Light-matter interactions lead to a variety of interesting phenomena, for example photosynthesis which is a process fundamental to life on earth. There exists many different spectroscopic methods to measure light-matter interactions, for example UV/Vis spectroscopy, that can provide information about electronically excited states. However, numerical methods and theory are important to model and gain understanding of these experiments. Quantum chemistry provides that understanding, giving the possibility to numerically calculate molecular properties like excitation energies. The aim of this thesis was to implement a reduced-space algorithm in Dalton, to solve an eigenvalue equation obtained by response theory, for the calculation of excitation energies of molecular systems. There already was a similar algorithm in Dalton, that was able to perform these calculations. However, in a different module of Dalton used mainly for complex response theory, an algorithm to obtain eigenvalues was missing. The new implementation was similar to the existing one, except for the division of the reduced space into even and odd parts used in the complex response module. The thesis starts with a quick introduction of light-matter interactions and proceeds with a description of many-body theory, including numerical methods used in that field. In the end of the theoretical part, the eigenvalue equation, used to calculate excitation energies, is derived. In the following section, the reduced-space algorithm is described. In the end of the thesis, numerical results obtained with the algorithm are presented, including a small basis set and method study. The comparison with the existing implementation of the similar algorithm verified the successful implementation of the algorithm presented in this thesis.

The problem of computing the minimum distance between two convex hulls has applications to many areas including robotics, computer graphics and path planning. Moreover, determining the minimum distance between two convex hulls plays a significant role in support vector machines (SVM). In this study, a new algorithm for finding the minimum distance between two convex hulls is proposed and investigated. A convergence of the algorithm is proved and applicability of the algorithm to support vector machines is demostrated. The performance of the new algorithm is compared with the performance of one of the most popular algorithms, the sequential minimal optimization (SMO) method. The new algorithm is simple to understand, easy to implement, and can be more efficient than the SMO method for many SVM problems.

Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And finally we show that for a Borel subset X of real numbers the product is CDH iff X is a G-delta zero-dimensional set or an interval.

Modern software systems have been pervasively concurrent to utilize parallel hardware and perform asynchronous tasks. The correctness of concurrent programming, however, has been challenging for real-world and large systems. As the concurrent events of a system can interleave arbitrarily, unexpected interleavings may lead the system to undefined states, resulting in denials of services, performance degradation, inconsistent data, security issues, etc. To detect such concurrency errors, concurrency testing repeatedly explores the interleavings of a system to find the ones that induce errors. Traditional systematic testing, however, suffers from the intractable number of interleavings due to the complexity in real-world systems. Moreover, each iteration in systematic testing adjusts the explored interleaving with a minimal change that swaps the ordering of two events. Such exploration may waste time in large homogeneous sub-spaces leading to the same testing result. Thus on real-world systems, systematic testing often performs poorly to reveal even simple errors within a limited time budget. On the other hand, randomized testing samples interleavings of the system to quickly surface simple errors with substantial chances, but it may as well explore equivalent interleavings that do not affect the testing results. Such redundancies weaken the probabilistic guarantees and performance of randomized testing to find any errors. Towards effective concurrency testing, this thesis leverages partial order semantics with randomized testing to find errors with strong probabilistic guarantees. First, we propose partial order sampling (POS), a new randomized testing framework to sample interleavings of a concurrent program with a novel partial order method. It effectively and simultaneously explores the orderings of all events of the program, and has high probabilities to manifest any errors of unexpected interleavings. We formally proved that our approach has exponentially better probabilistic guarantees to sample any partial orders of the program than state-of-the-art approaches. Our evaluation over 32 known concurrency errors in public benchmarks shows that our framework performed 2.6 times better than state-of-the-art approaches to find the errors. Secondly, we describe Morpheus, a new practical concurrency testing tool to apply POS to high-level distributed systems in Erlang. Morpheus leverages dynamic analysis to identify and predict critical events to reorder during testing, and significantly improves the exploration effectiveness of POS. We performed a case study to apply Morpheus on four popular distributed systems in Erlang, including Mnesia, the database system in standard Erlang distribution, and RabbitMQ, the message broker service. Morpheus found 11 previously unknown errors leading to unexpected crashes, deadlocks, and inconsistent states, demonstrating the effectiveness and practicalness of our approaches.

[pt] A generalização da equação de Riccati estudada neste trabalho é z′(t) = z(t)(n) + an−1(t)z(t)(n−1) + . . . + a1(t)z(t) + a0(t). A Aplicação de Avanço leva za em zb se o problema de valor inicial, com z(a) = za, tem solução definida em [a,b] com z(b) = zb. Quando a = 0 e b = 1, a Aplicação de Avanço é conhecida como Aplicação de Poincaré. O conjunto singular é o subconjunto da esfera de Riemann contendo as singularidades da aplicação de avanço. No caso genérico, o conjunto singular é a união de curvas com um número finito de descontinuidades: correspondentes às soluções que alcançam o infinito pelo menos duas vezes. Como consequência será apresentado um método, baseado na configuração do conjunto singular, para determinar o número de soluções periódicas. Será exibida uma família de equações não autônomas cuja Aplicação de Poincaré é a Identidade num aberto do plano complexo. / [en] The generalization of the Riccati equation studied in this work is z′(t) = z(t)n + an−1(t)z(t)n−1 + . . . + a1(t)z(t) + a0(t). The Advance Map takes za at zb if the initial value problem, with z(a) = za, has a solution defined on [a, b] with z(b) = zb. When a=0 and b=1 the Advance Map is known as Poincará Map. The singular set is the subset of the Riemann sphere containing the singularities of the advance map. In generic case, the singular set is the union of curves witha number finite discontinuities: corresponding solutions that reach infinity at least twice. As a consequence will be presented a method, based on configuration set singular, to determine the number of periodic solutions. A family of non-automous equations whose Poincaré Map is the Identity in a non-empty open subset of the complex plane will be presented.

