This thesis deals with the existence and properties of certain types of covering spaces. It contains the discussion of a generalization of the notion of simple connectedness and several well-known theorems depending on this. / Thesis / Master of Science (MS)

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The literature on the ethics of presenting as queer has been largely confined to a commonly acknowledged phenomenon called "passing," or fully concealing one's membership to a marginalized group. Often employed as a survival strategy, many are sympathetic to the idea that one should be able to pass if need be. With that said, many philosophers argue that it is inextricably tied to oppression in the sense that acts of passing are acts complicit with one's own oppression. Because of the usually drastic alteration to one's appearance or behavior passing encompasses, along with its connection to oppression, a larger problem has gone unnoticed: covering. Covering differs from passing as one's membership to a marginalized community is now background knowledge in any social interaction where one may cover. Covering, then, depicts the intentional editing of one's behavior to modify the way in which their marginalized status is communicated to an audience. It is because one has announced their status as a community member that this concept often surfaces without controversy. This, at first, is intuitive. Why should someone be able to permissibly hide the entirety of their identity if partial concealment is impermissible? In the end, the very reason that covering is often excluded from the ethical discourse – that one has already announced their status as a marginalized community member – is actually a reason for my claim, that covering is wrong but passing is not, rather than one against it. I begin my argument with a negative claim: there is no duty not to cover. After explaining why this is the case, I argue for my second, positive claim: there is a duty to refrain from covering. If successful, my argument should show that a duty not to pass, or to be out, is too demanding. This will offer a better starting point for a relocation of some duty, which I argue should be on covering. If it is placed on covering, then demandingness concerns are circumvented and the goal of a duty to be out is more tangible. / Master of Arts / The acts of passing and covering are socially constructed. To research them further, I spent a significant amount of time understanding others' experiences of passing and covering. This involved finding articles where one intentionally engaged in either act, as well as their commentary on how they believe others perceived these actions. Upon gaining more understanding on these aspects of passing and covering, I also spent time researching the social constructs that make these acts as important as they are. Specifically, it seems that queer people are often thought to represent the queer community, whether they want to or not, just because of how others view us. In light of this, it seemed especially important to understand why this is the case. This is where my research on standpoint epistemology entered the argument. Lastly, the goal of this project was not simply to understand these acts but to use that understanding in an attempt to alter the way the queer community is viewed. Because of this, I also researched how societal perceptions may change, specifically in reference to queer people. Here, I was able to find that those who do not accept the queer community are often unaccepting due to their lack of familiarity with the community, rather than because there was a logical issue that. In other words, it does not seem that rational persuasion is especially helpful in changing opinions on the queer community. This seems to provide good reason to begin to analyze the way the queer community is perceived by others.

In the past few years, there has been a strong and growing interest in evaluating the expected behavior of what we call combinatorial bin packing problems. A combinatorial bin packing problem consists of a number of items of various sizes and value ratios (value per unit of size) along with a collection of bins of fixed capacity into which the items are to be packed. The packing must be done in such a way that the sum of the sizes of the items into a given bin does not exceed the capacity of that bin. Moreover, an item must either be packed into a bin in its entirety or not at all: this "all or nothing" requirement is why these problems are characterized as being combinatorial. The objective of the packing is to optimize a given criterion Junction. Here optimize means either maximize or minimize, depending on the problem. We study two problems that fit into this framework: the Knapsack Problem and the Minimum Sum of Squares Problem. Both of these problems are known to be in the class of NP-hard problems and there is ample reason to suspect that these problems do not admit of efficient exact solution. We obtain results concerning the performance of heuristics under the assumption that the inputs are random samples from some distribution. For the Knapsack Problem, we develop four heuristics, two of which are on-line and two off-line. All four heuristics are shown to be asymptotically optimal in expectation when the item sizes and value ratios are assumed to be independent and uniform. One heuristic is shown to be asymptotically optimal in expectation when the item sizes are uniformly distributed and the value ratios are exponentially distributed. The amount of time required by these heuristics is no more than proportional to the amount of time required to sort the items in order of nonincreasing value ratios. For the Minimum Sum of Squares Problem, we develop two heuristics, both of which are off-line. Both of these heuristics are shown to be asymptotically optimal in expectation when the sizes of the items input are assumed uniformly distributed.

