Ab initio prediction of the electronic properties of solids is traditionally performed using groundstate Density Functional Theory. These methods are unreliable however, for a class of important problems involving weak attractive forces. These problems include (i) the energetics of hydrogen storage and metal interactions in graphene, (ii) cohesion properties of some polymer systems and (iii) possibly, the weak hydrophobic forces in biomolecules. For these cases a more powerful method than groundstate DFT are timedependent DFT (tdDFT) methods related to the Random-Phase Approximation (RPA). All of these methods proceed by looking at the dynamic density-density response function, whose long-ranged properties naturally lead to the weak forces referred to above. In this thesis we have tested these ideas by investigating electronic response and correlation on the predicted properties of spherical atoms. We have developed and tested a variety of approximations to the timedependent response function through approximations of the tdDFT class and a new method involving greater self-consistency in the screening equation, the inhomogenous STLS approach. Through the development of new methods and computer code, we have solved the response equation allowing us to test our approximations on atoms. Calculation of certain dynamic and static properties of a variety of atoms within our approximations generally agree well with known results. In this thesis we have calculated excitation energies of Helium, dipole polarisabilities and C6 van der Waals (vdW) coefficients of a variety of atoms, and groundstate correlation energies Ec of some atoms. The excitation spectra of Helium generated in our new PGG+c approximation are in good agreement with experiment. The dipole polarisabilities are generally in good agreement with known results, with the exception of Magnesium, Beryllium and Sodium. The C6 coefficients are a little poorer with the exception of Helium where they are nearly exact. Correlation energies are generally reasonable in the PGG+c approximation although they are considerably less accurate than the other properties we have calculated for all atoms other than He. The ISTLS correlation energy of Helium is within 5% suggesting that this method may perform well for larger atoms where our present numerical techniques require improvement. These generally positive results suggest that the approximations we have developed may be applied to more complicated systems such as those described above with good results.

In a strong magnetic field, the psuedospin chirality of bilayer graphene’s low energy bands results in degenerate, zero energy n=0 and n=1 Landau or- bital states. In this thesis, we find that in addition to endowing states with energetic broadness disorder strongly mixes the zero energy orbitals of the lowest Landau level. We study the dependence of mixing and conductivity on inter-layer bias. Quantum Hall ferromagnetic states emerge when electronic interactions are included at the mean-field level. We study the character of ground states and quasiparticle excitations in the context of orbital degen- eracy and in the presence of interactions with the filled Dirac sea of states. Lastly, we study the effect of interactions in ABA-stacked trilayer graphene, and discover a metal-insulator transition and a separate propensity to break mirror symmetry in certain areas of parameter space. / text

Prime numbers play an integral part in many upper level mathematics courses, most notably in Number Theory. Can a course or section on prime numbers be introduced at the secondary (high school) level? This report outlines a possible course in a manner suitable for grade level instruction. These topics include: an extended section on the complete number system, a brief history of primes, their cardinality, and both the Fundamental Theorem of Arithmetic and Prime Number Theorem, the applications of primes, and the impact of primes within perfect numbers will all be explored. A brief discussion on questions that still remain relating to prime numbers will conclude this report. / text

The simulation of the dynamics of quantum systems is very di cult, due to the fact that, in general, it cannot be calculated exactly for interacting many-body systems. Brute force simulations of quantum dynamics are simply not feasible, and approximations need to be made. In many instances a quantum system can be approximated as a quantum-classical system, where only a subsystem of interest is treated quantum mechanically, and the rest is considered as a classical bath. When energy is free to be exchanged between the subsystem and its environment, the dynamics that occur is said to be nonadiabatic. This type of dynamics is challenging to calculate on a computer, as it can lead to large statistical errors at long times. Hence, there is a need for improved algorithms for nonadiabatic dynamics. In this thesis, a recently introduced nonadiabatic sampling scheme [A. Sergi and F. Petruccione, Phys. Rev. E 81, 032101 (2010)] is used to calculate the long-time dynamics of a model system comprising a quantum spin coupled to a bath of harmonic oscillators. Also, various technical aspects of the algorithm are investigated. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.

Thesis (Ph. D.)--University of Texas at Austin. / Description based on print version record. Includes bibliographical references (leaves 164-166).

Thesis (Ph. D. in Physics)--Vanderbilt University, Aug. 2007. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 156-167).

We analyse the steady-state behaviour of two different models with collaborating queues: that is, models in which 'customers' can be served by many types of 'servers', and 'servers' can process many types of 'customers'. The first example is a large-scale service system, such as a call centre. Collaboration is the result of cross-trained staff attending to several different types of incoming calls. We first examine a load-balancing policy, which aims to keep servers in different pools equally busy. Although the policy behaves order-optimally over fixed time horizons, we show that the steady-state distribution may fail to be tight on the diffusion scale. That is, in a family of ever-larger networks whose arrival rates grow as O(r) (where r is a scaling parameter growing to infinity), the sequence of steady-state deviations from equilibrium scaled down by sqrt(r) is not tight. We then propose a different policy, for which we show that the sequence of invariant distributions is tight on the r (1/2+epsilon) scale, for any epsilon > 0. For this policy we conjecture that tightness holds on the diffusion scale as well. The second example models a limit order book, a pricing mechanism for a single-commodity market in which buyers (respectively sellers) are prepared to wait for the price to drop (respectively rise). We analyse the behaviour of a simplified model, in which the arrival events are independent of each other and the state of the limit order book. The system can be represented by a queueing model, with 'customers' and 'servers' corresponding to bids and asks; the roles of customers and servers are symmetric. We show that, with probability 1, the price interval breaks up into three regions. At small (respectively large) prices, only finitely many bid (respectively ask) orders ever get fulfilled, while in the middle region all orders eventually clear. We derive equations which define the boundaries between these regions, and solve them explicitly in the case of iid uniform arrivals to obtain numeric values of the thresholds. We derive a heuristic for the distribution of the highest bid (respectively lowest ask), and present simulation data confirming it.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1982. / Includes bibliographical references. / by Thierry Troudet. / Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1982.

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