This paper defines the concept of simple strategy and introduces three kinds of simple strategies: wealth-invariant, scale-invariant and "wealthier-accept more". For three commonly used utility function families: CARA, CRRA and DARA equivalent characterizations are obtained in terms of the corresponding simple strategy, in terms of the buying and selling price properties, and in terms of the utility function properties as expressed by Cauchy functional equations. Moreover, an extension of famous Pratt (1964) theorem is proved which involves buying price for a lottery as an alternative measure of comparative risk aversion. Additionally a number of propositions on both selling and buying price for a lottery and CRRA utility class are proved.

The invariant properties of the stability, reachability, observability and transfer matrices of positive linear electrical circuits with integer and fractional orders are investi- gated. It is shown that the stability, reachability, observability and transfer matrix of positive linear systems are invariant under their integer and fractional orders.

In search of the invariant semantics of the preposition “da”: a cognitive analysis of the predicative context – The purpose of this article is to verify whether the semantic invariant of the preposition da [starting point allowing physical or mental movement] in the nominal context remains valid in the context of the verb. The analysis of the content of predicates that link to the preposition da will help to answer the question of the extent to which the choice of a preposition is determined by the knowledge of the experienced activities and/or the predicate itself (its selective features) or if it is the result of convention.

Positively invariant sets play an important role in the theory and applications of dynamical systems. The stability in the sense of Lyapunov of the equilibrium x = 0 is equivalent to the existence of the ellipsoidal positively invariant sets. The constraints on the state and control vectors of dynamical systems can be formulated as polyhedral positively invariant sets in practical engineering problems. Numerical checking method of positive invariance of polyhedral sets is addressed in this paper. The validation of the positively invariant sets can be done by solving LPs which can be easily done numerically. It is illustrated by examples that our checking method is effective. Compared with the now existing algebraic methods, numerical checking method is an attractive method in that it’s easy to be implemented.

Achieving a reliable fault diagnosis for gears under variable operating conditions is a pressing need of industries to ensure productivity by averting unwanted breakdowns. In the present work, a hybrid approach is proposed by integrating Hu invariant moments and an artificial neural network for explicit extraction and classification of gear faults using time-frequency transforms. The Zhao-Atlas-Marks transform is used to convert the raw vibrations signals from the gears into time-frequency distributions. The proposed method is applied to a single-stage spur gearbox with faults created using electric discharge machining in laboratory conditions. The results show the effectiveness of the proposed methodology in classifying the faults in gears with high accuracy.