The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1;m2) in ZxZ : m1^2 + m2^2 <= x }, the number of integer lattice points inside a circle of radius sqrt(x). Gauss showed that P(x) = N(x)- pi * x satisfi es P(x) = O(sqrt(x)). Later Hardy and Landau independently proved that P(x) = Omega_(x1=4(log x)1=4). It is conjectured that inf{e in R : P(x) = O(x^e )}= 1/4. I. K atai showed that the integral from 0 to X of |P(x)|^2 dx = X^(3/2) + O(X(logX)^2). Similar results to those of the circle have been obtained for regions D in R^2 which contain the origin and whose boundary dD satis fies suff cient smoothness conditions. Denote by P_D(x) the similar error term to P(x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on dD, P_D(x) = Omega_(x1=4(log x)1=4) and that the integral from 0 to X of |P_D(x)|^2 dx = O(X^(3/2)). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, {(m1;m2) in Z^2 : gcd(m1;m2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.

The media is littered with various portrayals of aggression. This aggression has been shown to influence the attitudes, beliefs, and subsequent behaviors of its viewers (Bushman & Anderson, 2001). Relational aggression is a newer concern for researchers and has become more prevalent in recent research. Relational aggression is prevalent in the lives of adolescents. Using social cognitive theory (Bandura 2002), information processing theory (Huesmann, 1988), and the general aggression model (Anderson & Bushman, 2002) to justify how adolescents might be developing these relationally aggressive behaviors, this study seeks to expand the literature by evaluating the portrayals of relational aggression in popular teen movies; a genre primarily watched by adolescents. This thesis is a content analysis of the top 30 grossing teen movies for the 1980s, 1990s, and 2000s for a total of 90 movies. The study examines three types of relational aggression—direct, indirect, and nonverbal. The following variables were coded for each act of relational aggression: initiator and victim age, gender, sociometrics, attractiveness, SES, and role, their relationship to each other, the context, humor, and consequence of the act of relational aggression. Analysis revealed that relational aggression is extremely prevalent (94.4%) in teen movies. Direct relational aggression is more prevalent in teen movies than both indirect and nonverbal relational aggression. Results indicate that females are portrayed as the primary initiators of relational aggression in teen films. Initiators and victims of relational aggression are primarily portrayed as characters of average attractiveness, average popularity, and as having middle class incomes. Acts of relational aggression are portrayed as not justified and not humorous. However, acts of relational aggression were portrayed as rewarded. No significant differences across decade were found for amount of relational aggression shown or for what type of relational aggression was portrayed. Results showed there were more male aggressors in the 1980s than expected and more female aggressors in the 2000s than expected.

It is widely believed that the median is “usually” between the mean and the mode for skewed unimodal distributions. However, this inequality is not always true, especially with grouped data. Unavailability of complete raw data further necessitates the importance of evaluating this characteristic in grouped data. There is a gap in the current statistical literature on assessing mean–median–mode inequality for grouped data. The study aims to evaluate the relationship between the mean, median, and mode with unimodal grouped data; derive conditions for their inequalities; and present their application.

The concept of portfolio optimization has been widely studied in the academy and implemented in the financial markets since its introduction by Markowitz 70 years ago. The problem of the mean-variance optimization framework caused by input uncertainty has been one of the foci in the previous research. In this study, several models (linear shrinkage and Black-Litterman) based on Bayesian approaches are studied to improve the estimation of inputs. Moreover, a new framework based on robust optimization is presented to mitigate the input uncertainty further.  An out-of-sample test is specially designed, and the results show that Bayesian models in this study can improve the optimization results in terms of higher Sharpe ratios (the quotient between portfolio returns and their risks). Both covariance matrix estimators based on the linear shrinkage method contain less error and provide better optimization results, i.e. higher Sharpe ratios. The Black-Litterman model with a proper choice of inputs can significantly improve the portfolio return. The new framework based on the combination of shrinkage estimators, Black-Litterman, and robust optimization presents a better way for portfolio optimization than the classical framework of mean-variance optimization.

