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We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex., 12 pages, 2 figures

This study presents a fuzzy logic and binary-goal programming-based approach for solving the exam timetabling problem to create a balanced-exam schedule. To be able to address the practical challenges of the exam timetabling problem, the model is developed with and verified by a human expert for exam scheduling. We propose a fuzzy-criticality level identification methodology to assign the criticality levels of exams for the students using three pieces of information, namely, credits, success ratios, and types of the classes. It is noted that the computed criticality levels are close approximates for those of the human expert. We then present a goal programming model to schedule exams using these criticality levels as well as other general problem data. The result of the goal program is a balanced-exam schedule in terms of exam criticality levels. Final step includes room assignments using a simple algorithm. The significance of the study is the consideration of the exam criticalities, for not only the students of the same year but also the students with different levels of seniority, as well as an even distribution of exams for professors which make the problem more challenging for the human expert in practice. Using a real-life problem, we show that our approach creates an exam schedule that is more preferable than the one prepared by the human expert. Additionally, computational results show the potential of our model to be used in real-life problems of larger-size.

In this paper, inspired by two very different, successful metric theories such us the real view-point of Lowen's approach spaces and the probabilistic field of Kramosil and Michalek's fuzzymetric spaces, we present a family of spaces, called fuzzy approach spaces, that are appropriate to handle, at the same time, both measure conceptions. To do that, we study the underlying metric interrelationships between the above mentioned theories, obtaining six postulates that allow us to consider such kind of spaces in a unique category. As a result, the natural way in which metric spaces can be embedded in both classes leads to a commutative categorical scheme. Each postulate is interpreted in the context of the study of the evolution of fuzzy systems. First properties of fuzzy approach spaces are introduced, including a topology. Finally, we describe a fixed point theorem in the setting of fuzzy approach spaces that can be particularized to the previous existing measure spaces. The authors are very grateful to the Spanish Government for Grant DPI2012-30651 and to the Basque Government for Grant IT378-10. This paper has been partially supported by projects FQM-178, FQM-268 and FQM-265 of the Andalusian CICE, Spain.

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.

Spatial data is quite often is prone to uncertainty and imprecision. For this purpose, fuzzy regions have been developed: they basically consist of a fuzzy set over a two dimensional domain, allowing for both fuzzy regions and fuzzy points to be modelled. The model is extended to a level-2 fuzzy region to overcome some limitations, but this has an impact on operations. In this contribution, we will look into the construction of and combination of existing data to yield level-2 fuzzy regions.