Academic literature on the topic 'OPTIMAL NUMBERS'

In this thesis we study lower ramification numbers of power series tan- gent to the identity that are defined over fields of positive characteristics. Let f be such a series, then f has a fixed point at the origin and the corresponding lower ramification numbers of f are then, up to a constant, the multiplicity of zero as a fixed point of iterates of f. In this thesis we classify power series having ‘small’ ramification numbers. The results are then used to study ramification numbers of polynomials not tangent to the identity. We also state a few conjectures motivated by computer experiments that we performed.

The focus of this thesis paper is to study the impact the number of ants has on the found solution of the Ant Colony Optimization (ACO) metaheuristic when solving the Traveling Salesman Problem. The goal was to find out how the length of the computed tours change for different amounts of ants within a limited number of iterations. To study this, three well known versions of the ACO algorithm were implemented and tested: Min-Max Ant System (MMAS), Elitist Ant System (EliteAS) and Ranked Ant System (RankedAS). The results showed trends that were consistent over several test cases. EliteAS and RankedAS which both utilize specialist ants showed clear signs that the number of specialists had a large influence on the length of solutions. Meanwhile, normal ants did not affect the solutions as much. MMAS and EliteAS only had a small variation on the answer, with lower amount of ants being more favorable. On the other hand, RankedAS performed better by a large margin when working with five specialists and a number of ants equaling the number of cities in the problem.<br>Målet med denna rapport var att studera hur antalet myror som används av Ant Colony Optimization (ACO) påverkar resultatet vid lö- sandet av Traveling Salesman Problem (TSP). Hur ändras lösningens längd med olika antal myror, när antalet iterationer som får användas är begränsat? För att få fram ett svar på frågan implementerades och testades tre välkända ACO algoritmer: Min-Max Ant System (MMAS), Elitist Ant System (EliteAS) och Ranked Ant System (RankedAS). Efter implementering och utförlig testning så uppdagades trender som var konsistenta över flera testfall. För EliteAS och RankedAS, som bå- da förlitar sig på specialiserade myror, hade antalet specialister en stor påverkan på den funna längden. Normala myror hade istället en liten påverkan på slutresultatet. För MMAS och EliteAS så var skillnaden minimal, med en viss favör mot ett lägre antal myror. RankedAS hade en motsatt trend och hade bäst resultat med fem specialister och lika många normala myror som antalet städer i TSP instansen.

Wireless Local Area Networks (WLANs) have become a major success in telecommunications during the last few years, due to advantages such as mobility, flexibility, and easier maintenance. A device called an access point (AP) acts as a base station in WLAN for connecting a group of users to the network via radio signal. During the planning of such a network an important problem is to determine the optimal number of these devices and their placement/distribution so that coverage, capacity, and physical security are maximised at minimum cost. In this thesis we are using continuous optimisation techniques to optimise the number of APs and their distribution while cost of deployment is reduced and physical security of the network is enhanced. To find the number and placement of APs, we developed a multi-objective functions model based on path losses and power for free space environments. The two functions in the models are combined by using a balancing parameter. Since it is recognised that some of the objectives can be handled one at a time, in another approach, we followed a step-by-step procedure. We start with a novel optimisation model based on path losses for indoor environments including obstacles. Cost of deployment is saved by finding the minimum number of APs ensuring that the path loss at each test point/receiver is below the given maximum path loss. Next, the physical security of the network is enhanced by placing the APs far from places accessible to unauthorised users to reduce the risk of intrusion into the network. This is achieved in the framework of the model by introducing potential unauthorised users in unauthorised areas for whom coverage is minimised. Due to the presence of obstacles in indoor buildings, the path loss function is discontinuous. Therefore, the objective functions are very complicated and most of the existing optimisation algorithms cannot be applied to solve the problem. We use a global optimisation algorithm that is not used by other researchers to solve the same problem. To validate the accuracy of the optimisation model and performance of the numerical methods, we run tests on several indoor buildings and use wide range of WLAN parameters. The results demonstrate the quality of our model and algorithm. Based on the proposed model and algorithm, we developed a software to assist the network designers in planning wireless LANs.<br>Doctor of Philosophy

