Accelerated advancements in technology, the Internet of Things, and cloud computing have spurred an emergence of unstructured data that is contributing to rapid growth in data volumes. No human can manage to keep up with monitoring and analyzing these unbounded data streams and thus predictive and analytic tools are needed. By leveraging machine learning this data can be converted into insights which are enabling datadriven decisions that can drastically accelerate innovation, improve user experience, and drive operational efficiency. The purpose of this thesis is to design and implement a system for real-time outlier detection using unbounded data streams and machine learning. Traditionally, this is accomplished by using alarm-thresholds on important system metrics. Yet, a static threshold cannot account for changes in trends and seasonality, changes in the system, or an increased system load. Thus, the intention is to leverage machine learning to instead look for deviations in the behavior of the data not caused by natural changes but by malfunctions. The use-case driving the thesis forward is real-time outlier detection in a Content Delivery Network (CDN). The input data includes Http-error messages received by clients, and contextual information like region, cache domains, and error codes, to provide tailormade predictions accounting for the trends in the data. The outlier detection system consists of a data collection pipeline leveraging the technique of stream processing, a MiniBatchKMeans clustering model that provides online clustering of incoming data according to their similar characteristics, and an LSTM AutoEncoder that accounts for temporal nature of the data and detects outlier data points in the clusters. An important finding is that an outlier is defined as an abnormal amount of outlier data points all originating from the same cluster, not a single outlier data point. Thus, the alerting system will be implementing an outlier percentage threshold. The experimental results show that an outlier is detected within one minute from a cache break-down. This triggers an alert to the system owners, containing graphs of the clustered data to narrow down the search area of the cause to enable preventive action towards the prominent incident. Further results show that within 2 minutes from fixing the cause the system will provide feedback that the actions taken were successful. Considering the real-time requirements of the CDN environment, it is concluded that the short delay for detection is indeed real-time. Proving that machine learning is indeed able to detect outliers in unbounded data streams in a real-time manner. Further analysis shows that the system is more accurate during peakhours when more data is in circulation than during none peak-hours, despite the temporal LSTM layers. Presumably, an effect from the model needing to train on more data to better account for seasonality and trends. Future work necessary to put the outlier detection system in production thus includes more training to improve accuracy and correctness. Furthermore, one could consider implementing necessary functionality for a production environment and possibly adding enhancing features that can automatically avert incidents detected and handle the causes of them.

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在這篇論文裡，我們將要討論一類古典的問題，這類問題已經經由許多方法解決，例如：遞迴關係式、差分方程式、尤拉公式等等。接著我們歸納低維度的特性，並藉由定義出一組方程式-標準n維空間分割系統-來推廣這些特性到一般的$n$維度空間中。然後我們利用演算法來提供一個更直接的組合論證法。最後，我們會把問題再細分成有界區域與無界區域的個數。 / In this article, we will discuss a class of classical questions had been solved by Recurrence Relation, Difference Equation, and Euler's Formula, etc.. And then, we construct a system of equations -Standard Partition System of n-Dimensional Space- to generalize the properties of maximizing the number of regions made up by k partitioner in an n-dimensional space and look into the construction of each dimension. Also, we provide a more directly Combinatorial Argument by Algorithm for this kind of question. At last, we focus on the number of bounded regions and unbounded regions in sense of maximizing the number of regions.

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.

As cadeias estocásticas de memória ilimitada são uma generalização natural das cadeias de Markov, no caso em que as probabilidades de transição podem depender de todo o passado da cadeia. Estas cadeias, introduzidas, independentemente, por Onicescu e Mihoc em 1935 e Doeblin e Fortet em 1937, vêm recebendo uma atenção crescente na literatura probabilística, não só por serem uma classe mais rica que a classe das cadeias de Markov, como por suas capacidades práticas de modelagem de dados científicos em diversas áreas, indo da biologia à linguística. Neste trabalho, as utilizamos para modelar a interação entre sequências de disparos neuronais. Nosso objetivo principal é desenvolver novos resultados matemáticos acerca das cadeias de memória ilimitada. Inicialmente, estudamos as condições que garantem a existência e a unicidade de cadeias estacionárias compatíveis com uma família de probabilidades de transição descontínua. Em seguida, tratamos do entendimento da fenomenologia dos trens de disparos neuronais e usamos da informação dirigida para modelar a informação que flui de uma sequência de disparos a outra. Nesta ocasião, fixamos limites da concentração para estimação da informação dirigida. / Stochastic chains with unbounded memory are a natural generalization of Markov chains, in the sense that the transition probabilities may depend on the whole past. These process, introduced independently by Onicescu and Mihoc in 1935 and Doeblin and Fortet in 1937, have been receiving increasing attention in the probabilistic literature, because they form a class richer than the Markov chains and have practical capabilities modelling of scientific data in several areas, from biology to linguistics. In this work, we use them to model interactions between spike trains. Our main goal is to develop new mathematical results about stochastic chains with unbounded memory. First, we study conditions that guarantee the existence and uniqueness of stationary chains compatible with a discontinuous family of transition probabilities. Then, we address the understanding of the phenomenology of spike trains and we propose to use directed information to quantify the information flow from one neuron to another. In this occasion, we fix concentration bounds for directed information estimation.

