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The transition fundions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's Theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

The past few years have seen the rapid rise of all things “social” on the web from the growth of online social networks like Facebook, to real-time communication services like Twitter, to user-contributed content sites like Flickr and YouTube, to content aggregators like Digg. Beyond these popular Web 2.0 successes, the emer- gence of Social Information Systems is promising to fundamentally transform what information we encounter and digest, how businesses market and engage with their customers, how universities educate and train a new generation of researchers, how the government investigates terror networks, and even how political regimes interact with their citizenry. Users have moved from being passive consumers of information (via querying or browsing) to becoming active participants in the creation of data and knowledge artifacts, actively sorting, ranking, and annotating other users and artifacts. This fundamental shift to social systems places new demands on providing de- pendable capabilities for knowing whom to trust and what information to trust, given the open and unregulated nature of these systems. The emergence of large-scale user participation in Social Information Systems suggests the need for the development of user-centric approaches to information quality. As a step in this direction this research proposes an interaction-based approach for modeling the notion of user im- portance. The interaction-based model is centered around the uniquely social aspects of these systems, by treating who communicates with whom (an interaction) as a core building block in evaluating user importance. We first study the interaction characteristics of Twitter, one of the most buzzworthy recent Social Web successes, examining the usage statistics, growth patterns, and user interaction behavior of over 2 million participants on Twitter. We believe this is the first large-scale study of dynamic interactions on a real-world Social Information System. Based on the anal- ysis of the interaction structure of Twitter, the second contribution of this thesis research is an exploration of approaches for measuring user importance. As part of this exploration, we study several different approaches that build on the inherent interaction-based framework of Social Information Systems. We explore this model through an experimental study over an interaction graph consisting of 800,000 nodes and about 1.9 million interaction edges. The user importance modeling approaches that we present can be applied to any Social Information System in which interactions between users can be monitored.

Given p independent, symmetric random walks on d-dimensional integer lattice that are the domain of attraction for a stable distribution, we calculate the moderate deviation of the intersection of ranges of the random walks in the case where the walks intersect infinitely often as time goes to infinity. That is to say, we establish a weak law convergence of intersection of ranges to intersection local time of stable processes and use this convergence as a link to establish deviation results.

This study concludes two parts. In the first part, we establish the model of the interaction between two cell populations following the concept of the random-walk, and assume the cell movement is constrained by space limitation primarily. In the other part, the interaction model is deduced from the concept of the flux motion, and the movement is constrained by space limitation, too. Furthermore, we analyze two models to obtain the behavior of two cell populations as time is close to the initial state and far into the future. / This study concludes two parts. In the first part, we establish the model of the interaction between two cell populations following the concept of the random-walk, and assume the cell movement is constrained by space limitation primarily. In the other part, the interaction model is deduced from the concept of the flux motion, and the movement is constrained by space limitation, too. Furthermore, we analyze two models to obtain the behavior of two cell populations as time is close to the initial state and far into the future.

The ever increasing use of computer systems in society brings emergent challenges to companies and system designers. The reliability of software and hardware can be financially critical, and lives can depend on it. The growth in size and complexity of software, and increasing concurrency, compounds the problem. The potential for errors is greater than ever before, and the stakes are higher than ever before. Formal methods, particularly model checking, is an approach that attempts to prove mathematically that a model of the behaviour of a product is correct with respect to certain properties. Certain errors can therefore be proven never to occur in the model. This approach has tremendous potential in system development to provide guarantees of correctness. Unfortunately, in practice, model checking cannot handle the enormous sizes of the models of real-world systems. The reason is that the approach requires an exhaustive search of the model to be conducted. While there are exceptions, in general model checkers are said not to scale well. In this thesis, we deal with this scaling issue by using a guiding technique that avoids searching areas of the model, which are unlikely to contain errors. This technique is based on a process of model abstraction in which a new, much smaller model is generated that retains certain important model information but discards the rest. This new model is called a heuristic. While model checking using a heuristic as a guide can be extremely effective, in the worst case (when the guide is of no help), it performs the same as exhaustive search, and hence it also does not scale well in all cases. A second technique is employed to deal with the scaling issue. This technique is based on the concept of random walks. A random walk is simply a `walk' through the model of the system, carried out by selecting states in the model randomly. Such a walk may encounter an error, or it may not. It is a non-exhaustive technique in the sense that only a manageable number of walks are carried out before the search is terminated. This technique cannot replace the conventional model checking as it can never guarantee the correctness of a model. It can however, be a very useful debugging tool because it scales well. From this point of view, it relieves the system designer from the difficult task of dealing with the problem of size in model checking. Using random walks, the effort goes instead into looking for errors. The effectiveness of model checking can be greatly enhanced if the above two techniques are combined: a random walk is used to search for errors, but the walk is guided by a heuristic. This in a nutshell is the focus of this work. We should emphasise that the random walk approach uses the same formal model as model checking. Furthermore, the same heuristic technique is used to guide the random walk as a guided model checker. Together, guidance and random walks are shown in this work to result in vastly improved performance over conventional model checking. Verification has been sacrificed of course, but the new technique is able to find errors far more quickly, and deal with much larger models.

