Mean-field games (MFG) are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agents’ motion is hampered in high-density regions. First, we study radial solutions for first- and second-order stationary MFG with congestion on Rd. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. For the first case, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems. Next, we consider second-order stationary MFG with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFG with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians. Additionally, we study first-order stationary MFG with congestion with quadratic or power-like Hamiltonians. Using explicit examples, we illustrate two key difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFG with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we devise a discretization that is combined with optimization algorithms to numerically solve various MFG with congestion.

We use a combination of Density Functional Theory and Dynamical Mean Field Theory (DFT+DMFT) to study electronic correlations in unconventional superconductors, with a focus on nickelate analogs of cuprate superconductors. We study the infinite layer nickelate superconductor NdNiO₂ in parallel with the isostructural CaCuO₂. Our results point to superconductivity in the nickelate being cupratelike, with correlations dominated by a hybrid Ni-𝑑_{𝑥²-𝑦²} and O-𝑝 band, and with the extra bands not contributing substantially to the superconducting state. We find that the infinite layer nickelate NdNiO₂ and the trilayer nickelate Pr₄Ni₃O₈ are virtually identical in terms of correlation physics when compared at the same chemical doping, despite the differences in Fermiology, indicating that the number of layers can stand in for chemical doping for some properties related to electronic correlations. We find that as opposed to in narrow window DFT+DMFT, in wide window DFT+DMFT the choice of downfolding procedure leads to very different results. This is an important ambiguity in the method that must be resolved or the method is incomplete by itself. We also study Sr₂MoO₄ in parallel with the Hund's superconductor Sr₂RuO₄, and find that Sr₂MoO₄ is a particle-hole dual of Sr₂RuO₄ but without the van Hove singularity at the Fermi level, which disentangles the influence of the van Hove singularity from Hund's physics. We show that Sr₂MoO₄ has a characteristic Hund's peak on the occupied of the spectral function, indicating that the peak should be observable by photoemission experiments.

In this thesis, we consider the planning problem for first-order mean-field games (MFG). These games degenerate into optimal transport when there is no coupling between players. Our aim is to extend the concept of displacement convexity from optimal transport to MFGs. This extension gives new estimates for solutions of MFGs. First, we introduce the Monge-Kantorovich problem and examine related results on rearrangement maps. Next, we present the concept of displacement convexity. Then, we derive first-order MFGs, which are given by a system of a Hamilton-Jacobi equation coupled with a transport equation. Finally, we identify a large class of functions, that depend on solutions of MFGs, which are convex in time. Among these, we find several norms. This convexity gives bounds for the density of solutions of the planning problem.

A complex arrangement of electronic states within materials can manifest exotic quantum-mechanical effects. These systems are often referred to as quantum materials. Increased understanding of quantum materials has historically lead to the development of new technologies. It is therefore extremely important to develop and test precise methods for calculating the behaviour of electronic states within a material. For decades, the workhorse of electronic structure calculations has been density functional theory (DFT). DFT is often referred to as a first-principles method because it allows for the calculation of the distribution of electrons throughout a material with only specification of the lattice geometry and atomic components. From the results of a DFT calculation, it is possible to study the orbital character of electronic wavefunctions, topology of electronic band structure, and some aspects of superconductivity. This provides insight into many quantum properties of a system which may otherwise be difficult or impossible to ascertain from experiments. DFT is, however, sometimes limited by the approximations necessary for practical implementation. Further methods have been developed to systematically correct the limitations of DFT. In particular, the combination of DFT with dynamical mean-field theory (DFT+DMFT) is among the most widely accepted methods for correcting the inadequacy of DFT in handling strong electron-electron correlations. In this thesis, I use methods from DFT and DFT+DMFT to study the quantum properties of materials. / Thesis / Master of Science (MSc)

