Random walks describe diffusion processes, where movement at every time step is restricted only to neighbouring locations. Classical random walks are constructed using the non-relativistic Laplacian evolution operator and a coin toss instruction. In quantum theory, an alternative is to use the relativistic Dirac operator. That necessarily introduces an internal degree of freedom (chirality), which may be identified with the coin. The resultant walk spreads quadratically faster than the classical one, and can be applied to a variety of graph theoretical problems. We study in detail the problem of spatial search, i.e. finding a marked site on a hypercubic lattice in d-dimensions. For d=1, the scaling behaviour of classical and quantum spatial search is the same due to the restriction on movement. On the other hand, the restriction on movement hardly matters for d ≥ 3, and scaling behaviour close to Grover’s optimal algorithm(which has no restriction on movement) can be achieved. d=2 is the borderline critical dimension, where infrared divergence in propagation leads to logarithmic slow down that can be minimised using clever chirality flips. In support of these analytic expectations, we present numerical simulation results for d=2 to d=9, using a lattice implementation of the Dirac operator inspired by staggered fermions. We optimise the parameters of the algorithm, and the simulation results demonstrate that the number of binary oracle calls required for d= 2 and d ≥ 3 spatial search problems are O(√NlogN) and O(√N) respectively. Moreover, with increasing d, the results approach the optimal behaviour of Grover’s algorithm(corresponding to mean field theory or d → ∞ limit). In particular, the d = 3 scaling behaviour is only about 25% higher than the optimal value.

ABSTRACT DETERMINATION OF STOCHASTIC MODEL PARAMETERS OF INERTIAL SENSORS &Uuml / nver, Alper PhD, Department of Electric Electronic Engineering Supervisor: Prof. Dr. M&uuml / beccel Demirekler January 2013, 82 pages Gyro and accelerometer systematic errors due to biases, scale factors, and misalignments can be compensated via an on-board Kalman filtering approach in a Navigation System. On the other hand, sensor random noise sources such as Quantization Noise (QN), Angular Random Walk (ARW), Flicker Noise (FN), and Rate Random Walk (RRW) are not easily estimated by an on-board filter, due to their random characteristics. In this thesis a new method based on the variance of difference sequences is proposed to compute the powers of the above mentioned noise sources. The method is capable of online or offline estimation of stochastic model parameters of the inertial sensors. Our aim in this study is the estimation of ARW, FN and RRW parameters besides the quantization and the Gauss-Markov noise parameters of the inertial sensors. The proposed method is tested both on the simulated and the real sensor data and the results are compared with the Allan variance method. Comparison shows very satisfactory results for the performance of the method. Computational load of the new method is less than the computational load of the Allan variance on the order of tens. One of the usages of this method is the individual noise characterization. A noise, whose power spectral density has a constant slope, can be identified accurately by the proposed method. In addition to this, the parameters of the GM noise can also be determined. Another idea developed here is to approximate the overall error source as a combination of ARW and some number of GM sources only. The reasons of selecting such a structure is the feasibility of using these models in a Kalman filter framework for error propagation as well as their generality of modeling other noise sources.

Background: The value of stock recommendations have been debated for a century andthe debate has escalated since Alfred Cowles (1933) published his research in “Can StockMarket Forecasters Forecast”? As of late, savings in stocks has increased and the householdsare managing their savings more actively. The consequence of the increased interestin stocks has resulted in a growing market for stock recommendations. Not just financialmedia but daily newspapers have embraced this new found interest, hence stock recom-mendations can be found in almost all large newspapers in Sweden. Furthermore, this phe-nomenon has also lead to intensified research within stock recommendations. Researchers have under the 20th and 21st century investigated a wide number of issues concerning stock recommendations. The most common issue with different angles has been whether stockrecommendations generate abnormal returns or not? Earlier research on this issue havehowever mostly been based on the American stock market. According to us, not enoughnational research has been done on whether stock recommendations generate abnormal returnsor not on the Swedish stock market. Further we think that there is a lack of research with regards to a short time span. Purpose: The purpose of the study is to investigate whether stock recommendations on Nasdaq OMXSLarge-Mid-Small CAP, First North and Aktietorget