1. Drawing balls without replacement (20 pts). Suppose that we have an urn containing one red ball and n − 1 white balls. Each time we draw a ball at random from the urn without replacement (so after the n-th drawing, there is no ball left in the urn). X i = 1 if the i-th ball is red, 0 otherwise.

The models surveyed include generalized Pólya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.

We test the hypothesis that superficial knowledge interdependence is more effective in fostering individual learning from collaboration than the true knowledge interdependence often realized by jigsaw-type collaboration arrangements. Based on research on group informationprocessing, we argue for the benefits of distributing only contextual information, but not core principles between learners, ...

Urn automata are a new class of automata consisting of an input tape, a finite-state controller, and an urn containing tokens with a finite set of colors, where the finite-state controller can sample and replace tokens in the urn but cannot control which tokens it receives. We consider the computational power of urn automata, showing that an urn automaton with O(f(n)) tokens can, with high prob...

Abstract. A general analytical treatment for the free energy of an urn model with quenched disorder is developed. It is revealed that the disordered urn model has similar mathematical structure to a random field Ising model. From this similarity, the free energy of the disordered urn model in the thermodynamic limit can be calculated from the canonical partition function by the replica analysis...

Urn models play an important role to express various basic ideas in probability theory. Here we extend this urn model with tubes. An contains coloured balls, which can be drawn probabilities proportional the numbers of balls each colour. For colour a tube is assumed. These tubes have different sizes (lengths). The idea that after drawing ball from it dropped corresponding We consider two associ...

Laguerre geometry of surfaces in R is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in Rn. For any umbilical free hypersurface x : M → Rn with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invarian...

The Generalized Pólya Urn (GPU) is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we propose a sequential estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide spectrum of applications. Because the proposed urn model depends on sequential estimations of unknown...