This thesis is an attempt at studying quantum coherence effects coupled via plasmons. After introducing the quantum coherence in atomic systems in Chapter 1, we utilize it in Chapter 2 to demonstrate a new technique of detection of motion of single atoms or irons inside an optical cavity. By taking into account the interaction of coherences with surface plasmonic waves excited in metal nanoparticles, we provide a theoretical model along with experimental data in Chapter 3 to describe the modification of Raman spectra near metal nanoparticles. We show in chapter 4 that starting from two emitters, coupled via a plasmonic field, the symmetry breaking occurs, making detectable the simultaneous existence of the fast super-radiance and the slow sub-radiance emission of dye fluorescence near a plasmonic surface. In Chapter 5, we study the photon statistics of a group of emitters coupled via plasmons and by the use of quantum regression theorem, we provide a theoretical model to fully investigate the dependence of photon bunching and anti-bunching effects to the interaction between atoms, fields and surrounding mediums.

In der Arbeit befassen wir uns mit der Potenznutzenmaximierung des Endvermögens, wenn die Aktienpreise stetigen Semimartingaldynamiken genügen und die Strategien des Agenten Investitions- und Informationsrestriktionen unterworfen sind. Hauptaugenmerk liegt auf der stochastischen Rückwärtsgleichung (BSDE) für den dynamischen Wertprozess und auf der Übertragung von neuen Ergebnissen zu quadratischen Semimartingal-BSDEs auf das Investitionsproblem. Dieses gelingt unter der Annahme endlicher exponentiellen Momente des Mean-Variance Tradeoff und verallgemeinert frühere Resultate, die Beschränktheit fordern. Wir betrachten dabei zunächst die Beziehung zwischen den Dualitäts- und BSDE-Ansätzen zur Lösung des Problems und gehen dann über zum Studium der quadratischen Semimartingal-BSDE, wenn der Marktpreis des Risikos vom BMO-Typ ist. Wir zeigen, dass es stets ein Kontinuum verschiedener BSDE-Lösungen mit quadratisch integrierbarem Martingalteil gibt. Wir stellen dann eine neue scharfe Bedingung an geeignete dynamische exponentielle Momente vor, die die Beschränktheit der BSDE-Lösungen in einer allgemeinen Filtration garantiert. In weiterer Folge weisen wir Existenz-, Eindeutigkeits-, Stabilitäts- und Maßwechselresultate für allgemeine quadratische stetige BSDEs unter exponentiellen Momenten nach. Diese Ergebnisse verwenden wir, um das Investitionsproblem für den Fall konischer Investitionsrestriktionen zu untersuchen. Ausgehend von der Zerlegung von Elementen des dualen Gebietes erhalten wir die zugehörige BSDE und beweisen, dass der Wertprozess in einem Raum liegt, in dem Lösungen quadratischer BSDEs eindeutig sind. Als Folgerung aus dem Stabilitätsresultat für BSDEs erhalten wir die Stetigkeit der Optimierer in der Semimartingaltopologie in den Parametern des Modells. Schließlich betrachten wir das Investitionsproblem unter exponentiellen Momenten, kompakten Handelsrestriktionen und eingeschränkter Information. Hierbei benutzen wir ausschließlich BSDE-Resultate. / In this thesis we consider the problem of maximizing the power utility from terminal wealth when the stocks have continuous semimartingale dynamics and there are investment and information constraints on the agent''s strategies. The main focus is on the backward stochastic differential equation (BSDE) that encodes the dynamic value process and on transferring new results on quadratic semimartingale BSDEs to the portfolio choice problem. This is accomplished under the assumption of finite exponential moments of the mean-variance tradeoff, generalizing previous results which require boundedness. We first recall the relationship between the duality and BSDE approaches to solving the problem and then study the associated quadratic semimartingale when the market price of risk is of BMO type. We show that there is always a continuum of distinct solutions to this BSDE with square-integrable martingale part. We then provide a new sharp condition on the dynamic exponential moments of the mean-variance tradeoff which guarantees the boundedness of BSDE solutions in a general filtration. In a subsequent step we establish existence, uniqueness, stability and measure change results for general quadratic continuous BSDEs under an exponential moments condition. We use these results to study the portfolio selection problem when there are conic investment constraints. Building on a decomposition result for the elements of the so-called dual domain we derive the associated BSDE and show that the value process is contained in a specific space in which BSDE solutions are unique. A consequence of the stability result for BSDEs is then the continuity of the optimizers with respect to the input parameters of the model in the semimartingale topology. Finally, we study the optimal investment problem under exponential moments, compact constraints and restricted information. This is done by referring to BSDE results only.

