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The concept of function is critical in mathematics in general and abstract algebra in particular. We observe, however, that much of the research on functions in abstract algebra (1) reports widespread student difficulties, and (2) focuses on specific types of functions, including binary operation, homomorphism, and isomorphism. Direct, detailed examinations of the function concept itself–and such fundamental properties as well-definedness and everywhere-definedness–are scarce. To this end, in this paper we examine non-examples of function in abstract algebra by conducting a textbook analysis and semi-structured interviews with abstract algebra instructors. In doing so, we propose four key categories based upon the definitive function properties of well-definedness and everywhere-definedness. These categories identify specific characteristics of the kinds of non-examples of function that abstract algebra instruction should emphasize, enabling us to hypothesize how students might be able to develop a robust view of function and explain in greater detail the nature of the reported difficulties that students experience.

The purpose of this contribution is to highlight an underexplored property of the directional distance function, a recently introduced generalization of the Shephard distance function. It diagnoses in detail the economic conditions under which infeasibilities may occur for the case of directional distance functions and explores whether there exist any solutions that remedy the problem in an economically meaningful way. This discussion is linked to determinateness as a property in index theory and is illustrated by analyzing the Luenberger total factor productivity indicator, based upon directional distance functions. This indicator turns out to be impossible to compute under certain weak conditions. A fortiori, the same problems can also occur for less general productivity indicators and indexes.

Time Petri nets describe the state of a timed system through a marking and a set of clocks. If clocks take values in a dense domain, state space analysis must rely on equivalence classes. These support verification of logical sequencing and quantitative timing of events, but they are hard to be enriched with a stochastic characterization of nondeterminism necessary for performance and dependability evaluation. Casting clocks into a discrete domain overcomes the limitation, but raises a number of problems deriving from the intertwined effects of concurrency and timing. We present a discrete-time variant of time Petri nets, called stochastic preemptive time Petri nets, which provides a unified solution for the above problems through the adoption of a maximal step semantics in which the logical location evolves through the concurrent firing of transition sets. We propose an analysis technique, which integrates the enumeration of a succession relation among sets of timed states with the calculus of their probability distribution. This enables a joint approach to the evaluation of performance and dependability indexes as well as to the verification of sequencing and timeliness correctness. Expressive and analysis capabilities of the model are demonstrated with reference to a real-time digital control system.