Department of Mathematics and Statistics Graduate Theses
http://hdl.handle.net/1974/758
20171211T17:10:41Z

Exploring the Evolutionary Relationship Between Virulence and Drug Resistance
http://hdl.handle.net/1974/22761
Exploring the Evolutionary Relationship Between Virulence and Drug Resistance
van den Hoogen, JosÃ©e
One of the many possible negative outcomes of pathogen evolution is the emergence of a virulent drug resistant strain. A pathogen may exploit plasmidmediated 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 plasmidmediated 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 nonsuperinfection effects and superinfection effects.
Our model reveals that plasmidmediated 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 Memory
http://hdl.handle.net/1974/22701
Channel Optimized Scalar Quantization over Orthogonal Multiple Access Channels with Memory
Preusser, Kiraseya
In this thesis, the joint sourcechannel coding method, channel optimized scalar quantization, is applied to realvalued, correlated data. The data is sent over the orthogonal multiple access channel, with nonbinary noisy discrete channels with memory as the two subchannels. 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 endtoend 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 nonbinary noisy discrete channel can be used to model memory and the $2^q$ary output allows for performance improvement via softdecision decoding. It is observed that by taking the source correlation into consideration, significant signaltodistortion 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 signaltodistortion ratio due to source correlation is 10.90 dB.

Pushsum Algorithm on Timevarying Random Graphs
http://hdl.handle.net/1974/22039
Pushsum Algorithm on Timevarying Random Graphs
Rezaeinia, Pouya
In this thesis, we study the problem of achieving average consensus over a random timevarying sequence of directed graphs by extending the class of socalled 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 timevarying Birreducible sequence of probability matrices, we establish convergence rates for the proposed pushsum algorithm.

Global Fluctuations of Random Matrices and the SecondOrder Cauchy Transform
http://hdl.handle.net/1974/22033
Global Fluctuations of Random Matrices and the SecondOrder 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 secondorder free probability perspective, putting a particular emphasis on block Gaussian matrices. Our main contributions are threefold.
First, we provide a formula for the secondorder Cauchy transform of block Gaussian matrices in terms of the corresponding (matrixvalued) Cauchy transform. In order to do this, we introduce a (matrixvalued) secondorder conditional expectation, in the spirit of the conditional expectations in operatorvalued 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 secondorder 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 secondorder 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 secondorder Cauchy transform. In this context, we make explicit the relation between the notion of secondorder limit distribution and the asymptotic Gaussianity of continuously differentiable linear statistics.