The Theses deals with applications of mathematical knowledge in teaching biology and it is divided into four chapters. Each chapter is dedicated to another application: 1. Application of conditional probability in medical diagnostics, 2. Application of exponential function in population ecology, 3. Application of logic functions in mathematical modelation of neuron and 4. Aplication of binomial theorem and binomial distribution in genetics. Each application contains solved problems, a worksheet for students and a solution for each worksheet. Two application (1. and 2.) have been tested in teaching and as an assessment of my lessons students filled questionnaires. Results of these questionnaires are processed in the end of these chapters. This Thesis can be used in teaching or self-studying.

現今市場競爭愈來愈激烈，迫使廠商紛紛至海外尋求產品消費市場，在跨國銷售的背景之下，需要有更多可以確定國家選擇、預測銷售及估計需求的方法。而其中可以滿足這些需求的方法之中，就是研究產品跨國擴散型態，藉以瞭解後進國家與領先國家中新產品如何擴散且會如何互相影響 (Douglas and Craig, 1992)。 在眾多的跨國產品中，本研究選擇好萊塢電影做為實證分析的對象。 經由集群分析，本研究發現（一）台灣高首週票房且口碑佳的電影，會遇到假日人潮、有很高的美國總票房、以及很高的美國首週票房；（二）美國影片在美國及台灣映演的每週票房趨勢有差異存在；（三）片商沒有做好影片在台灣映演的檔期歸劃；（四）三群電影中，在影片類型沒有明顯地區別。 經由十二個新產品銷售預測模型的建立：對數線性迴歸模式（LN-Regression Model）（不考慮新產品領先國擴散經驗）（以OLS估計）、卜瓦松迴歸模式(Poisson Regression Model) （不考慮新產品領先國擴散經驗）（以MLE估計）、負二項分配迴歸模式(Negative Binomial Distribution Regression Model) （不考慮新產品領先國擴散經驗）（以MLE估計）、Exponential Decay模式（以OLS估計）＋迴歸方程式體系（不考慮新產品領先國擴散經驗）（以SUR估計）、Exponential Decay模式（以OLS估計）＋迴歸方程式體系（考慮新產品領先國擴散經驗）（以SUR估計）、Exponential Decay模式＋層級貝氏迴歸模式（考慮新產品領先國擴散經驗）、Bass連續型擴散模式（以NLS估計）＋迴歸方程式體系（不考慮新產品領先國擴散經驗（以SUR估計）、Bass連續型擴散模式（以NLS估計）＋迴歸方程式體系（考慮新產品領先國擴散經驗（以SUR估計）、Bass離散型擴散模式（以OLS估計）＋迴歸方程式體系（不考慮新產品領先國擴散經驗）（以SUR估計）、Bass離散型擴散模式（以OLS估計）＋迴歸方程式體系（考慮新產品領先國擴散經驗）（以SUR估計）、層級貝氏BASS離散型擴散模式＋迴歸方程式體系（不考慮新產品領先國擴散經驗）（以SUR估計）、層級貝氏BASS離散型擴散模式＋迴歸方程式體系（考慮新產品領先國擴散經驗）（以SUR估計）。本研究發現：（一）在考慮影響後進國的新產品擴散速度時，領先國的擴散經驗為絕對必要的考慮因子；（二）必須使用Bass連續型擴散模式做為建構新產品銷售預測模型的基礎；（三）必須使用Bass連續型擴散模式的NLS估計法估計Bass模型的創新係數p、模仿係數q及市場潛量m。

