [ABSTRACT FROM AUTHOR]