Department of Mathematics and Statistics Graduate Theseshttp://hdl.handle.net/1974/7582017-10-17T16:20:11Z2017-10-17T16:20:11ZExploring the Evolutionary Relationship Between Virulence and Drug Resistancevan den Hoogen, Joséehttp://hdl.handle.net/1974/227612017-09-27T07:16:59ZExploring the Evolutionary Relationship Between Virulence and Drug Resistance
van den Hoogen, Josée
One of the many possible negative outcomes of pathogen evolution is the emergence of a virulent drug resistant strain. A pathogen may exploit plasmid-mediated gene transfer to gain drug resistance and virulence genes in an attempt to become the exclusive infection. To answer questions on the possible emergence of a virulent drug resistant strain we constructed a nine dimensional SIS model with treatment. We used plasmid-mediated gene transfer by way of superinfection to distribute virulence and drug resistance genes. By examining the resulting system of differential equations at the two extremes of treatment level (%0 and %100), we determined the stability of the single strain endemic infection equilibrium points of interest. In the absence of treatment we determined a set of ancestral conditions which provided constraints on the model parameters. In the presence of treatment we focused on the ability of the
virulent drug resistant (AB) strain to invade the avirulent drug resistant (B) strain. We were able to express these invasion conditions in terms of novel reproductive ratios that incorporated both non-superinfection effects and superinfection effects.
Our model reveals that plasmid-mediated gene transfer by way of superinfection increases the parameter space allowing for virulence to evolve in a drug resistant population. However, virulence in a drug resistant population is not always more
likely to evolve in comparison to virulence in a drug sensitive population.
Channel Optimized Scalar Quantization over Orthogonal Multiple Access Channels with MemoryPreusser, Kiraseyahttp://hdl.handle.net/1974/227012017-09-19T15:53:07ZChannel Optimized Scalar Quantization over Orthogonal Multiple Access Channels with Memory
Preusser, Kiraseya
In this thesis, the joint source-channel coding method, channel optimized scalar quantization, is applied to real-valued, correlated data. The data is sent over the orthogonal multiple access channel, with non-binary noisy discrete channels with memory as the two sub-channels. Three different schemes are compared for this system: in the first scheme encoding and decoding are performed independently, in the second scheme encoding is done independently and joint decoding is carried out, and the third scheme is with jointly optimized encoders and joint decoding. The goal is to derive optimality conditions that will result in a lower end-to-end distortion. To this end, necessary optimality conditions for the two latter schemes are fully derived and implemented for the bivariate Gaussian and bivariate Laplacian distributions of varying correlation.\\
The first and second methods are then further compared, by implementing them for an image transmission system. Here the images are first processed with the 2 dimensional discrete cosine transform, and then encoded using channel optimized scalar quantization. At the decoder, two different methods are used, the independent and joint decoder.\\
In addition to comparing the different coding methods, various channels characteristics are exploited. For example, the non-binary noisy discrete channel can be used to model memory and the $2^q$-ary output allows for performance improvement via soft-decision decoding. It is observed that by taking the source correlation into consideration, significant signal-to-distortion ratio gains can be achieved. For example, the highest gain incurred from the third scheme is when the bivariate Gaussian is compressed at rate 2, where the gain in signal-to-distortion ratio due to source correlation is 10.90 dB.
Push-sum Algorithm on Time-varying Random GraphsRezaeinia, Pouyahttp://hdl.handle.net/1974/220392017-09-15T13:19:11ZPush-sum Algorithm on Time-varying Random Graphs
Rezaeinia, Pouya
In this thesis, we study the problem of achieving average consensus over a random time-varying sequence of directed graphs by extending the class of so-called push- sum algorithms to such random scenarios. Provided that an ergodicity notion, which we term the directed infinite flow property, holds and the auxiliary states of nodes are uniformly bounded away from zero infinitely often, we prove the almost sure convergence of the evolutions of this class of algorithms to the average of initial states. Moreover, for a random sequence of graphs generated using a time-varying B-irreducible sequence of probability matrices, we establish convergence rates for the proposed push-sum algorithm.
Global Fluctuations of Random Matrices and the Second-Order Cauchy TransformDiaz Torres, Mariohttp://hdl.handle.net/1974/220332017-09-15T13:19:27ZGlobal Fluctuations of Random Matrices and the Second-Order Cauchy Transform
Diaz Torres, Mario
In this thesis we study the global fluctuations of random matrices (i.e., the covariance of two traces) from a second-order free probability perspective, putting a particular emphasis on block Gaussian matrices. Our main contributions are threefold.
First, we provide a formula for the second-order Cauchy transform of block Gaussian matrices in terms of the corresponding (matrix-valued) Cauchy transform. In order to do this, we introduce a (matrix-valued) second-order conditional expectation, in the spirit of the conditional expectations in operator-valued free probability theory.
Second, we establish a criterion, based on the positivity of a cluster function, that guarantees the existence of a particular integral representation for the second-order Cauchy transform. This approach is based on some relations between the moments and fluctuation moments of a measure on the plane.
Finally, we prove that under some conditions the covariance of resolvents converges to the second-order Cauchy transform. In fact, these conditions also ensure the convergence of the covariance of analytic linear statistics to a certain contour integral depending on the second-order Cauchy transform. In this context, we make explicit the relation between the notion of second-order limit distribution and the asymptotic Gaussianity of continuously differentiable linear statistics.