For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

The term aggregation refers to overdispersion and both are used interchangeably in this thesis. In addressing the problem of prevalence of infectious parasite species faced by most rural livestock farmers, we model the distribution of faecal egg counts of 15 parasite species (13 internal parasites and 2 ticks) common in sheep and goats. Aggregation and excess zeroes is addressed through the use of generalised linear models. The abundance of each species was modelled using six different distributions: the Poisson, negative binomial (NB), zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), zero-altered Poisson (ZAP) and zero-altered negative binomial (ZANB) and their fit was later compared. Excess zero models (ZIP, ZINB, ZAP and ZANB) were found to be a better fit compared to standard count models (Poisson and negative binomial) in all 15 cases. We further investigated how distributional assumption a↵ects aggregation and zero inflation. Aggregation and zero inflation (measured by the dispersion parameter k and the zero inflation probability) were found to vary greatly with distributional assumption; this in turn changed the fixed-effects structure. Serial autocorrelation between adjacent observations was later taken into account by fitting observation driven time series models to the data. Simultaneously taking into account autocorrelation, overdispersion and zero inflation proved to be successful as zero inflated autoregressive models performed better than zero inflated models in most cases. Apart from contribution to the knowledge of science, predictability of parasite burden will help farmers with effective disease management interventions. Researchers confronted with the task of analysing count data with excess zeroes can use the findings of this illustrative study as a guideline irrespective of their research discipline. Statistical methods from model selection, quantifying of zero inflation through to accounting for serial autocorrelation are described and illustrated. / Statistics / M.Sc. (Statistics)

For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

In this study, we explore models for optimal bidding in auctions on the bond market using data gathered from the Bloomberg Fixed Income Trading platform and MIFID II reporting. We define models that aim to fulfill two purposes. The first is to hit the best competitor price, such that a dealer can win the trade with the lowest possible margin. This model should also take into account the phenomenon of the Winner's Curse, which states that the winner of a common value auction tends to be the bidder who overestimated the value. We want to avoid this since setting a too aggressive bid could be unprofitable even when the dealer wins. The second aim is to define a model that estimates a quote that allows the dealer to win a certain target ratio of trades. We define three novel models for these purposes that are based on the best competitor prices for each trade, modeled by a Skew Exponential Power distribution. Further, we define a proxy for the Winner's Curse, represented by the distance of the estimated price from a reference price for the trade calculated by Bloomberg which is available when the request for quote (RFQ) arrives. Relevant covariates for the trades are also included in the models to increase the specificity for each trade. The novel models are compared to a linear regression and a random forest regression method using the same covariates. When trying to hit the best competitor price, the regression models have approximately equal performance to the expected price method defined in the study. However, when incorporating the Winner's Curse proxy, our Winner's Curse adjusted models are able to reduce the effect of the Winner's Curse as we define it, which the regression methods cannot. The results of the models for hitting a target ratio show that the actual hit ratio falls within an interval of 5% of the desired target ratio when running the model on the test data. The inclusion of covariates in the models does not impact the results as much as expected, but still provide improvements with respect to some measures. In summary, the novel methods show promise as a first step towards building algorithmic trading for bonds, but more research is needed and should incorporate more of the growing data set of RFQs and MIFID II recorded transaction prices. / I denna studie utforskar vi modeller för optimal budgivning för auktioner på obligationsmarknaden med hjälp av data som samlats in från plattformen Bloomberg Fixed Income Trading och MIFID II-rapportering. Vi definierar modeller som ämnar att uppfylla två syften. Det första är att träffa det bästa konkurrentpriset så att en handlare kan vinna auktionen med minsta möjliga marginal. Denna modell bör också ta hänsyn till fenomenet Winner's Curse, som innebär att vinnaren av en så kallad common value auction tenderar att vara den budgivare som överskattat värdet. Vi vill undvika detta eftersom det kan vara olönsamt att skicka ett alltför aggressivt bud även om handlaren vinner. Det andra syftet är att definiera en modell som uppskattar ett pris som gör det möjligt för handlaren att vinna en viss andel av sina obligationsaffärer. Vi definierar tre nya modeller för dessa ändamål som bygger på de bästa konkurrentpriserna för varje transaktion vi har data på. Dessa modelleras av en Skew Exponential Power-fördelning. Vidare definierar vi en variabel som indirekt mäter fenomenet Winner's Curse, representerad av budprisets avstånd från ett referenspris för transaktionen beräknad av Bloomberg som är tillgänglig när en request for quote (RFQ) anländer. Relevanta kovariat för transaktionen implementeras också i modellerna för att öka specificiteten för varje transaktion. De nya modellerna jämförs med en linjärregression och en random forest-regression som använder samma kovariat. När målet är att träffa det bästa konkurrentpriset ger regressionsmodellerna ungefär samma resultat som expected price-modellen som definieras i denna studie. När man däremot integrerar effekten av Winner's Curse med den definierade indirekta variablen kan vår Winner's Curse-justerade modell minska effekten av Winner's Curse, vilket regressionsmetoderna inte kan. Resultaten av modellerna som ämnar vinna en förbestämd andel av transaktionerna visar att den faktiska andelen transaktioner som man vinner faller inom ett intervall på 5% kring den önskade andelen när modellen körs på testdata. Att inkludera kovariat i modellerna påverkar inte resultaten till den grad som uppskattades, men ger mindre förbättringar med avseende på vissa mättal. Sammanfattningsvis visar de nya metoderna potential som ett första steg mot att bygga algoritmisk handel för obligationer, men mer forskning behövs och bör utnyttja mer av den växande datamängden av RFQs och MIFID II-rapporterade transaktionspriser.