[pt] Nesta teses, nós estudamos H-superfícies, isto é, superfícies tendo curvatura media constante, imersas em variedades homogêneas simplesmente conexas de dimensão 3. Nós focamos nossa atenção no estudo de existência de H multigráficos. Também estudamos a H-superfícies invariantes por um grupo a um parâmetro de isometrias que estão imersas no espaço PSL(2) (R, T). / [en] In this thesis we study H-surfaces, that is, surfaces having constant mean curvature, immersed in homogeneous simply connected 3-manifold. We focus our attention in the study of existence of H multigraphs. We also study the H-surfaces invariant by one-parameter group of isometries which are immersed in the space]PSL2(R, T).

In this study within quantitative portfolio optimization, stochastic programming is investigated as an investment decision tool. This research takes the direction of scenario based Mean-Absolute Deviation and is compared with the traditional Mean-Variance model and widely used Risk Parity portfolio. Furthermore, this thesis is done in collaboration with the First Swedish National Pension Fund, AP1, and the implemented multi-asset portfolios are thus tailored to match their investment style. The models are evaluated on two different fund management levels, in order to study if the portfolio performance benefits from a more restricted feasible domain. This research concludes that stochastic programming over the investigated time period is inferior to Risk Parity, but outperforms the Mean-Variance Model. The biggest aw of the model is its poor performance during periods of market stress. However, the model showed superior results during normal market conditions. / I denna studie inom kvantitativ portföljoptimering undersöks stokastisk programmering som ett investeringsbeslutsverktyg. Denna studie tar riktningen för scenariobaserad Mean-Absolute Deviation och jämförs med den traditionella Mean-Variance-modellen samt den utbrett använda Risk Parity-portföljen. Avhandlingen görs i samarbete med Första AP-fonden, och de implementerade portföljerna, med era tillgångsslag, är därför skräddarsydda för att matcha deras investeringsstil. Modellerna utvärderas på två olika fondhanteringsnivåer för att studera om portföljens prestanda drar nytta av en mer restrektiv optimeringsmodell. Den här undersökningen visar att stokastisk programmering under undersökta tidsperioder presterar något sämre än Risk Parity, men överträffar Mean-Variance. Modellens största brist är dess prestanda under perioder av marknadsstress. Modellen visade dock något bättre resultat under normala marknadsförhållanden.

Background: Individuals with cerebral palsy (CP) are prone to sedentary life with limited recreational- and physical activity. RaceRunning (RR) is a sport for individuals with disabilities such as motor impairment that allow them to move independently and enjoy physical activity. The aim of the study was to investigate muscle fatigue of RR athletes with different classiffcation of CP during six minuteRaceRunning test (6MRRT). The Cerebral Palsy International Sports and RecreationAssociation has the goal to make RR a Paralympic sport. For a sport to be included in the Paralympics a classiffcation system has to be made. Information about muscle function during RR could give vital physical information about athletes with CP and contribute to the development of such classiffcation system. Method: Median frequency of the power spectrum from surface electromyography(sEMG) of four muscles (vastus lateralis, biceps femoris, gluteus medius, gastrocnemius lateralis) was computed along with amplitude changes (root mean square) in order to assess muscle fatigue. Results: Indications of muscle fatigue were observed from one out of five athletes for athlete's less affected side. Further sEMG studies with more subjects have to be conducted on CP RR athletes to give more conclusive results of muscle fatigue.