We study the relationship between subdivision rules, 3-dimensional manifolds, and circle packings. We find explicit subdivision rules for closed right-angled hyperbolic manifolds, a large family of hyperbolic manifolds with boundary, and all 3-manifolds of the E^3,H^2 x R, S^2 x R, SL_2(R), and S^3 geometries (up to finite covers). We define subdivision rules in all dimensions and find explicit subdivision rules for the n-dimensional torus as an example in each dimension. We define a graph and space at infinity for all subdivision rules, and use that to show that all subdivision rules for non-hyperbolic manifolds have mesh not going to 0. We provide an alternate proof of the Combinatorial Riemann Mapping Theorem using circle packings (although this has been done before). We provide a new definition of conformal for subdivision rules of unbounded valence, show that the subdivision rules for the Borromean rings complement are conformal and show that barycentric subdivision is almost conformal. Finally, we show that subdivision rules can be degenerate on a dense set, while still having convergent circle packings.

Unbounded domains often appear in engineering applications, such as acoustic or elastic wave radiation from a body immersed in an infinite medium. To simulate the unboundedness of the domain special boundary conditions have to be imposed: the Sommerfeld radiation condition. In the present work we focused on steady-state wave propagation. The objective of this research is to obtain accurate prediction of phenomena occurring in exterior acoustics and elastodynamics and ensure the quality of the solutions even for high wavenumbers. To achieve this aim, we develop higher-order domain-based schemes: Spectral Element Method (SEM) coupled to Dirichlet-to-Neumann (DtN ), Perfectly Matched Layer (PML) and Infinite Element (IEM) methods. Spectral elements combine the rapid convergence rates of spectral methods with the geometric flexibility of the classical finite element methods. The interpolation is based on Chebyshev and Legendre polynomials. This work presents an implementation of these techniques and their validation exploiting some benchmark problems. A detailed comparison between the DtN, PML and IEM is made in terms of accuracy and convergence, conditioning and computational cost.

We study asymptotic properties of evolution partial differential equations posed in unbounded spatial domain in the context of locally uniform spaces. This context allows the use of non-integrable data and carries an inherent non-compactness and non-separability. We establish the existence of a lo- cally compact attractor for non-local parabolic equation and weakly damped semilinear wave equation and provide an upper bound on the Kolmogorov's ε-entropy of these attractors and the attractor of strongly damped wave equation in the subcritical case using the method of trajectories. Finally we also investigate infinite dimensional exponential attractors of nonlinear reaction-diffusion equation in its natural energy setting. 1

We consider fast computation methods for simulation of dynamics of a collection of particles dispersed in an unbounded Stokesian suspension. Stokesian suspensions are of great practical interest in the manufacturing and processing of various commercial products. The most popular dynamic simulation method for these kind of suspensions was developed by Brady and Bossis (Brady and Bossis [1988]). This method uses a truncated multipole expansion to represent the fluid traction on particle surfaces. The hydrodynamic interactions in Stoke-sian suspension are long ranged in nature, resulting in strong coupled motion of all particles. For an N particle system, this method imposes an O(N3) computational cost, thus posing limitations to the number of particles that may be simulated. More recent methods (Sierou and Brady [2001], Scintilla, Darve and Shaqfeh [2005]) have attempted to solve this problem using Particle Mesh Ewald summation techniques by distributing the moments on a grid and using Fast Fourier Transform algorithms, resulting in an O(N log N) computational cost. We review these methods and propose a version that we believe is some-what superior. In the course of this study, we have identified and corrected errors in previous studies that maybe of some importance in determining the bulk properties of suspensions. Finally, we show the utility of our method in determining certain properties of suspensions and compare them to existing analytical results for the same.

Made available in DSpace on 2015-05-15T11:46:19Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 903709 bytes, checksum: 4a8dba984b00ee5480eecf90097b2745 (MD5) Previous issue date: 2014-06-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study the Navier-Stokes flow in R2 8> >>>>>><> >>>>>>: @u @t − ⌫!u + (u ·r)u + rp = f em ⌦ ⇥ [0,+1) , divu = r· u = 0 em ⌦⇥ [0,+1) , u = 0 sobre @⌦ ⇥ [0,+1) , u(·, 0) = u0 em ⌦, in an unbounded domain such that the Poincar´e s inequality is holds, i.e., there is a constant #1 > 0 such that we have the following inequality Z⌦ $2dx  1 #1 Z⌦ |r$|2dx, for all $ 2 H1 0 (⌦). We show the existence of global attractor in the natural phases spaces for this system exploring the energy equation of the problem / Neste trabalho, estudamos o sistema de equa¸c oes de Navier-Stokes em R2 8> >>>>>><> >>>>>>: @u @t − ⌫!u + (u ·r)u + rp = f em ⌦ ⇥ [0,+1) , divu = r· u = 0 em ⌦⇥ [0,+1) , u = 0 sobre @⌦ ⇥ [0,+1) , u(·, 0) = u0 em ⌦, em dom´ınios ilimitados sob os quais vale a desigualdade de Poincar´e, isto ´e, existe uma constante #1 > 0 tal que Z⌦ $2dx  1 #1 Z⌦ |r$|2dx, para todo $ 2 H1 0 (⌦). Provamos a exist encia de atrator global no espa¸co de fases natural para este sistema explorando a equa¸c ao de energia do problema.