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Abstract: Let if , and if . We conjecture that the θ-orbit of every nonnegative rational number ends in 0. A weaker conjecture asserts that there are no positive rational fixed points for any map in the semigroup Λ generated by the maps and . In this paper, we prove that the asymptotic density of the set of maps in Λ that have rational fixed points is zero. Moreover, we prove that certain types of elements in the semigroup Λ cannot have rational fixed points. [Copyright &y& Elsevier]

Abstract: In this paper we propose a novel, game-theoretic approach for finding multiple instances of an object category as sets of mutually coherent votes in a generalized Hough space. Existing Hough-voting based detection systems have to inherently apply parameter-sensitive non-maxima suppression (NMS) or mode detection techniques for finding object center hypotheses. Moreover, the voting origins contributing to a particular maximum are lost and hence mostly bounding boxes are drawn to indicate the object hypotheses. To overcome these problems, we introduce a two-stage method, applicable on top of any Hough-voting based detection framework. First, we define a Hough environment, where the geometric compatibilities of the voting elements are captured in a pairwise fashion. Then we analyze this environment within a game-theoretic setting, where we model the competition between voting elements as a Darwinian process, driven by their mutual geometric compatibilities. In order to find multiple and possibly overlapping objects, we introduce a new enumeration method inspired by tabu search. As a result, we obtain locations and voting element compositions of each object instance while bypassing the task of NMS. We demonstrate the broad applicability of our method on challenging datasets like the extended TUD pedestrian crossing scene. [Copyright &y& Elsevier]

Abstract: A system of distinct representatives (SDR) of a family is a sequence of distinct elements with for . Let denote the number of SDRs of a family ; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer , a family is called a -family if the union of any sets in the family contains at least elements. The famous Hall’s theorem says that if and only if is a -family. Denote by the minimum number of SDRs in a -family. The problem of determining and those families containing exactly SDRs was first raised by Chang [G.J. Chang, On the number of SDR of a -family, European J. Combin. 10 (1989) 231–234]. He solved the cases when and gave a conjecture for . In this paper, we solve the conjecture. [Copyright &y& Elsevier]

In many cases Bayes' theorem is an appropriate algorithm for the aggregation of probabilistic evidence. As with other statistical procedures, there are restrictions that must be taken into account. In the present paper we shall comment on several approaches that have been devoted to one of these restrictions; the incorporation of uncertainty about the true state of a datum. A datum is a variable which can be partitioned into equivalence classes. These classes represent the possible data states which will be also called events. In Bayes' theorem an event is an item of information which will be used for revising the opinion about the relative likelihood of hypotheses. In any specific situation only one of the possible events will be the true event. An event may come from a source whose reporting or observational accuracy is not perfect. An example may illustrate the issue. A medical doctor wants to come to a diagnosis. To achieve this he considers several data. One datum might be the result of a medical test, which has three possible states: positive, negative, inconclusive. If the doctor reports the state of the datum to be positive, he may be wrong by whatever reasons. That is, the report of an event must not necessarily coincide with the actual or true event of the datum under consideration. This kind of uncertainty about the true event in any specific situation is a characteristic of the source. In most cases it will reduce the diagnostic impact of an event. Whenever the report of an event and the true event coincide imperfectly, measures of source inaccuracy must be incorporated into Bayes' theorem. The task can be considered as two stage probabilistic induction. The first step is induction from the reported to the actual event, the second is induction from the actual event to the creditation of hypotheses. This is the reason why Gettys and Willke (1969) speak about "cascaded inference." [ABSTRACT FROM AUTHOR]