Syftet med denna rapport är att undersöka de medelvärdesåtervändande egenskaperna i aktiekursen för noterade fastighetsbolag med utgångspunkt för bolagets extraordinära substansvärde, samt hur väl information kan användas för investeringsbeslut.Resultatet av den empiriska utfallsstudien som utförts visar på att det finns ett samband mellan aktiekurs och substansvärden, samt att en investeringsstrategi baserat på medelvärdesåtervändande egenskaper i dessa variabler fungerar. Av de tre dynamiska portföljerna som tagits fram och som strävat efter att allokera störst vikt till de bolag som fundamentalt framstått som billigast jämfört med sitt genomsnittliga substansvärde, överträffar samtliga portföljer ett aktieindex av samma bolag. / This study investigates mean reversion in publicly traded property stocks with respect to the company’s extraordinary discounts to the net asset value; furthermore, to what extent this knowledge can be applied to investment strategies.The result of the empirical survey shows that there is a relationship between the stock prices and the net asset value, also, that an investment strategy utilizing mean reversion in these two variables works. Of the three dynamic portfolios, aiming to allocate the largest portfolio weight to the fundamentally cheapest stocks, based on their current discounts and premiums to the net asset value with respect to the company average, all three outperform an index of the same companies.

Emergent phenomena in condensed matter physics like, e.g., magnetism, superconductivity, or non-trivial topology often come along with a surprise and exert great fascination to researchers up to this day. Within this thesis, we are concerned with the analysis of associated types of order that arise due to strong electronic interactions and focus on the high-\(T_c\) cuprates and Kondo systems as two prime candidates. The underlying many-body problem cannot be solved analytically and has given rise to the development of various approximation techniques to tackle the problem. In concrete terms, we apply the auxiliary particle approach to investigate tight-binding Hamiltonians subject to a Hubbard interaction term to account for the screened Coulomb repulsion. Thereby, we adopt the so-called Kotliar-Ruckenstein slave-boson representation that reduces the problem to non-interacting quasiparticles within a mean-field approximation. Part I provides a pedagogical review of the theory and generalizes the established formalism to encompass Gaussian fluctuations around magnetic ground states as a crucial step to obtaining novel results. Part II addresses the two-dimensional one-band Hubbard model, which is known to approximately describe the physics of the high-\(T_c\) cuprates that feature high-temperature superconductivity and various other exotic quantum phases that are not yet fully understood. First, we provide a comprehensive slave-boson analysis of the model, including the discussion of incommensurate magnetic phases, collective modes, and a comparison to other theoretical methods that shows that our results can be massively improved through the newly implemented fluctuation corrections. Afterward, we focus on the underdoped regime and find an intertwining of spin and charge order signaled by divergences of the static charge susceptibility within the antiferromagnetic domain. There is experimental evidence for such inhomogeneous phases in various cuprate materials, which has recently aroused interest because such correlations are believed to impact the formation of Cooper pairs. Our analysis identifies two distinct charge-ordering vectors, one of which can be attributed to a Fermi-surface nesting effect and quantitatively fits experimental data in \(\mathrm{Nd}_{2-\mathrm{x}}\mathrm{Ce}_\mathrm{x}\mathrm{CuO}_4\) (NCCO), an electron-doped cuprate compound. The other resembles the so-called Yamada relation implying the formation of periodic, double-occupied domain walls with a crossover to phase separation for small dopings. Part III investigates Kondo systems by analyzing the periodic Anderson model and its generalizations. First, we consider Kondo metals and detect weakly magnetized ferromagnetic order in qualitative agreement with experimental observations, which hinders the formation of heavy fermions. Nevertheless, we suggest two different parameter regimes that could host a possible Kondo regime in the context of one or two conduction bands. The part is concluded with the study of topological order in Kondo insulators based on a three-dimensional model with centrosymmetric spin-orbit coupling. Thereby, we classify topologically distinct phases through appropriate \(\mathbb{Z}_2\) invariants and consider paramagnetic and antiferromagnetic mean-field ground states. Our model parameters are chosen to specifically describe samarium hexaboride (\(\mbox{SmB}_6\)), which is widely believed to be a topological Kondo insulator, and we identify topologically protected surface states in agreement with experimental evidence in that material. Moreover, our theory predicts the emergence of an antiferromagnetic topological insulator featuring one-dimensional hinge-states as the signature of higher-order topology in the strong coupling regime. While