Many problems in applied mathematics and physics are formulated most naturally in terms of matrices, and can be solved by computing functions of these matrices. For example, in quantum mechanics, the coherent dynamics of physical systems is described by the matrix exponential of their Hamiltonian. In state of the art experiments, one can now observe such unitary evolution of many-body systems, which is of fundamental interest in the study of many-body quantum phenomena. On the other hand the theoretical simulation of such non-equilibrium many-body dynamics is very challenging. In this thesis, we develop a symbolic approach to matrix functions and quantum dynamics based on a novel algebraic structure we identify for sets of walks on graphs. We begin by establishing the graph theoretic equivalent to the fundamental theorem of arithmetic: all the walks on any finite digraph uniquely factorise into products of prime elements. These are the simple paths and simple cycles, walks forbidden from visiting any vertex more than once. We give an algorithm that efficiently factorises individual walks and obtain a recursive formula to factorise sets of walks. This yields a universal continued fraction representation for the formal series of all walks on digraphs. It only involves simple paths and simple cycles and is thus called a path-sum. In the second part, we recast matrix functions into path-sums. We present explicit results for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We introduce generalised matrix powers which extend desirable properties of the Drazin inverse to all powers of a matrix. In the third part, we derive an intermediary form of path-sum, called walk-sum, relying solely on physical considerations. Walk-sum describes the dynamics of a quantum system as resulting from the coherent superposition of its histories, a discrete analogue to the Feynman path-integrals. Using walk-sum we simulate the dynamics of quantum random walks and of Rydberg-excited Mott insulators. Using path-sum, we demonstrate many-body Anderson localisation in an interacting disordered spin system. We give two observable signatures of this phenomenon: localisation of the system magnetisation and of the linear magnetic response function. Lastly we return to the study of sets of walks. We show that one can construct as many representations of series of walks as there are ways to define a walk product such that the factorisation of a walk always exist and is unique. Illustrating this result we briefly present three further methods to evaluate functions of matrices. Regardless of the method used, we show that graphs are uniquely characterised, up to an isomorphism, by the prime walks they sustain.

Η παρούσα μεταπτυχιακή διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής Θεωρίας Αποφάσεων και ειδικότερα στην εκτίμηση των παραμέτρων στο μοντέλο της διπαραμετρικής εκθετικής κατανομής με παράμετρο θέσης μ και παράμετρο κλίμακος σ. Θεωρούμε το πρόβλημα εκτίμησης των παραμέτρων κλίμακας μ και θέσης σ, όταν μ≤c, όπου c είναι μία γνωστή σταθερά. Αποδεικνύουμε ότι σε σχέση με το κριτήριο του Μέσου Τετραγωνικού Σφάλματος (ΜΤΣ), οι βέλτιστοι αναλλοίωτοι εκτιμητές των μ και σ, είναι μη αποδεκτοί όταν μ≤c, και προτείνουμε βελτιωμένους. Επίσης συγκρίνουμε του εκτιμητές αυτούς σε σχέση με το κριτήριο του Pitman. Επιπλέον, προτείνουμε εκτιμητές που είναι καλύτεροι από τους βέλτιστους αναλλοίωτους εκτιμητές, όταν μ≤c, ως προς την συνάρτηση ζημίας LINEX. Τέλος, η θεωρία που αναπτύσσεται εφαρμόζεται σε δύο ανεξάρτητα δείγματα προερχόμενα από εκθετική κατανομή. / The present master thesis deals with the estimation of the location parameter μ and the scale parameter σ of the two-parameter exponential distribution. We consider the problem of estimation of locasion parameter μ and the scale parameter σ, when it is known apriori that μ≤c, where c is a known constant. We establish that with respect to the mean square error (mse) criterion the best affine estimators of μ and σ in the absence of information μ≤c are inadmissible and we propose estimators which are better than these estimators. Also, we compare these estimators with respect to the Pitman Nearness criterion. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choise of LINEX loss. Finally, the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.