generate abnormal returns? Central for the study is to investigate if stock recommendation generates abnormal returns in the short-term. Secondary the study will also investigate stock recommendations effect on stock prices in the long run. Method: First step was to create databases from high frequency data, 10 minute OHLCVbars, and from closing prices. Thereafter use these as a base for statistical calculations. Conclusion: The results from this study show that analysts have succeeded with generat-ing significant abnormal returns with their recommendations. Above all it is the business journals buy recommendations that generate the highest abnormal return. Sell recommen-dations didn’t give high significant results in our measurements. Our results also show thatabnormal returns from buy recommendations are sufficiently high and the transaction costsare sufficiently low so that investors can capitalize on the analyses. We found that buy recommendations effect companies noted on Small Cap etc to a higher extent than companies noted on Large- and Mid-Cap because of the lower degree of transparency in the smaller companies. Our results also indicate that the weekly magazines generate approximatelyequal abnormal returns. The monthly generates the lowest abnormal returns in our study. In our research we also found that buy recommendations that are published when thestock market has a positive momentum perform better than when the stock market has anegative momentum. We also found that investor should closely monitor the information flow before the recommendation is published, since a positive information flow can have afavorable outcome on the size of the abnormal return. Last but not least, we observe a correlationbetween the numbers of investors the recommendation reaches and the magnitudeof the observed abnormal return. / Bakgrund: Värdet av aktierekommendationer har debatterats i snart ett sekel, och debat-ten har eskalerat ända sedan Alfred Cowles (1933) publicerade sin forskning ”Can StockMarket Forecasters Forecast?" De senaste åren har sparandet i aktier ökat och hushållenblir allt aktivare i sitt pensionssparande. Konsekvensen av det ökade intresset för sparande iaktier har resulterat i en växande marknad för aktierekommendationer. Inte bara finansiellmedia utan också dagliga tidskrifter har tagit till sig detta ökade intresse och nästan samtligastörre dagstidningar publicerar numera aktierekommendationer. Vidare har detta fenomen också lett till att forskningen kring aktierekommendationer intensifierats de senaste decennierna.Forskare har under 1900-talet och 2000-talet undersökt en mängd olika problem-ställningar kring aktierekommendationer. Varav den vanligaste problemställningen med varierandevinklar, har varit huruvida aktierekommendationer genererar överavkastning ellerej? Tidigare forskning har dock framförallt baserats på den amerikanska aktiemarknaden. I vår mening råder det brist på nationell forskning som visar huruvida aktierekommendatio-ner genererar överavkastning på bolag noterade på svenska marknadsplatser. Vidare råderdet brist på forskning som tar hänsyn till det korta tidsperspektivet. Syfte: Syftet med studien är att undersöka huruvida aktierekommendationer på Nasdaq OMXS Large-Mid-Small CAP och First North samt Aktietorget genererar överavkastning? Centralt för studien är attundersöka om aktierekommendationer är kursdrivande på kort sikt, Sekundärt kommer också studienundersöka aktierekommendationers inverkan på aktiekurser på lång sikt. Metod: Första steget var att konstruera databaser från högfrekvensdata, 10M OHLCVbars, och stängningskurser. Därefter använda dessa som grund för statistiska beräkningar. Slutsats: Resultat från vår studie visar att ekonomijournalister har lyckats med att genererasignifikant överavkastning med deras rekommendationer. Framför allt är det affärstidskrifternasköprekommendationer som genererar högst överavkastning. Säljrekommendationergav ett lägre signifikant utslag i våra mätningar. Våra resultat indikerar också på att överavkastningen från köprekommendationer är tillräckligt hög och transaktionskostnaderna tillräckligtlåga för att investerare skall kunna kapitalisera på rekommendationerna. Vi finneratt köprekommendationer är mer kursdrivande på Small Cap etc. än bolag noterade på Lar-ge- och Mid Cap på grund av den lägre graden av transparens i de mindre bolagen. Våra resultatpekar också på att Affärsvärlden, Börsveckan & Veckans affärer genererar ungefärlika stor överavkastning. Aktiespararen & Privata Affärer genererar den lägsta överavkastningeni våra mätningar. I våra efterforskningar ser vi också en stark tendens för att aktiersom rekommenderas när börsen har ett positivt momentum presterar bättre än aktier somrekommenderas när börsen har ett negativt momentum. Vi finner också att investerare börbeakta informationsflödet innan rekommendationstidpunkten då ett positivt informations-flöde kan får en gynnsam effekt på överavkastningens storlek. Sist men inte minst observe-rar vi en korrelation mellan antalet investerare rekommendationen når och magnituden påden observerade överavkastningen.