Clustering problems often arise in the fields like data mining, machine learning and computational biology to group a collection of objects into similar groups with respect to a similarity measure. For example, clustering can be used to group genes with related expression patterns. Covering problems are another important class of problems, where the task is to select a subset of objects from a larger set, such that the objects in the subset "cover" (or contain) a given set of elements. Covering problems have found applications in various fields including wireless and sensor networks, VLSI, and image processing. For example, covering can be used to find placement locations of the minimum number of mobile towers to serve all the customers of a region. In this dissertation, we consider an interesting collection of geometric clustering and covering problems, which are modeled as optimization problems. These problems are known to be $\mathsf{NP}$-hard, i.e. no efficient algorithms are expected to be found for these problems that return optimal solutions. Thus, we focus our effort in designing efficient approximation algorithms for these problems that yield near-optimal solutions. In this work, we study three clustering problems: $k$-means, $k$-clustering and Non-Uniform-$k$-center and one covering problem: Metric Capacitated Covering. $k$-means is one of the most studied clustering problems and probably the most frequently used clustering problem in practical applications. In this problem, we are given a set of points in an Euclidean space and we want to choose $k$ center points from the same Euclidean space. Each input point is assigned to its nearest chosen center, and points assigned to a center form a cluster. The cost per input point is the square of its distance from its nearest center. The total cost is the sum of the costs of the points. The goal is to choose $k$ center points so that the total cost is minimized. We give a local search based algorithm for this problem that always returns a solution of cost within $(1+\eps)$-factor of the optimal cost for any $\eps > 0$. However, our algorithm uses $(1+\eps)k$ center points. The best known approximation before our work was about 9 that uses exactly $k$ centers. The result appears in Chapter \ref{sec:kmeanschap}. $k$-clustering is another popular clustering problem studied mainly by the theory community. In this problem, each cluster is represented by a ball in the input metric space. We would like to choose $k$ balls whose union contains all the input points. The cost of each ball is its radius to the power $\alpha$ for some given paramater $\alpha \ge 1$. The total cost is the sum of the costs of the chosen $k$ balls. The goal is to find $k$ balls such that the total cost is minimized. We give a probabilistic metric partitioning based algorithm for this problem that always returns a solution of cost within $(1+\eps)$-factor of the optimal cost for any $\eps > 0$. However, our algorithm uses $(1+\eps)k$ balls, and the running time is quasi-polynomial. The best known approximation in polynomial time is $c^{\alpha}$ that uses exactly $k$ balls, where $c$ is a constant. The result appears in Chapter \ref{sec:kcluster}. Non-Uniform-$k$-center is another clustering problem, which was posed very recently. Like in $k$-clustering here also each cluster is represented by a ball. Additionally, we are given $k$ integers $r_1,\ldots,r_k$, and we want to find the minimum dilation $\alpha$ and choose $k$ balls with radius $\alpha\cdot r_i$ for $1\le i\le k$ whose union contains all the input points. This problem is known to be notoriously hard. No approximation is known even in the special case when $r_i$'s belong to a set of three integers. We give an LP rounding based algorithm for this special case that always returns a solution of cost within a constant factor of the optimal cost. However, our algorithm uses $(2+\eps)k$ balls for some constant $\epsilon$. We also show that this special case can be solved in polynomial time under a practical assumption. Moreover, we prove that the Euclidean version of the problem is also as hard as the general version. These results appear in Chapter \ref{sec:nukc}. Capacitated Covering is a generalization of the classical set cover problem. In the Metric Capacitated Covering problem, we are given a set of balls and a set of points in a metric space. Additionally, we are given an integer that is referred to as the capacity. The goal is to find a minimum subset of the input set of balls, such that each point can be assigned to the chosen balls in a manner so that the number of points assigned to each ball is bounded by the capacity. We give an LP rounding based algorithm for this problem that always returns a solution of cost within a constant factor of the optimal cost. However, we assume that we are allowed to expand the balls by a fairly small constant. If no expansion is allowed, then the problem is known to not admit any constant approximation. We discuss our findings in Chapter \ref{sec:capa}. As mentioned above, for many of the problems we consider, we obtain results that improve the best known approximation bounds. Our findings make significant progress towards better understanding the internals of these problems, which have impact across the disciplines. Also, during the course of our work, we have designed tools and techniques, which might be of independent interest for solving similar optimization problems. Finally, in Chapter \ref{sec:conclude}, we conclude our discussion and pose some open questions, which we consider as our potential future work.

