Geometrical interpretations of Bäcklund transformations and certain types of partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University

Page 37 is missing from the original copy.Gauss' Theorema Egregium contains a partial differential equation relating the Gaussian curvature K to components of the metric tensor and its derivatives. Well known partial differential equations such as the Schrödinger equation and the sine-Gordon equation correspond to this PDE for special choices of K and special coördinate systems. The sine-Gordon equation, for example, can be derived via Gauss' equation for K = –1 using the Tchebychef net as a coördinate system. In this thesis we consider a special class of Bäcklund Transformations which correspond to coördinate transformations on surfaces having a specified Gaussian curvature. These transformations lead to Gauss' PDE in different forms and provide a method for solving certain classes of non-linear second order partial differential equations. In addition, we develop a more systematic way to obtain a coordinate system for a more general class of PDE, such that this PDE corresponds to the Gauss equation

ASSESSMENT OF POSITION OF MENTAL FORAMEN, INTERMENTAL FORAMEN DISTANCE AND ITS CORRELATION WITH AGE AND GENDER AMONG PEDIATRIC PATIENTS

The aim of this study was to assess the position of mental foramen, inter-mental foramen distance, and its correlation with age and gender between 9 to 12 years of age

Adaptive Differential Feedback in Time-Varying Multiuser MIMO Channels

In the context of a time-varying multiuser multiple-input-multiple-output (MIMO) system, we design recursive least squares based adaptive predictors and differential quantizers to minimize the sum mean squared error of the overall system. Using the fact that the scalar entries of the left singular matrix of a Gaussian MIMO channel becomes almost Gaussian distributed even for a small number of transmit antennas, we perform adaptive differential quantization of the relevant singular matrix entries. Compared to the algorithms in the existing differential feedback literature, our proposed quantizer provides three advantages: first, the controller parameters are flexible enough to adapt themselves to different vehicle speeds; second, the model is backward adaptive i.e., the base station and receiver can agree upon the predictor and variance estimator coefficients without explicit exchange of the parameters; third, it can accurately model the system even when the correlation between two successive channel samples becomes as low as 0.05. Our simulation results show that our proposed method can reduce the required feedback by several kilobits per second for vehicle speeds up to 20 km/h (channel tracker) and 10 km/h (singular vector tracker). The proposed system also outperforms a fixed quantizer, with same feedback overhead, in terms of bit error rate up to 30 km/h.Comment: IEEE 22nd International Conference on Personal, Indoor and Mobile Radio Communications (2011

Analyzing the Reduced Required BS Density due to CoMP in Cellular Networks

In this paper we investigate the benefit of base station (BS) cooperation in the uplink of coordinated multi-point (CoMP) networks. Our figure of merit is the required BS density required to meet a chosen rate coverage. Our model assumes a 2-D network of BSs on a regular hexagonal lattice in which path loss, lognormal shadowing and Rayleigh fading affect the signal received from users. Accurate closed-form expressions are first presented for the sum-rate coverage probability and ergodic sum-rate at each point of the cooperation region. Then, for a chosen quality of user rate, the required density of BS is derived based on the minimum value of rate coverage probability in the cooperation region. The approach guarantees that the achievable rate in the entire coverage region is above a target rate with chosen probability. The formulation allows comparison between different orders of BS cooperation, quantifying the reduced required BS density from higher orders of cooperation.Comment: Accepted for presentation in IEEE Globecom Conf., to be held in Atlanta, USA, Dec. 2013. arXiv admin note: text overlap with arXiv:1302.159