Wireless Local Area Networks (WLANs) have become a major success in telecommunications during the last few years, due to advantages such as mobility, flexibility, and easier maintenance. A device called an access point (AP) acts as a base station in WLAN for connecting a group of users to the network via radio signal. During the planning of such a network an important problem is to determine the optimal number of these devices and their placement/distribution so that coverage, capacity, and physical security are maximised at minimum cost. In this thesis we are using continuous optimisation techniques to optimise the number of APs and their distribution while cost of deployment is reduced and physical security of the network is enhanced. To find the number and placement of APs, we developed a multi-objective functions model based on path losses and power for free space environments. The two functions in the models are combined by using a balancing parameter. Since it is recognised that some of the objectives can be handled one at a time, in another approach, we followed a step-by-step procedure. We start with a novel optimisation model based on path losses for indoor environments including obstacles. Cost of deployment is saved by finding the minimum number of APs ensuring that the path loss at each test point/receiver is below the given maximum path loss. Next, the physical security of the network is enhanced by placing the APs far from places accessible to unauthorised users to reduce the risk of intrusion into the network. This is achieved in the framework of the model by introducing potential unauthorised users in unauthorised areas for whom coverage is minimised. Due to the presence of obstacles in indoor buildings, the path loss function is discontinuous. Therefore, the objective functions are very complicated and most of the existing optimisation algorithms cannot be applied to solve the problem. We use a global optimisation algorithm that is not used by other researchers to solve the same problem. To validate the accuracy of the optimisation model and performance of the numerical methods, we run tests on several indoor buildings and use wide range of WLAN parameters. The results demonstrate the quality of our model and algorithm. Based on the proposed model and algorithm, we developed a software to assist the network designers in planning wireless LANs.<br>Doctor of Philosophy

Wireless Local Area Networks (WLANs) have become a major success in telecommunications during the last few years, due to advantages such as mobility, flexibility, and easier maintenance. A device called an access point (AP) acts as a base station in WLAN for connecting a group of users to the network via radio signal. During the planning of such a network an important problem is to determine the optimal number of these devices and their placement/distribution so that coverage, capacity, and physical security are maximised at minimum cost. In this thesis we are using continuous optimisation techniques to optimise the number of APs and their distribution while cost of deployment is reduced and physical security of the network is enhanced. To find the number and placement of APs, we developed a multi-objective functions model based on path losses and power for free space environments. The two functions in the models are combined by using a balancing parameter. Since it is recognised that some of the objectives can be handled one at a time, in another approach, we followed a step-by-step procedure. We start with a novel optimisation model based on path losses for indoor environments including obstacles. Cost of deployment is saved by finding the minimum number of APs ensuring that the path loss at each test point/receiver is below the given maximum path loss. Next, the physical security of the network is enhanced by placing the APs far from places accessible to unauthorised users to reduce the risk of intrusion into the network. This is achieved in the framework of the model by introducing potential unauthorised users in unauthorised areas for whom coverage is minimised. Due to the presence of obstacles in indoor buildings, the path loss function is discontinuous. Therefore, the objective functions are very complicated and most of the existing optimisation algorithms cannot be applied to solve the problem. We use a global optimisation algorithm that is not used by other researchers to solve the same problem. To validate the accuracy of the optimisation model and performance of the numerical methods, we run tests on several indoor buildings and use wide range of WLAN parameters. The results demonstrate the quality of our model and algorithm. Based on the proposed model and algorithm, we developed a software to assist the network designers in planning wireless LANs.<br>Doctor of Philosophy

This empirical study has shown that optimal portfolios need approximately 10 securities to diversify away the unsystematic risk. This challenges previous studies of randomly chosen portfolios which states that at least 30 securities are needed. The result of this study sheds light upon the difference in risk diversification between random portfolios and optimal portfolios and is a valuable contribution for investors. The study suggests that a major part of the unsystematic risk in a portfolio can be diversified away with fewer securities by using portfolio optimization. Individual investors especially, who usually have portfolios consisting of few securities, benefit from these results. There are today multiple user-friendly software applications that can perform the computations of portfolio optimization without the user having to know the mathematics behind the program. Microsoft Excel’s solver function is an example of a well-used software for portfolio optimization. In this study however, MATLAB was used to perform all the optimizations. The study was executed on data of 140 stocks on NASDAQ Stockholm during 2000-2014. Multiple optimizations were done with varying input in order to yield a result that only depended on the investigated variable, that is, how many different stocks that are needed in order to diversify away the unsystematic risk in a portfolio.<br><p>Osäker på examinatorns namn, tog namnet på den person som skickade mejl om betyg.</p>