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the Higman-Sims and Held groups. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

Nežinomo ploto dengimas yra aktuali ir paplitusi problema. NPD sprendimas realiuose robotuose susiduria su daviklių ir mechanizmų netikslumu. Atliktame darbe yra pateiktas „Bouncing“ NPD algoritmo sprendimas robotui, turinčiam mažo tikslumo daviklius ir neprecizinius valdiklius. Taip pat atliktas darbas parodė sudėtingus roboto kūrimo aspektus ir galimus sprendimus. Sukurtas robotas dėl pigumo ir nesudėtingos realizacijos gali būti naudojamas kaip platforma kitokių algoritmų tyrimui. / The problem of unknown area coverage with mobile robots has received considerable attention over the past years. This problem is a common challenge in many applications, including automatic lawn mowing and vacuum cleaning. However, most of the approaches find difficult to implement in real life because of problems of environment data reading. In this paper we consider the problem of robust area covering algorithm implementation in mobile robot. The chosen approach is based on simple and robust algorithm for uncertain environment and simple robot platform. The results showed robustness, reliability of chosen method of control. The constructed robot has shown simplicity, cheapness of creation and possibility for different algorithm testing. The significance of this paper lies in the practical solution for robust mobile robot area coverage, suitable for noisy environment and low precisions robot sensors.

Covering arrays with row limit, CARLs, are a new family of combinatorial objects which we introduce as a generalization of group divisible designs and covering arrays. In the same manner as their predecessors, CARLs have a natural application as combinatorial models for interaction test suites. A CARL(N;t,k,v:w), is an N×k array with some empty cells. A component, which is represented by a column, takes values from a v-set called the alphabet. In each row, there are exactly w non-empty cells, that is the corresponding components have an assigned value from the alphabet. The parameter w is called the row limit. Moreover, any N×t subarray contains every of v^t distinct t-tuples of alphabet symbols at least once. This thesis is concerned with the bounds on the size and with the construction of CARLs when the row limit w(k) is a positive integer valued function of the number of columns, k. Here we give a lower bound, and probabilistic and algorithmic upper bounds for any CARL. Further, we find improvements on the upper bounds when w(k)ln(w(k)) = o(k) and when w(k) is a constant function. We also determine the asymptotic size of CARLs when w(k) = Θ(k) and when w(k) is constant. Next, we study constructions of CARLs. We provide two combinatorial constructions of CARLs, which we apply to construct families of CARLs with w(k)=ck, where c<1. Also, we construct optimal CARLs when t=2 and w=4, and prove that there exists a constant δ, such that for any v and k≥4, an optimal CARL(2,k,v:4) differs from the lower bound by at most δ rows, with some possible exceptions. Finally, we define a packing array with row limit, PARL(N;t,k,v:w), in the same way as a CARL(N;t,k,v:w) with the difference that any t-tuple is contained at most once in any N×t subarray. We find that when w(k) is a constant function, the results on the asymptotic size of CARLs imply the results on the asymptotic size of PARLs. Also, when t=2, we consider a transformation of optimal CARLs with row limit w=3 to optimal PARLs with w=3.

Covering arrays with row limit, CARLs, are a new family of combinatorial objects which we introduce as a generalization of group divisible designs and covering arrays. In the same manner as their predecessors, CARLs have a natural application as combinatorial models for interaction test suites. A CARL(N;t,k,v:w), is an N×k array with some empty cells. A component, which is represented by a column, takes values from a v-set called the alphabet. In each row, there are exactly w non-empty cells, that is the corresponding components have an assigned value from the alphabet. The parameter w is called the row limit. Moreover, any N×t subarray contains every of v^t distinct t-tuples of alphabet symbols at least once. This thesis is concerned with the bounds on the size and with the construction of CARLs when the row limit w(k) is a positive integer valued function of the number of columns, k. Here we give a lower bound, and probabilistic and algorithmic upper bounds for any CARL. Further, we find improvements on the upper bounds when w(k)ln(w(k)) = o(k) and when w(k) is a constant function. We also determine the asymptotic size of CARLs when w(k) = Θ(k) and when w(k) is constant. Next, we study constructions of CARLs. We provide two combinatorial constructions of CARLs, which we apply to construct families of CARLs with w(k)=ck, where c<1. Also, we construct optimal CARLs when t=2 and w=4, and prove that there exists a constant δ, such that for any v and k≥4, an optimal CARL(2,k,v:4) differs from the lower bound by at most δ rows, with some possible exceptions. Finally, we define a packing array with row limit, PARL(N;t,k,v:w), in the same way as a CARL(N;t,k,v:w) with the difference that any t-tuple is contained at most once in any N×t subarray. We find that when w(k) is a constant function, the results on the asymptotic size of CARLs imply the results on the asymptotic size of PARLs. Also, when t=2, we consider a transformation of optimal CARLs with row limit w=3 to optimal PARLs with w=3.