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Single-particle tracking (SPT) is a method used to study the diffusion of various molecules within the cell. SPT involves tagging proteins with optical labels and observing their individual two-dimensional trajectories with a microscope. The analysis of this data provides important information about protein movement and mechanism, and is used to create multistate biological models. One of the challenges in SPT analysis is the variety of complex environments that contribute to heterogeneity within movement paths. In this thesis, we explore the limitations of current methods used to analyze molecular movement, and adapt analytical methods used in animal movement analysis, such as correlated random walks and first-passage time variance, to SPT data of leukocyte function-associated antigen-1 (LFA-1) integral membrane proteins. We discuss the consequences of these methods in understanding different types of heterogeneity in protein movement behaviour, and provide support to results from current experimental work. / Applied Mathematics

A estimação da estrutura da taxa de juros é relevante por duas razões fundamentais: em primeiro lugar é considerado como um indicador antecipado de política, sendo uma das principais ferramentas para os bancos centrais como instrumento de política monetária; em segundo lugar, através da curva de rendimentos é possível fazer valoração de ativos financeiros. A causa da sua relevância, tanto na área macroeconômica e como no campo financeiro, uma ampla literatura dedicada a estimá-la se desenvolveu. Neste sentido, o objetivo deste documento é a previsão da curva de rendimentos da Colômbia através da metodologia de Spectrum Singular Analysis (SSA) durante o período 2006-2014. Para a previsão são usados parâmetros diários estimados pelo modelo de fatores de Nelson e Siegel (1987). Os resultados indicam ganhos na acurácia preditiva fora da amostra da abordagem de MSSA em relação ao modelo Random Walk e outros benchmarks amplamente usados na literatura, principalmente nos horizontes de previsão mais curtos. Os resultados são estatisticamente significantes. Assim mesmo, observasse que o MSSA se ajusta melhor que os modelos competidores em todos os horizontes para as previsões das menores maturidades. / The estimation of the Yield curve is relevant because of two fundamental reasons: firstly, it is considered an anticipated indicator of economic policies, being one of the principal central banks tools as instrument of monetary policy; secondly, through this estimation it is possible to valuate financial assets. Due to its relevance in the macroeconomics area and the financial field, an extensive literature has been dedicated to its estimation. Concerning that, the goal of this document is to get a prediction of Colombia’s yield curve through the Spectrum Singular Analysis (SSA) from 2006 to 2014. Daily estimated parameters by Nelson and Siegel (1987) factors model are used to obtain the prognostication. Results are statistically significant and indicate gains of the MMSA on the accuracy of previsions out of the sample in relation to the Random Walk competitor model and other benchmarks widely used in literature, mainly on short term previsions. Likewise, we observe that the MSSA method is better adjusted than competitors’ models in all the horizons for the previsions where maturity is lower. / La estimación de la curva de rendimientos es relevante por dos razones fundamentales: en primer lugar es considerado como un indicador anticipado de política económica, siendo una de las principales herramientas para los bancos centrales como instrumento de política monetaria; en segundo lugar, a través de esta es posible realizar valoración de activos financieros. Dada su relevancia tanto en el área macroeconómica como en el campo financiero una amplia literatura ha sido dedicada a su estimación. En este sentido, el objetivo de este documento es la previsión de la curva de rendimientos de Colombia a través de la metodología de Spectrum Singular Analysis (SSA) durante noviembre de 2006 a diciembre de 2014. Para su pronóstico son usados los parámetros diarios estimados por el modelo de factores de Nelson e Siegel (1987). Los resultados son estadísticamente significativos e indican ganancias del método MSSA en la precisión de las previsiones fuera de la muestra principalmente en horizontes de previsión más cortos en relación al Random Walk y otros benchmarks ampliamente usados en la literatura. Así mismo, se observa que el método MSSA se ajusta mejor que los modelos competidores en todos los horizontes para las previsiones donde el vencimiento es menor.

