A Measure of Distance for the Unit Root Hypothesis

This paper proposes and analyses a measure of distance for the unit root hypothesis tested against stochastic stationarity. It applies over a family of distributions, for any sample size, for any specification of deterministic components and under additional autocorrelation, here parameterised by a finite order moving-average. The measure is shown to obey a set of inequalities involving the measures of distance of Gibbs and Su (2002) which are also extended to include power. It is also shown to be a convex function of both the degree of a time polynomial regressors and the moving average parameters. Thus it is minimisable with respect to either. Implicitly, therefore, we find that linear trends and innovations having a moving average negative unit root will necessarily make power small. In the context of the Nelson and Plosser (1982) data, the distance is used to measure the impact that specification of the deterministic trend has on our ability to make unit root inferences. For certain series it highlights how imposition of a linear trend can lead to estimated models indistinguishable from unit root processes while freely estimating the degree of the trend yields a model very different in character.

Small gain theorems for large scale systems and construction of ISS Lyapunov functions

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.Comment: provisionally accepted by SIAM Journal on Control and Optimizatio

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Computing optical properties of large systems

In recent years, time-dependent density-functional theory (TDDFT) has been the method of choice for calculating optical excitations in medium sized to large systems, due to its good balance between computational cost and achievable accuracy. In this thesis, TDDFT is reformulated to fit the framework of the linear-scaling density-functional theory (DFT) code ONETEP. The implementation relies on representing the optical response of the system using two sets of localised, atom centered, in situ optimised orbitals in order to ideally describe both the electron and the hole wavefunctions of the excitation. This dual representation approach requires only a minimal number of localised functions, leading to a very efficient algorithm. It is demonstrated that the method has the capability of computing low energy excitations of systems containing thousands of atoms in a computational effort that scales linearly with system size. The localised representation of the response to a perturbation allows for the selective convergence of excitations localised in certain regions of a larger system. The excitations of the whole system can then be obtained by treating the coupling between different subsystems perturbatively. It is shown that in the limit of weakly coupled excitons, the results obtained with the coupled subsystem approach agree with a full treatment of the entire system, with a large reduction in computational cost. The strengths of the methodology developed in this work are demonstrated on a number of realistic test systems, such as doped p-terphenyl molecular crystals and the exciton coupling in the Fenna-Matthews-Olson complex of bacteriochlorophyll. It is shown that the coupled subsystem TDDFT approach allows for the treatment of system sizes inaccessible by previous methods.Open Acces

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

Glosarium Matematika

273 p.; 24 cm