The Fibonacci Numbers are one of the most intriguing sequences in mathematics. I present generalizations of this well known sequence. Using combinatorial proofs, I derive closed form expressions for these generalizations. Then using Markov Chains, I derive a second closed form expression for these numbers which is a generalization of Binet’s formula for Fibonacci Numbers. I expand further and determine the generalization of Binet’s formula for any kth order linear recurrence.

My Masters research aims to deepen our understanding of the behaviour of robust methods in logistic regression. Logistic regression is a special case of Generalized Linear Modelling (GLM), which is a powerful and popular technique for modelling a large variety of data. Robust methods are useful in reducing the effect of outlying values in the response variable on parameter estimates. A literature survey shows that we are still at the beginning of being able to detect extreme observations in logistic regression analyses, to apply robust methods in logistic regression and to present informatively the results of logistic regression analyses. In Chapter 1 I have made a basic introduction to logistic regression, with an example, and to robust methods in general. In Chapters 2 through 4 of the thesis I have described traditional methods and some relatively new methods for presenting results of logistic regression using powerful visualization techniques as well as the concepts of outliers in binomial data. I have used different published data sets for illustration, such as the Prostate Cancer data set, the Damaged Carrots data set and the Recumbent Cow data set. In Chapter 4 I summarize and report on the modem concepts of graphical methods, such as central dimension reduction, and the use of graphics as pioneered by Cook and Weisberg (1999). In Section 4.6 I have then extended the work of Cook and Weisberg to robust logistic regression. In Chapter 5 I have described simulation studies to investigate the effects of outlying observations on logistic regression (robust and non-robust). In Section 5.2 I have come to the conclusion that, in the case of classical or robust multiple logistic regression with no outliers, robust methods do not necessarily provide more reasonable estimates of the parameters for the data that contain no st~ong outliers. In Section 5.4 I have looked into the cases where outliers are present and have come to the conclusion that either the breakdown method or a sensitivity analysis provides reasonable parameter estimates in that situation. Finally, I have identified areas for further study.

<p>The aim of this study was to evaluate if a population suffering from generalized anxiety disorder could benefit from an Internet based self-help treatment guided via email contact with a therapist. The treatment was based on established cognitive behavioral principles. It was hypothesized that significant improvements would be found as measured by eight self report questionnaires, absence of a clinical diagnoses and global clinical improvement. A total of 89 participants were included and 44 were randomized to a treatment condition and 45 were assigned to a waitlist control. The controls received similar treatment after the first post treatment assessment, conducted eight weeks after the beginning or treatment of the first group. The results showed statistically significant improvements for the treatment group. No changes were observed in the waiting-list control group, with the exception of a minor decrease in depression scores. Large effect sizes were found both within the treatment group and between the two groups in favor of the treatment. In conclusion, Internet treatment can be an efficacious format for treating generalized anxiety disorder.</p>

<p>Sedan årsskiftet 2006/2007 har antalet medlemmar i arbetslöshetskassorna minskat drastiskt. Under samma period har ett flertal reformer genomförts på arbetslöshetsförsäkringens område som bland annat resulterat i höjda medlemsavgifter för de flesta a-kassorna. Syftet med denna uppsats är att undersöka huruvida det över tid går att finna något samband mellan förändringar i medlemsantal och medlemsavgifter. För att undersöka detta måste man förutom avgifterna även ta hänsyn till andra variabler kopplade till arbetslöshetsförsäkringen. Dessa övriga variabler är grundbelopp, högsta dagpenning, ersättningsgrad och arbetslöshet. Vi formulerar en modell för sambandet mellan medlemsantal och dessa variabler och skattar denna genom metoden Generalized Method of Moments med hjälp av data från 2000-2009. Våra resultat visar i enlighet med teori och tidigare forskning på ett negativt samband mellan medlemsavgifter och antalet medlemmar i a-kassan. Detta samband visar sig vara starkt, särskilt på lång sikt. För att tydigare se hur avgiftsförändringar påverkar olika typer av individer i olika grad har vi även undersökt huruvida medlemsantalet i a-kassor kopplade till tjänstemanna- respektive arbetarförbund är olika känsliga för förändringar i avgiften. Våra resultat visar i kontrast till tidigare studier att a-kassorna kopplade till tjänstemannaförbunden (TCO och Saco) är mer känsliga för förändringar jämfört med arbetarförbunden (LO). Detta skapar anledning att tro att det finns andra faktorer än avgifter och de övriga variablerna som inkluderats i vår modell vilka påverkar anslutningsgraden och som kan förklara skillnaden mellan de olika grupperna.</p>

Sedan årsskiftet 2006/2007 har antalet medlemmar i arbetslöshetskassorna minskat drastiskt. Under samma period har ett flertal reformer genomförts på arbetslöshetsförsäkringens område som bland annat resulterat i höjda medlemsavgifter för de flesta a-kassorna. Syftet med denna uppsats är att undersöka huruvida det över tid går att finna något samband mellan förändringar i medlemsantal och medlemsavgifter. För att undersöka detta måste man förutom avgifterna även ta hänsyn till andra variabler kopplade till arbetslöshetsförsäkringen. Dessa övriga variabler är grundbelopp, högsta dagpenning, ersättningsgrad och arbetslöshet. Vi formulerar en modell för sambandet mellan medlemsantal och dessa variabler och skattar denna genom metoden Generalized Method of Moments med hjälp av data från 2000-2009. Våra resultat visar i enlighet med teori och tidigare forskning på ett negativt samband mellan medlemsavgifter och antalet medlemmar i a-kassan. Detta samband visar sig vara starkt, särskilt på lång sikt. För att tydigare se hur avgiftsförändringar påverkar olika typer av individer i olika grad har vi även undersökt huruvida medlemsantalet i a-kassor kopplade till tjänstemanna- respektive arbetarförbund är olika känsliga för förändringar i avgiften. Våra resultat visar i kontrast till tidigare studier att a-kassorna kopplade till tjänstemannaförbunden (TCO och Saco) är mer känsliga för förändringar jämfört med arbetarförbunden (LO). Detta skapar anledning att tro att det finns andra faktorer än avgifter och de övriga variablerna som inkluderats i vår modell vilka påverkar anslutningsgraden och som kan förklara skillnaden mellan de olika grupperna.