The origin of cosmic magnetic fields is a important area of astrophysics. The process by which they are created falls under the heading of dynamo theory, and is the topic of this thesis. Our focus for the location of where these magnetic fields operate is one the most ubiquitous objects in the universe, the accretion disk. By studying the accretion disk and the dynamo process that occurs there we wish to better understand both the accretion process and the dynamo process in stars and galaxies as well. We analyse the output from a stratified zero net flux shearing box simulation performed using the ATHENA MHD code in collaboration with Shane Davis. The simulation has turbulence which is naturally forced by the presence of a linear instability called the magnetorotational instability (MRI). We utilise Fourier filtering and the tools of mean field dynamo theory to establish a connection between the calculated EMF and the model predictions of the dynamically quenched alpha model. We find a positive correlation for both components parallel to the large scale magnetic field and the azimuthal components. We have explored many aspects of the theory including additional contributions from magnetic buoyancy and an effect arising from the large scale shear and the current density. We also directly measure the turbulent correlation time for the velocity and magnetic fields both large scale and small. We can also observe the effects of the dynamo cycle, with the azimuthal component of the large scale magnetic field flipping sign in this analysis. We find a positive correlation between the divergence of the eddy scale magnetic helicity flux and the component of the electromotive force parallel to the large scale magnetic field. This correlation directly links the transfer of magnetic helicity to the dynamo process in a system with naturally driven turbulence. This highlights the importance of magnetic helicity and its conservation even in a system with triply periodic boundary conditions. / Thesis / Doctor of Philosophy (PhD)

No description available.

The purpose of this thesis is to broaden the tools and knowledge available for understanding the behavior of metals under irradiation to aid in the pursuit of advanced materials for deployment in Generation IV (Gen-IV) nuclear reactor designs. A mean-field study is conducted on a body-centered cubic (BCC) A-B binary metal alloy system. The performance of the simulated metal system is measured by assessing the degree of segregation that occurs at the grain boundary (GB) in the center of the one-dimensional simulation box. This mean-field method was developed using rate theory equations to observe the diffusion of defects and solute atoms in the binary BCC alloy modeled after a section of planes in the <100> direction of α-iron. The method in this thesis is adapted from a previous radiation-induced segregation (RIS) study that was similarly validated against thermal segregation isotherms. This adapted simulation code was used to study RIS by varying the initial values and conditions across ranges relevant to Generation IV reactor designs. The simulations run with this code were centered around segregation energy and the diffusion coefficient relationships between defects and solute atoms. The most influential conditions applied to both the segregation energy and diffusion coefficient relationship test suites were the temperature and dose rate. The interplay of the various segregation energies, manipulated diffusion coefficients, temperatures, and dose rates is explored in this thesis. The code used in this thesis is presented as a modular framework for further parameter study with a clear direction for more complex alloys. / Master of Science / The growing electricity demand for more efficient, safe, reliable, and sustainable means of power generation requires research and subsequent implementation of advanced Generation IV (Gen-IV) nuclear reactor designs. These proposed designs operate under significantly more strenuous conditions from the perspective of materials used in constructing the reactor. Materials inside the reactor will experience temperatures, pressures, and radiation doses greatly exceeding those of previous generations: Gen II through III+. Metals are employed in almost every component inside a reactor and are particularly susceptible to the demanding conditions due to their tendency to lose their ductility under these stressors. This thesis presents a diffusion-based code that models a binary metal alloy under conditions similar to those expected in Gen-IV reactors. The results of the code give insight into the prevalence of a phenomenon known as radiation induced segregation (RIS) in metals under these Gen-IV relevant conditions. The values input into the code have significant effects on the resulting RIS behavior of the metal alloy. This thesis presents correlations between the initial parameters and the amount of segregation this alloy experiences. The results of this thesis allow a sort of mapping of material parameters and operating conditions so that materials can be designed for optimal performance over the lifespan of the next generation of nuclear reactors. The code in this thesis was developed with the expectation that its modularity would be expanded upon to apply to more complex alloys under a broader range of initial conditions.