A classic problem in physics concerns normal versus anomalous diffusion. Fractal analysis of random walks with memory aims at quantitatively describing the complex phenomenology observed in economic, ecological, biological and physical systems. Markov processes exhaustively account for random walks with short-range memory. In contrast, long-range memory typically gives rise to non-Markovian walks. The most extreme case of a non-Markovian random walk corresponds to a stochastic process with dependence on the entire history of the system. We study a recently proposed non-Markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence. We report numerical and analytical results showing the complete phase diagram, consisting of 4 phases, for this system: (i) classical nonpersistence, (ii) classical persistence (iii) log-periodic nonpersistence and (iv) log-periodic persistence driven by negative feedback. The first two phases possess continuous scale invariance symmetry, however log-periodicity breaks this symmetry. Instead, log-periodic motion satisfies discrete scale invariance symmetry, with complex rather than real fractal dimensions. We find for log-periodic persistence evidence not only of statistical but also of geometric self-similarity. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Um clássico problema em física consiste em difusão normal versus anômala. Análise fractal de caminhadas aleatórias com memória, sugere descrever quantitativamente uma fenomenologia complexa observada em economia, ecologia, biologia, e física. Processos Markovianos estão representados em caminhadas aleatórias com memória de curto alcance. Em contraste, memória de longo alcance surge tipicamente em caminhadas não-Markovianas. O caso mais extremo de uma caminhada não-Markoviana corresponde a um processo estocástico com dependência em sua história completa. Estudamos uma proposta recente de caminhada não-Markoviana caracterizada por perda de memória do passado recente e persistência induzida amnesicamente. Apresento resultados analíticos mostrando um diagrama de fase completo, consistindo de 4 fases. (i) não-persistente clássico, (ii) persistente clássico controlado por feedback positivo, (iii) não-persistente log-periódico e (iv) persistente log-periódico controlado por feedback negativo. As primeiras duas fases apresentam invariância de escala em simetria contínua. Em compensação, movimento log-periódico apresenta invariância de escala em simetria discreta, com dimensão complexa maior do que a dimensão fractal real. É mostrado evidências de persistência log-periódica não somente estatísticas, mas devido também a auto-similaridade geométrica. Obtivemos os resultados numéricos e analíticos para seis expoentes críticos, que juntos caracterizam completamente as propriedades das transições.

A estimação da estrutura da taxa de juros é relevante por duas razões fundamentais: em primeiro lugar é considerado como um indicador antecipado de política, sendo uma das principais ferramentas para os bancos centrais como instrumento de política monetária; em segundo lugar, através da curva de rendimentos é possível fazer valoração de ativos financeiros. A causa da sua relevância, tanto na área macroeconômica e como no campo financeiro, uma ampla literatura dedicada a estimá-la se desenvolveu. Neste sentido, o objetivo deste documento é a previsão da curva de rendimentos da Colômbia através da metodologia de Spectrum Singular Analysis (SSA) durante o período 2006-2014. Para a previsão são usados parâmetros diários estimados pelo modelo de fatores de Nelson e Siegel (1987). Os resultados indicam ganhos na acurácia preditiva fora da amostra da abordagem de MSSA em relação ao modelo Random Walk e outros benchmarks amplamente usados na literatura, principalmente nos horizontes de previsão mais curtos. Os resultados são estatisticamente significantes. Assim mesmo, observasse que o MSSA se ajusta melhor que os modelos competidores em todos os horizontes para as previsões das menores maturidades. / The estimation of the Yield curve is relevant because of two fundamental reasons: firstly, it is considered an anticipated indicator of economic policies, being one of the principal central banks tools as instrument of monetary policy; secondly, through this estimation it is possible to valuate financial assets. Due to its relevance in the macroeconomics area and the financial field, an extensive literature has been dedicated to its estimation. Concerning that, the goal of this document is to get a prediction of Colombia’s yield curve through the Spectrum Singular Analysis (SSA) from 2006 to 2014. Daily estimated parameters by Nelson and Siegel (1987) factors model are used to obtain the prognostication. Results are statistically significant and indicate gains of the MMSA on the accuracy of previsions out of the sample in relation to the Random Walk competitor model and other benchmarks widely used in literature, mainly on short term previsions. Likewise, we observe that the MSSA method is better adjusted than competitors’ models in all the horizons for the previsions where maturity is lower. / La estimación de la curva de rendimientos es relevante por dos razones fundamentales: en primer lugar es considerado como un indicador anticipado de política económica, siendo una de las principales herramientas para los bancos centrales como instrumento de política monetaria; en segundo lugar, a través de esta es posible realizar valoración de activos financieros. Dada su relevancia tanto en el área macroeconómica como en el campo financiero una amplia literatura ha sido dedicada a su estimación. En este sentido, el objetivo de este documento es la previsión de la curva de rendimientos de Colombia a través de la metodología de Spectrum Singular Analysis (SSA) durante noviembre de 2006 a diciembre de 2014. Para su pronóstico son usados los parámetros diarios estimados por el modelo de factores de Nelson e Siegel (1987). Los resultados son estadísticamente significativos e indican ganancias del método MSSA en la precisión de las previsiones fuera de la muestra principalmente en horizontes de previsión más cortos en relación al Random Walk y otros benchmarks ampliamente usados en la literatura. Así mismo, se observa que el método MSSA se ajusta mejor que los modelos competidores en todos los horizontes para las previsiones donde el vencimiento es menor.