The aim of this study was to evaluate if a population suffering from generalized anxiety disorder could benefit from an Internet based self-help treatment guided via email contact with a therapist. The treatment was based on established cognitive behavioral principles. It was hypothesized that significant improvements would be found as measured by eight self report questionnaires, absence of a clinical diagnoses and global clinical improvement. A total of 89 participants were included and 44 were randomized to a treatment condition and 45 were assigned to a waitlist control. The controls received similar treatment after the first post treatment assessment, conducted eight weeks after the beginning or treatment of the first group. The results showed statistically significant improvements for the treatment group. No changes were observed in the waiting-list control group, with the exception of a minor decrease in depression scores. Large effect sizes were found both within the treatment group and between the two groups in favor of the treatment. In conclusion, Internet treatment can be an efficacious format for treating generalized anxiety disorder.

This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case <i>n</i> = 3<i>k</i> + 1 and <i>k</i> greater than or equal to 1. We do this mainly by means of an induction which takes us from <i>GP</i>(3<i>k</i> + 1, <i>k</i>) to <i>GP</i>(3(<i>k</i> + 2) + 1, <i>k</i> + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call <i>rotors</i> to obtain a Hamilton path in the larger graph.

Nowadays in the world of mass consumption there is big demand for distributioncenters of bigger size. Managing such a center is a very complex and difficult taskregarding to the different processes and factors in a usual warehouse when we want tominimize the labor costs. Most of the workers’ working time is spent with travelingbetween source and destination points which cause deadheading. Even if a worker knowsthe structure of a warehouse well and because of that he or she can find the shortest pathbetween two points, it is still not guaranteed that there won’t be long traveling timebetween the locations of two consecutive tasks. We need optimal assignments betweentasks and workers.In the scientific literature Generalized Assignment Problem (GAP) is a wellknownproblem which deals with the assignment of m workers to n tasks consideringseveral constraints. The primary purpose of my thesis project was to choose a heuristics(genetic algorithm, tabu search or ant colony optimization) to be implemented into SAPExtended Warehouse Management (SAP EWM) by with task assignment will be moreeffective between tasks and resources.After system analysis I had to realize that due different constraints and businessdemands only 1:1 assingments are allowed in SAP EWM. Because of that I had to use adifferent and simpler approach – instead of the introduced heuristics – which could gainbetter assignments during the test phase in several cases. In the thesis I described indetails what ware the most important questions and problems which emerged during theplanning of my optimized assignment method.

This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case <i>n</i> = 3<i>k</i> + 1 and <i>k</i> greater than or equal to 1. We do this mainly by means of an induction which takes us from <i>GP</i>(3<i>k</i> + 1, <i>k</i>) to <i>GP</i>(3(<i>k</i> + 2) + 1, <i>k</i> + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call <i>rotors</i> to obtain a Hamilton path in the larger graph.

Empirically-based models of landslide distribution and susceptibility are currently the most commonly used approach for mapping probabilities of landslide initiation and analyzing their association with natural and anthropogenic environmental factors. In general, these models statistically estimate susceptibility based on the predisposition of an area to experience a landslide given a range of environmental factors, which may include land use, topography, hydrology and other spatial attributes. Novel statistical approaches include the generalized additive model (GAM), a non-parametric regression technique, which is used in this study to explore the relationship of landslide initiation to topography, rainfall and forest land cover and logging roads on Vancouver Island, British Columbia. The analysis is centered on an inventory of 639 landslides of winter 2006/07. Data sources representing potentially relevant environmental conditions of landslide initiation are based on: terrain analysis derived from a 20-m CDED digital elevation model; forest land cover classified from Landsat TM scenes for the summer before the 2006 rainy season; geostatistically interpolated antecedent rainfall patterns representing different temporal scales of rainfall (a major storm, winter and annual rainfall); and the main lithological units of surface geology. In order to assess the incremental effect of these data sources to predict landslide susceptibility, predictive performances of models based on GAMs are compared using spatial cross-validation estimates of the area under the ROC curve (AUROC), and variable selection frequencies are used to determine the prevalence of non-parametric associations to landslides. In addition to topographic variables, forest land cover (e.g., deforestation), and logging roads showed a strong association with landslide initiation, followed by rainfall patterns and the very general lithological classification as less important controls of landscape-scale landslide activity in this area. Annual rainfall patterns are found not to contribute significantly to model prediction improvement and may lead to model overfitting. Comparisons to generalized linear models (i.e., logistic regression) indicate that GAMs are significantly better for modeling landslide susceptibility. Overall, based on the model predictions, the most susceptible 4% of the study area had 29 times higher density of landslide initiation points than the least susceptible 73% of the study area (0.156 versus 0.005 landslides/km2).