the nature of the true ground state is still under debate, corresponding long-range magnetic order has been observed in pressurized or alloyed \(\mbox{SmB}_6\), and recent experimental findings point towards non-trivial topology under these circumstances. The ability to understand and control topological systems brings forth promising applications in the context of spintronics and quantum computing. / Emergente Phänomene in der Physik der kondensierten Materie, wie z. B. Magnetismus, Supraleitung oder nicht-triviale Topologie gehen oft mit Überraschungen einher und faszinieren Wissenschaftler bis heute. Innerhalb dieser Arbeit befassen wir uns mit der Analyse damit assoziierter Art von Ordnung, die durch starke elektronische Wechselwirkungen entsteht und konzentrieren uns auf die Kuprat-Hochtemperatursupraleiter und Kondo-Systeme als zwei prominente Kandidaten. Das zugrunde liegende Vielteilchenproblem kann nicht analytisch gelöst werden und hat zur Entwicklung vielfältiger Näherungsverfahren geführt, um das Problem anzugehen. Konkret wenden wir den Hilfsteilchenansatz an, um tight-binding Hamiltonoperatoren zu untersuchen, die einen Hubbard-Wechselwirkungsterm aufweisen, um die abgeschirmte Coulomb-Abstoßung zu berücksichtigen. Dabei benutzen wir die sogenannte Kotliar-Ruckenstein-Slave-Boson-Darstellung, die das Problem im Rahmen einer Molekularfeldnäherung auf nicht-wechselwirkende Quasiteilchen zurückführt. Teil I beinhaltet eine pädagogisch aufgearbeitete Zusammenfassung der Theorie und verallgemeinert durch die Berücksichtigung Gaußscher Fluktuationen um magnetische Grundzustände den etablierten Formalismus, was sich als entscheidender Schritt herausstellt, um neuartige Ergebnisse erzielen zu können. Teil II befasst sich mit dem zweidimensionalen Einband-Hubbard-Modell, von dem bekannt ist, dass es näherungsweise die Physik der Kuprat-Hochtemperatursupraleiter beschreibt, welche Hochtemperatursupraleitung und verschiedene andere exotische Quantenphasen aufweisen, die noch nicht vollständig verstanden sind. Zunächst machen wir eine ausführliche Slave-Boson-Analyse des Modells, einschließlich der Diskussion inkommensurabler magnetischer Phasen, kollektiver Moden und eines Vergleichs mit anderen theoretischen Methoden, der zeigt, dass unsere Ergebnisse durch die neu implementierten Fluktuationskorrekturen massiv verbessert werden können. Danach konzentrieren wir uns auf den unterdotierten Bereich und finden eine Verflechtung von Spin- und Ladungsordnung, die durch Divergenzen der statischen Ladungssuszeptibilität innerhalb der antiferromagnetischen Domäne signalisiert wird. Es gibt experimentelle Hinweise auf derartige inhomogene Phasen in verschiedenen Kuprat-Materialien, was in letzter Zeit vermehrt Interesse geweckt hat, da angenommen wird, dass entsprechende Korrelationen die Bildung von Cooper-Paaren beeinflussen. Unsere Analyse identifiziert zwei unterschiedliche Ladungsordnungsvektoren, von denen einer einem Fermi-Flächeneffekt zugeschrieben werden kann und quantitativ zu experimentellen Daten von \(\mathrm{Nd}_{2-\mathrm{x}}\mathrm{Ce}_\mathrm{x}\mathrm{CuO}_4\) (NCCO), einer elektronendotierten Kupratverbindung, passt. Der andere erinnert an die sogenannte Yamada-Beziehung und impliziert die Bildung von periodischen, doppelt besetzten Domänenwänden und einem Übergang zu Phasenseperation für kleine Dotierungen. Teil III untersucht Kondo-Systeme durch Analyse des periodischen Anderson-Modells und seiner Verallgemeinerungen. Zunächst betrachten wir Kondo-Metalle und finden schwach magnetisierte ferromagnetische Ordnung in qualitativer Übereinstimmung mit experimentellen Beobachtungen, welche die Bildung von schweren Fermionen hemmt. Dennoch identifizieren wir zwei verschiedene Parameterbereiche, die ein mögliches Kondo-Regime im Kontext von einem oder zwei Leitungsbändern beherbergen könnten. Der Teil wird mit der Untersuchung topologischer Ordnung in Kondo-Isolatoren basierend auf einem dreidimensionalen Modell mit zentrosymmetrischer Spin-Bahn-Kopplung abgeschlossen. Dabei klassifizieren wir topologisch unterscheidbare Phasen durch geeignete \(\mathbb{Z}_2\)-Invarianten und betrachten paramagnetische und antiferromagnetische Molekularfeld-Grundzustände. Unsere Modellparameter wurden gewählt, um insbesondere Samariumhexaborid (\(\mbox{SmB}_6\)) zu beschreiben, von dem allgemein angenommen wird, dass es sich um einen topologischen Kondo-Isolator handelt, und wir identifizieren topologisch geschützte Oberflächenzustände in Übereinstimmung mit experimentellen Befunden in diesem Material. Darüber hinaus sagt unsere Theorie die Emergenz eines antiferromagnetischen topologischen Isolators mit eindimensionalen Randzuständen als Merkmal von Topologie höherer Ordnung im Parameterbereich starker Korrelationen voraus. Während das Wesen des korrekten Grundzustands noch umstritten ist, wurde eine entsprechende langreichweitige magnetische Ordnung in unter Druck stehendem oder legiertem \(\mbox{SmB}_6\) beobachtet und kürzliche experimentelle Befunde weisen unter diesen Umständen auf nicht-triviale Topologie hin. Die Fähigkeit, topologische Systeme zu verstehen und zu kontrollieren, bringt vielversprechende Anwendungen im Kontext von Spintronik und Quantencomputing hervor.