On s'intéresse dans cette thèse au modèle de la marche aléatoire branchante, un système de particules qui évoluent au court du temps en se déplaçant et se reproduisant de façon indépendante. Le but est d'étudier le rythme auquel ces particules se déplacent, dans deux variantes particulières de marches aléatoires branchantes. Dans la première variante, la façon dont les individus se déplacent et se reproduisent dépend du temps. Ce modèle a été introduit par Fang et Zeitouni en 2010. Nous nous intéresserons à trois types de dépendance en temps : une brusque modification du mécanisme de reproduction des individus après un temps long ; une lente évolution de ce mécanisme à une échelle macroscopique ; et des fluctuations aléatoires à chaque génération. Dans la seconde variante, le mécanisme de reproduction est constant, mais les individus subissent un processus de sélection darwinien. La position d'un individu est interprétée comme son degré d'adaptation au milieu, et le déplacement d'un enfant par rapport à son parent représente l'héritage des gènes. Dans un tel processus, la taille maximale de la population est fixée à une certaine constante N, et à chaque étape, seuls les N plus à droite sont conservés. Ce modèle a été introduit par Brunet, Derrida, Mueller et Munier, et étudié par Bérard et Gouéré en 2010. Nous nous sommes intéressés dans un premier temps à une variante de ce modèle, qui autorise quelques grands sauts. Dans un second temps, nous avons considéré que la taille totale N de la population dépend du temps. / In this thesis, we take interest in the branching random walk, a particles system, in which particles move and reproduce independently. The aim is to study the rhythm at which these particles invade their environment, a quantity which often reveals information on the past of the extremal individuals. We take care of two particular variants of branching random walk, that we describe below.In the first variant, the way individuals behave evolves with time. This model has been introduced by Fang and Zeitouni in 2010. This time-dependence can be a slow evolution of the reproduction mechanism of individuals, at macroscopic scale, in which case the maximal displacement is obtained through the resolution of a convex optimization problem. A second kind of time-dependence is to sample at random, at each generation, the way individuals behave. This model has been introduced and studied in an article in collaboration with Piotr Mi\l{}os.In the second variant, individuals endure a Darwinian selection mechanism. The position of an individual is understood as its fitness, and the displacement of a child with respect to its parent is associated to the process of heredity. In such a process, the total size of the population is fixed to some integer N, and at each step, only the N fittest individuals survive. This model was introduced by Brunet, Derrida, Mueller and Munier. In a first time, we took interest in a mechanism of reproduction which authorises some large jumps. In the second model we considered, the total size N of the population may depend on time.

<p>This thesis studies non-nearest-neighbor random walks in random environments (RWRE) on the integers and on the d-dimensional integer lattic that are drawn in an i.i.d. way according to a Dirichlet distribution. We complete a characterization of recurrence and transience in a given direction for random walks in Dirichlet environments (RWDE) by proving directional recurrence in the case where the Dirichlet parameters are balanced and the annealed drift is zero. As a step toward this, we prove a 0-1 law for directional transience of i.i.d. RWRE on the 2-dimensional integer lattice with bounded jumps. Such a 0-1 law was proven by Zerner and Merkl for nearest-neighbor RWRE in 2001, and Zerner gave a simpler proof in 2007. We modify the latter argument to allow for bounded jumps. We then characterize ballisticity, or nonzero liiting velocity, of transienct RWDE on the integers. It turns out that ballisticity is controlled by two parameters, kappa0 and kappa1. The parameter kappa0, which controls finite traps, is known to characterize ballisticity for nearest-neighbor RWDE on the d-dimensional integer lattice for dimension d at least 3, where transient walks are ballistic if and only if kappa0 is greater than 1. The parameter kappa1, which controls large-scale backtracking, is known to characterize ballisticity for nearest-neighbor RWDE on the one-dimensional integer lattice, where transient walks are ballistic if and only if the absolute value of kappa1 is greater than 1. We show that in our model, transient walks are ballistic if and only if both parameters are greater than 1. Our characterization is thus a mixture of known characterizations of ballisticity for nearest-neighbor one-dimensional and higher-dimensional cases. We also prove more detailed theorems that help us better understand the phenomena affecting ballisticity.</p>