In this dissertation, we consider a generalization of the historically significant problem posed in 1850 by Reverend Thomas Kirkman which asked whether it was possible for 15 schoolgirls to walk in lines of three to school for seven days so that no two of them appear in the same line on multiple days. This puzzle spawned the study of what we now call resolvable pairwise balanced designs, which balance pair coverage of points within blocks while also demanding that the blocks can be grouped in such a way that each group partitions the point-set. Our generalization aims to relax this condition slightly, so that each group of blocks balances point-wise coverage but each point occurs in each group σ times (instead of just once). We call these objects thickly-resolvable designs. Here we show that the necessary divisibility conditions for the existence of thickly-resolvable designs are also sufficient when the size of the point set is large enough. A few variations of this problem are considered as well. / Graduate

O principal objetivo deste trabalho é apresentar um estudo sistemático da teoria dos espaços topológicos resolúveis e irresolúveis. Enfocaremos diversas propriedades inerentes aos mesmos, incluindo uma exposição meticulosa de técnicas utilizadas na construção de espaços topológicos irresolúveis e sem pontos isolados. Dado um cardinal \\kappa > 1, exibiremos exemplos de espaços topológicos que são \\kappa-resolúveis, mas que não são \\kappa^{+}-resolúveis. Mostraremos, ainda, que se um espaço topológico for n-resolúvel, para todo número natural n, o mesmo será \\omega-resolúvel. Provaremos, contudo, que se \\lambda é um cardinal tal que \\omega < cf(\\lambda) = \\lambda, existe um espaço topológico que é \\mu-resolúvel, para todo cardinal \\mu < \\lambda, mas que não é \\lambda-resolúvel. O cerne desta dissertação refere-se à construção, em ZFC, de um subespaço enumerável, denso e submaximal de 2^c. / The main purpose of this work is to study the theory of resolvable and irresolvable topological spaces. We shall introduce many properties of these spaces and we shall give special attention to some techniques used in the construction of irresolvable topological spaces without isolated points. Given a cardinal \\kappa > 1, we will present some examples of topological spaces which are \\kappa-resolvable, but not \\kappa^{+}-resolvable. Besides, we will show that if a topological space is n-resolvable, for every natural number n > 1, then it is \\omega-resolvable too. Nevertheless, we shall prove that if \\lambda is a cardinal with \\omega < cf(\\lambda) = \\lambda, there is a topological space which is \\mu-resolvable, for each cardinal \\mu < \\lambda, but that is not \\lambda-resolvable. The backbone of this dissertation is the construction, in ZFC, of a countable, dense and submaximal subspaces of 2^c.

O principal objetivo deste trabalho é apresentar um estudo sistemático da teoria dos espaços topológicos resolúveis e irresolúveis. Enfocaremos diversas propriedades inerentes aos mesmos, incluindo uma exposição meticulosa de técnicas utilizadas na construção de espaços topológicos irresolúveis e sem pontos isolados. Dado um cardinal \\kappa > 1, exibiremos exemplos de espaços topológicos que são \\kappa-resolúveis, mas que não são \\kappa^{+}-resolúveis. Mostraremos, ainda, que se um espaço topológico for n-resolúvel, para todo número natural n, o mesmo será \\omega-resolúvel. Provaremos, contudo, que se \\lambda é um cardinal tal que \\omega < cf(\\lambda) = \\lambda, existe um espaço topológico que é \\mu-resolúvel, para todo cardinal \\mu < \\lambda, mas que não é \\lambda-resolúvel. O cerne desta dissertação refere-se à construção, em ZFC, de um subespaço enumerável, denso e submaximal de 2^c. / The main purpose of this work is to study the theory of resolvable and irresolvable topological spaces. We shall introduce many properties of these spaces and we shall give special attention to some techniques used in the construction of irresolvable topological spaces without isolated points. Given a cardinal \\kappa > 1, we will present some examples of topological spaces which are \\kappa-resolvable, but not \\kappa^{+}-resolvable. Besides, we will show that if a topological space is n-resolvable, for every natural number n > 1, then it is \\omega-resolvable too. Nevertheless, we shall prove that if \\lambda is a cardinal with \\omega < cf(\\lambda) = \\lambda, there is a topological space which is \\mu-resolvable, for each cardinal \\mu < \\lambda, but that is not \\lambda-resolvable. The backbone of this dissertation is the construction, in ZFC, of a countable, dense and submaximal subspaces of 2^c.

This work examines various asymptotic edge-decomposition problems on graphs. A G-group divisible design (G-GDD) of type [g_1, ..., g_u] and index lambda is a decomposition of the edges of the complete lambda-fold multipartite graph H, with groups (maximal independent sets) G_1, ..., G_n, |G_i| = g_i, into graphs (blocks) isomorphic to G. We shall also examine special types of G-GDDs (such as G-frames) and prove that, given all parameters except u, these structures exist for all asymptotically large u satisfying the necessary conditions. Our primary technique is to invoke a useful theorem of Lamken and Wilson on edge-colored graph decompositions. The basic construction for k-RGDDs shall be outlined at the end of the thesis.

This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpi&#324;ski, Vainštein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and G&#322;&#261;b.

Approved for public release, distribution is unlimited / In this study, the use of multi narrow-band filters was explored to correlate or fuse information to improve detection and recognition for specific targets of interest. To fully understand the viability and limitations of narrow band filters in thermal imaging, laboratory test procedures were first designed to characterize thermal images using common performance parameters, particularly the Noise Equivalent Temperature Difference (NETD), the Modulation Transfer Function (MTF) and the Minimum Resolvable Temperature (MRT). An available thermal imager, the Cincinnati Electronics IRRIS-256LN, and a newly purchased thermal imager, the Indigo Systems Merlin InSb Laboratory Camera, were investigated and compared. The MRT measurement showed a superior cut-off spatial frequency of 1.33 cycle/mrad for the Merlin camera, compared with 1.18 cycle/mrad for the Cincinnati, with similar sensitivity of about 0.1 ʻC for both cameras. Below cut-off the Modulation Transfer for the Merlin was approximately twice that of the Cincinnati. NETD measurements for both systems were inhibited by excess noise in the measurement system. The Merlin camera was consequently employed for further research on narrow band images. Three commercially available narrow band filters, 4110/4720 nm, 4540/4720 nm and 3750/4020 nm, were selected to isolate the "red" and "blue" spike signatures of plume emission and aerodynamic heating found in the signature of fast moving air targets. Similar sensor characterization was carried out with these filters, however, quantitative performance parameter measurements were hindered by excessive measurement system noise. As a preliminary study in multi-spectral image analysis, spectral features of hydrocarbon combustion were extracted from correlated narrow-band images of a laboratory propane flame. Digital analysis using software statistical tools showed potential for discrimination of hydrocarbon emission from other thermal sources by correlating pixel number and pixel radiant intensity. / Captain, Singapore Armed Forces