The ever increasing use of computer systems in society brings emergent challenges to companies and system designers. The reliability of software and hardware can be financially critical, and lives can depend on it. The growth in size and complexity of software, and increasing concurrency, compounds the problem. The potential for errors is greater than ever before, and the stakes are higher than ever before. Formal methods, particularly model checking, is an approach that attempts to prove mathematically that a model of the behaviour of a product is correct with respect to certain properties. Certain errors can therefore be proven never to occur in the model. This approach has tremendous potential in system development to provide guarantees of correctness. Unfortunately, in practice, model checking cannot handle the enormous sizes of the models of real-world systems. The reason is that the approach requires an exhaustive search of the model to be conducted. While there are exceptions, in general model checkers are said not to scale well. In this thesis, we deal with this scaling issue by using a guiding technique that avoids searching areas of the model, which are unlikely to contain errors. This technique is based on a process of model abstraction in which a new, much smaller model is generated that retains certain important model information but discards the rest. This new model is called a heuristic. While model checking using a heuristic as a guide can be extremely effective, in the worst case (when the guide is of no help), it performs the same as exhaustive search, and hence it also does not scale well in all cases. A second technique is employed to deal with the scaling issue. This technique is based on the concept of random walks. A random walk is simply a `walk' through the model of the system, carried out by selecting states in the model randomly. Such a walk may encounter an error, or it may not. It is a non-exhaustive technique in the sense that only a manageable number of walks are carried out before the search is terminated. This technique cannot replace the conventional model checking as it can never guarantee the correctness of a model. It can however, be a very useful debugging tool because it scales well. From this point of view, it relieves the system designer from the difficult task of dealing with the problem of size in model checking. Using random walks, the effort goes instead into looking for errors. The effectiveness of model checking can be greatly enhanced if the above two techniques are combined: a random walk is used to search for errors, but the walk is guided by a heuristic. This in a nutshell is the focus of this work. We should emphasise that the random walk approach uses the same formal model as model checking. Furthermore, the same heuristic technique is used to guide the random walk as a guided model checker. Together, guidance and random walks are shown in this work to result in vastly improved performance over conventional model checking. Verification has been sacrificed of course, but the new technique is able to find errors far more quickly, and deal with much larger models.

The ever increasing use of computer systems in society brings emergent challenges to companies and system designers. The reliability of software and hardware can be financially critical, and lives can depend on it. The growth in size and complexity of software, and increasing concurrency, compounds the problem. The potential for errors is greater than ever before, and the stakes are higher than ever before. Formal methods, particularly model checking, is an approach that attempts to prove mathematically that a model of the behaviour of a product is correct with respect to certain properties. Certain errors can therefore be proven never to occur in the model. This approach has tremendous potential in system development to provide guarantees of correctness. Unfortunately, in practice, model checking cannot handle the enormous sizes of the models of real-world systems. The reason is that the approach requires an exhaustive search of the model to be conducted. While there are exceptions, in general model checkers are said not to scale well. In this thesis, we deal with this scaling issue by using a guiding technique that avoids searching areas of the model, which are unlikely to contain errors. This technique is based on a process of model abstraction in which a new, much smaller model is generated that retains certain important model information but discards the rest. This new model is called a heuristic. While model checking using a heuristic as a guide can be extremely effective, in the worst case (when the guide is of no help), it performs the same as exhaustive search, and hence it also does not scale well in all cases. A second technique is employed to deal with the scaling issue. This technique is based on the concept of random walks. A random walk is simply a `walk' through the model of the system, carried out by selecting states in the model randomly. Such a walk may encounter an error, or it may not. It is a non-exhaustive technique in the sense that only a manageable number of walks are carried out before the search is terminated. This technique cannot replace the conventional model checking as it can never guarantee the correctness of a model. It can however, be a very useful debugging tool because it scales well. From this point of view, it relieves the system designer from the difficult task of dealing with the problem of size in model checking. Using random walks, the effort goes instead into looking for errors. The effectiveness of model checking can be greatly enhanced if the above two techniques are combined: a random walk is used to search for errors, but the walk is guided by a heuristic. This in a nutshell is the focus of this work. We should emphasise that the random walk approach uses the same formal model as model checking. Furthermore, the same heuristic technique is used to guide the random walk as a guided model checker. Together, guidance and random walks are shown in this work to result in vastly improved performance over conventional model checking. Verification has been sacrificed of course, but the new technique is able to find errors far more quickly, and deal with much larger models.

In this expository paper we study the stationary measures of a stochastic process called nearest neighbor random walk on Z, and further we describe conditions for these measures to have the stronger property of reversibility. We consider both the cases of symmetric and non-symmetric random walk.

The diploma thesis is focused on the comparison of goodwill values of two construction companies. The theoretical part describes the basic knowledge and principles they are required to terminated the value of goodwill. Furthermore, terms such as marketing planning and questionnaire creation are explained. In the practical part, selected methods are used to terminate the value of goodwill. Subsequently, five questions are set and the questions are tested. Based on the analysis results is proposed marketing strategy at the end of the thesis.