In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a countable union of Haar null sets in a Polish group is Haar null. We show the openly Haar null ideal may be distinct from the Haar null ideal, which resolves an uncertainty of Solecki. We show that compact sets are always Haar null in S? and in any countable product of locally compact non-compact groups, which extends the domain of a result of Dougherty. We show that any countable product of locally compact non-compact groups decomposes into the disjoint union of a meager set and a Haar null set, which gives a partial positive answer to a question of Darji. We display a translation property in the homeomorphism group Homeo+[0,1] which is impossible in any non-trivial locally compact group. Other related results are peppered throughout.

With the increasing availability of large amount of data in a wide range of applications, no matter for industry or academia, it becomes crucial to understand the nature of complex raw data, in order to gain more values from data engineering. Although many problems have been successfully solved by some mature machine learning techniques, the problem of learning from imbalanced data continues to be one of the challenges in the field of data engineering and machine learning, which attracted growing attention in recent years due to its complexity. In this thesis, we focus on four aspects of imbalanced data learning and propose solutions to the key problems. The first aspect is about ensemble methods for imbalanced data classification. Ensemble methods, e.g. bagging and boosting, have the advantages to cure imbalanced data by integrated with sampling methods. However, there are still problems in the integration. One problem is that undersampling and oversampling are complementary each other and the sampling ratio is crucial to the classification performance. This thesis introduces a new method HSBagging which is based on bagging with hybrid sampling. Experiments show that HSBagging outperforms other state-of-the-art bagging method on imbalanced data. Another problem is about the integration of boosting and sampling for imbalanced data classification. The classifier weights of existing AdaBoost-based methods are inconsistent with the objective of class imbalance classification. In this thesis, we propose a novel boosting optimization framework GOBoost. This framework can be applied to any boosting-based method for class imbalance classification by simply replacing the calculation of classifier weights. Experiments show that the GOBoost-based methods significantly outperform the corresponding boosting-based methods. The second aspect is about online learning for imbalanced data stream with concept drift. In the online learning scenario, if the data stream is imbalanced, it will be difficult to detect concept drifts and adapt the online learner to them. The ensemble classifier weights are hard to adjust to achieve the balance between the stability and adaptability. Besides, the classier built on samples in size-fixed chunk, which may be highly imbalanced, is unstable in the ensemble. In this thesis, we propose Adaptive Chunk-based Dynamic Weighted Majority (ACDWM) to dynamically weigh the individual classifiers according to their performance on the current data chunk. Meanwhile, the chunk size is adaptively selected by statistical hypothesis tests. Experiments on both synthetic and real datasets with concept drift show that ACDWM outperforms both of the state-of-the-art chunk-based and online methods. In addition to imbalanced data classification, the third aspect is about clustering on imbalanced data. This thesis studies the key problem of imbalanced data clustering called uniform effect within the k-means-type framework, where the clustering results tend to be balanced. Thus, this thesis introduces a new method called Self-adaptive Multi-prototype-based Competitive Learning (SMCL) for imbalanced clusters. It uses multiple subclusters to represent each cluster with an automatic adjustment of the number of subclusters. Then, the subclusters are merged into the final clusters based on a novel separation measure. Experimental results show the efficacy of SMCL for imbalanced clusters and the superiorities against its competitors. Rather than a specific algorithm for imbalanced data learning, the final aspect is about a measure of class imbalanced dataset for classification. Recent studies have shown that imbalance ratio is not the only cause of the performance loss of a classifier in imbalanced data classification. To the best of our knowledge, there is no any measurement about the extent of influence of class imbalance on the classification performance of imbalanced data. Accordingly, this thesis proposes a data measure called Bayes Imbalance Impact Index (B1³) to reflect the extent of influence purely by the factor of imbalance for the whole dataset. As a result we can therefore use B1³ to judge whether it is worth using imbalance recovery methods like sampling or cost-sensitive methods to recover the performance loss of a classifier. The experiments show that B1³ is highly consistent with improvement of F1score made by the imbalance recovery methods on both synthetic and real benchmark datasets. Two ensemble frameworks for imbalanced data classification are proposed for sampling rate selection and boosting weight optimization, respectively. 2. •A chunk-based online learning algorithm is proposed to dynamically adjust the ensemble classifiers and select the chunk size for imbalanced data stream with concept drift. 3. •A multi-prototype competitive learning algorithm is proposed for clustering on imbalanced data. 4. •A measure of imbalanced data is proposed to evaluate how the classification performance of a dataset is influenced by the factor of imbalance.

A subset S⊆V in a graph G=(V,E) is a [j,k]-set if, for every vertex vεV\-S, j≤|N(v)\∩S|≤k for non-negative integers j and k, that is, every vertex vεV\-S is adjacent to at least j but not more than k vertices in S. In this paper, we focus on small j and k, and relate the concept of [j,k]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G, the restrained domination number is equal to the domination number of G.

A subset S⊆V in a graph G=(V,E) is a [j,k]-set if, for every vertex vεV\-S, j≤|N(v)\∩S|≤k for non-negative integers j and k, that is, every vertex vεV\-S is adjacent to at least j but not more than k vertices in S. In this paper, we focus on small j and k, and relate the concept of [j,k]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G, the restrained domination number is equal to the domination number of G.

We say that a function f:V→{0,1,...,diam(G)} is a broadcast if for every vertex v∈V, f(v)≤e(v), where diam(G) denotes the diameter of G and e(v) denotes the eccentricity of v. The cost of a broadcast is the value f(V)=∑v∈Vf(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.