Abstract Resource allocation and interference management in wireless small cell networks have been areas of key research interest in the past few years. Although a large number of research studies have been carried out, the needs for high capacity, reliability, and energy efficiency in the emerging fifth-generation (5G) networks warrants the development of methodologies focusing on ultra-dense and self-organizing small cell network (SCN) scenarios. In this regard, the prime motivation of this thesis is to propose an array of distributed methodologies to solve the problem of joint resource allocation and interference management in SCNs pertaining to different network architectures. The present dissertation proposes and investigates distributed control mechanisms for wireless SCNs mainly in three cases: a backhaul-aware interference management mechanism of the uplink of wireless SCNs, a dynamic cluster-based approach for maximizing the energy efficiency of dense wireless SCNs, and a joint power control and user scheduling mechanism for optimizing energy efficiency in ultra-dense SCNs. Optimizing SCNs, especially in the ultra-dense regime, is extremely challenging due to the severe coupling in interference and the dynamics of both queues and channel states. Moreover, due to the lack of inter-base station/cluster communications, smart distributed learning mechanisms are required to autonomously choose optimal transmission strategies based on local information. To overcome these challenges, an array of distributed algorithms are developed by combining the tools from machine learning, Lyapunov optimization and mean-field theory. For each of the above proposals, extensive sets of simulations have been carried out to validate the performance of the proposed methods compared to conventional models that fail to account for the limitations due to network scale, dynamics of queue and channel states, backhaul heterogeneity and capacity constraints, and the lack of coordination between network elements. The results of the proposed methods yield significant gains of the proposed methods in terms of energy savings, rate improvements, and delay reductions compared to the conventional models studied in the existing literature. / Tiivistelmä Langattomien piensoluverkkojen resurssien allokointi ja häiriön hallinta on ollut viime vuosina tärkeä tutkimuskohde. Tutkimuksia on tehty paljon, mutta uudet viidennen sukupolven (5G) verkot vaativat suurta kapasiteettia, luotettavuutta ja energiatehokkuutta. Sen vuoksi on kehitettävä menetelmiä, jotka keskittyy ultratiheisiin ja itseorganisoituviin piensoluverkkoihin. (SCN). Tämän väitöskirjan tärkein tavoite onkin esittää joukko hajautettuja menetelmiä piensoluverkkojen yhteisten resurssien allokointiin ja häiriön hallintaan, kun käytössä on erilaisia verkkoarkkitehtuureja. Tässä väitöskirjassa ehdotetaan ja tutkitaan hajautettuja menetelmiä langattomien piensoluverkkojen hallintaan kolmessa eri tilanteessa: välityskanavan huomioiva häiriönhallinta menetelmä langattomissa piensoluverkoissa, dynaamisiin klustereihin perustuva malli tiheiden langattomien piensoluverkkojen energiatehokkuuden maksimointiin ja yhteinen tehonsäädön ja käyttäjien allokaatio menetelmä ultratiheiden piensoluverkkojen energiatehokkuuden optimointiin. Ultratiheiden piensoluverkkojen optimointi on erittäin haastavaa häiriön sekä jonojen ja kanavatilojen vahvojen kytkösten vuoksi. Lisäksi, koska klustereilla/tukiasemilla ei ole kommunikaatiota, tarvitaan hajautettuja oppimisalgoritmeja, jotta saadaan itsenäisesti valittua optimaaliset lähetys menetelmät hyödyntäen vain paikallista tietoa. Tämän vuoksi kehitetään useita hajautettuja algoritmeja, jotka hyödyntävät koneoppimista, Lyapunov optimointia ja mean-field teoriaa. Kaikki yllä olevat esitetyt menetelmät on validoitu laajoilla simulaatioilla, joilla on voitu todentaa niiden suorituskyky perinteisiin malleihin verrattuna. Perinteiset mallit eivät pysty ottamaan huomioon verkon laajuuden, jonon ja kanavatilojen dynamiikan, eri välityskanavien ja rajallisen kapasiteetin asettamia rajoituksia sekä verkon elementtien välisen koordinoinnin puuttumista. Esitetyt menetelmät tuottavat huomattavia parannuksia energiansäästöön, siirtonopeuteen ja viiveiden vähentämiseen verrattuna perinteisiin malleihin, joita kirjallisuudessa on tarkasteltu.

published_or_final_version / abstract / Physics / Doctoral / Doctor of Philosophy

No description available.