On the Determination of Minimal Facets and Edges of a Polyhedral Set

In this paper we investigate the image of a polyhedral set under a linear map. Moreover, we present an algorithm for the determination of so-called minimal facets and certain minimal irredundant proper edges of a convex polyhedral set in ℝ³. © 1997 John Wiley & Sons, Ltd.     

Rough Sets and 3-Valued Logics

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator.     

Mathematical foundations of consciousness

We employ the Zermelo–Fränkel Axioms that characterize sets as mathematical primitives. The Anti-foundation Axiom plays a significant role in our development, since among other of its features, its replacement for the Axiom of Foundation in the Zermelo–Fränkel Axioms motivates Platonic interpretations. These interpretations also depend on such allied notions for sets as pictures, graphs, decorations, labelings and various mappings that we use. A syntax and semantics of operators acting on sets is developed. Such features enable construction of a theory of non-well-founded sets that we use to frame mathematical foundations of consciousness. To do this we introduce a supplementary axiomatic system that characterizes experience and consciousness as primitives. The new axioms proceed through characterization of so-called consciousness operators. The Russell operator plays a central role and is shown to be one example of a consciousness operator. Neural networks supply striking examples of non-well-founded graphs the decorations of which generate associated sets, each with a Platonic aspect. Employing our foundations, we show how the supervening of consciousness on its neural correlates in the brain enables the framing of a theory of consciousness by applying appropriate consciousness operators to the generated sets in question.     

An examination of indexes for determining the number of clusters in binary data sets

The problem of choosing the correct number of clusters is as old as cluster analysis itself. A number of authors have suggested various indexes to facilitate this crucial decision. One of the most extensive comparative studies of indexes was conducted by Milligan and Cooper (1985). The present piece of work pursues the same goal under different conditions. In contrast to Milligan and Cooper's work, the emphasis here is on high-dimensional empirical binary data. Binary artificial data sets are constructed to reflect features typically encountered in real-world data situations in the field of marketing research. The simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration. Index results are evaluated and their performance is compared and analyzed.Author names are listed in alphabetical order.This piece of research was supported by the Austrian Science Foundation (FWF) under grant SFB#010 (Adaptive Information Systems and Modeling in Economics and Management Science).The authors would like to thank the anonymous reviewers and especially the associate editor for their helpful comments and suggestions.     

Fuzzy partition models for fitting a set of partitions

Methodology is described for fitting a fuzzy consensus partition to a set of partitions of the same set of objects. Three models defining median partitions are described: two of them are obtained from a least-squares fit of a set of membership functions, and the third (proposed by Pittau and Vichi) is acquired from a least-squares fit of a set of joint membership functions. The models are illustrated by application to both a set of hard partitions and a set of fuzzy partitions and comparisons are made between them and an alternative approach to obtaining a consensus fuzzy partition proposed by Sato and Sato; a discussion is given of some interesting differences in the results.We are grateful to Dr. M.G. Pittau for carrying out the analyses of the macroeconomic data using the method of Sato and Sato (1994).     

When “decoy effect” meets gender bias: The role of choice set composition in hiring decisions

A large body of research has found evidence that hiring decisions are frequently subject to strong gender bias and has explored factors that help to predict and prevent such a bias from occurring. In this paper, we explore a novel factor that has received only little attention: the composition of the choice set. Drawing on prior research on the attraction effect of decoys in consumer choice and personnel decisions, we posit that when decision makers need to decide whether to hire a male or a female applicant for a stereotypically male position, the presence of a third applicant whose profile is asymmetrically dominated by one of the two applicants can in many circumstances strongly increase the odds that the male applicant will be selected, but will not be beneficial for the female applicant. We test our hypotheses in five experimental studies with different designs, experimental settings, and participant pools—including managers with professional experience in hiring decisions. Our results provide robust evidence demonstrating the strong effects of choice set composition on the emergence of gender bias. In addition, we found that the presence of asymmetrically dominated applicants makes decision makers more confident in their biased decisions and more likely to implement them immediately without searching for further information. Finally, our results also provide some initial evidence that our results for stereotypically male positions will be reversed when hiring decisions are made for stereotypically female positions where the presence of decoys instead gives an advantage to female over male applicants.