In this thesis, Density Functional Theory (DFT) and Dynamical Mean-Field Theory (DMFT) approaches are applied to study the magnetic properties of transition metal nanosystems of different sizes and compositions. In particular, in order to take into account dynamical electron correlation effects (time-resolved local charge interactions), we have adopted the DFT+DMFT formalism and made it suitable for application to nanostructures. Preliminary application of this DFT+DMFT approach, using available codes, to study the magnetic properties of small (2 to 5-atom) Fe and FePt clusters provide meaningful results: dynamical effects lead to a reduction of the cluster magnetic moment as compared to that obtained from DFT or DFT+U (U being the Coulomb repulsion parameter). We have subsequently developed our own nanoDFT+DMFT code and applied it to examine the magnetization of iron particles containing10-147 atoms. Our results for the cluster magnetic moments are in a good agreement with experimental data. In particular, we are able to reproduce the oscillations in magnetic moment with size as observed in the experiments. Also, DFT+DMFT does not lead to an overestimation of magnetization for the clusters in the size range of 10-27 atoms found with DFT and DFT+U. On application of the nanoDFT+DMFT approach to systems with mixed geometry – Fe2O3 film, which are periodic (infinitely extended), in two directions, and finite in the third. Similar to DFT+U, we find that the surface atom magnetic moments are smaller compared to the bulk. However, the absolute values of the surface atoms magnetic moments are smaller in DFT+DMFT. In parallel, we have carried out a systematic study of magnetic anisotropy in bimetallic L10 FePt nanoparticles (20-484 atoms) by using two DFT-based approaches: direct and the torque method. We find that the magnetocrystalline anisotropy (MCA) of FePt clusters is larger than that of the pure Fe and Pt ones. We explain this effect by a large hybridization of 3d Fe- and 5d Pt-atom orbitals, which lead to enhancement of the magnetic moment of the Pt atom, and hence to a larger magnetic anisotropy because of large spin-orbit coupling of Pt atoms. In addition, we find that particles whose (large) central layer consists of Pt atoms, rather than Fe, have larger MCA due to stronger hybridization effects. Such 'protected' MCA, which does not require protective cladding, can be used in modern magnetic technologies.