QSpace Community: Queen's UniversityInformation
http://hdl.handle.net/1974/6
Queen's UniversityInformation2015-01-30T18:35:26ZAsymptotic Liberation in Free Probability
http://hdl.handle.net/1974/12711
Title: Asymptotic Liberation in Free Probability
Authors: Vazquez Becerra, Josue Daniel
Abstract: Recently G. Anderson and B. Farrel presented the notion of asymptotic liberation on sequences of families of random unitary matrices. They showed that asymptotically liberating sequences of families of random unitary matrices, when used for conjugation, delivers asymptotic freeness, a fundamental concept in free probability theory. Furthermore, applying the Fibonacci-Whitle inequality together with some combinatorial manipulations, they established sufficient conditions on a sequence of families of random unitary matrices in order to be asymptotically liberating.
On the other hand, a theorem by J. Mingo and R. Speicher states that given a graph
$G=(E,V,s,r)$ there exists an optimal rational number $ \mathfrak{r}_{G}$,
depending only on the structure of $G$, such that for any collection of
$n\times n$ complex matrices
$\{ A_{e}=( A_{e} (i_{s(e)},i_{r(e)}) ) \mid e\in E \}$ we have%
\begin{equation*}
\left\vert
\sum_{ i_{v_{1}},\ldots i_{v_{m}} =1 }^{n}
\left( \prod_{e\in E} A_{e} (i_{s(e)},i_{r(e)}) \right)
\right\vert
\leq
n^{\mathfrak{r}_{G}}\prod\limits_{e\in E}\left\Vert A_{e}\right\Vert
\end{equation*}
where $V=\left\{ v_{1},\ldots,v_{m}\right\} $ and $\left\Vert \cdot
\right\Vert $ denotes the operator norm.
In this report we show how to use the latter inequality to prove the same result as G. Anderson and B. Farrell
regarding sufficient conditions for asymptotic liberation.2015-01-26T05:00:00ZThe evolution of altruistic behaviour in homogeneous deme-structured populations
http://hdl.handle.net/1974/12710
Title: The evolution of altruistic behaviour in homogeneous deme-structured populations
Authors: Cabral, Michael2015-01-26T05:00:00ZOptimal Binary Signaling for Correlated Sources over the Orthogonal Gaussian Multiple-Access Channel
http://hdl.handle.net/1974/12637
Title: Optimal Binary Signaling for Correlated Sources over the Orthogonal Gaussian Multiple-Access Channel
Authors: Mitchell, Tyson
Abstract: Optimal binary communication, in the sense of minimizing symbol error rate, with nonequal probabilities has been derived in [1] under various signalling configurations for the single-user case with a given average energy E. This work extends a subset of the results in [1] to a two-user orthogonal multiple access Gaussian channel (OMAGC) transmitting a pair of correlated sources, where the modulators use a single phase or basis function and have given average energies E1 and E2, respectively. These binary modulation schemes fall in one of two categories: (1) transmission sig- nals are both nonnegative, or (2) one transmission signal is positive and the other negative. To optimize the energy allocations for the transmitters in the two-user OMAGC, the maximum a posteriori detection rule, probability of error, and union error bound are derived. The optimal energy allocations are determined numerically and analytically. Both results show that the optimal energy allocations coincide with corresponding results from [1]. It is demonstrated in Chapter 3 that three parameters are needed to describe the source. The optimized OMAGC is compared to three other schemes with varying knowledge about the source statistics, which influence the optimal energy allocation. A gain of at least 0.73 dB is achieved when E1 = E2 or 2E1 =E2. When E1 ≫ E2 again of at least 7 dB is observed.2014-12-03T05:00:00ZApplications of Multitaper Spectral Analysis to Nonstationary Data
http://hdl.handle.net/1974/12584
Title: Applications of Multitaper Spectral Analysis to Nonstationary Data
Authors: Rahim, KARIM
Abstract: This thesis is concerned with changes in the spectrum over time observed in Holocene climate data as recorded in the Burgundy grape harvest date series. These changes represent nonstationarities, and while spectral estimation techniques are relatively robust in the presence of nonstationarity--that is, they are able to detect significant contributions to power at a given frequency in cases where the contribution to power at that given frequency is not constant over time--estimation and prediction can be improved by considering nonstationarity. We propose improving spectral estimation by considering such changes. Specifically, we propose estimating the level of change in frequency over time, detecting change-point(s) and sectioning the time series into stationary segments. We focus on locating a change in frequency domain in time, and propose a graphical technique to detect spectral changes over time. We test the estimation technique in simulation, and then apply it to the Burgundy grape harvest date series. The Burgundy grape harvest date series was selected to demonstrate the introduced estimator and methodology because the time series is equally spaced, has few missing values, and a multitaper spectral analysis, which the methodology proposed in this thesis is based on, of the grape harvest date series was recently published. In addition, we propose a method using a test for goodness-of-fit of autoregressive estimators to aid in assessment of change in spectral properties over time.
This thesis has four components: (1) introduction and study of a level-of-change estimator for use in the frequency domain change-point detection, (2) spectral analysis of the Burgundy grape harvest date series, (3) goodness-of-fit estimates for autoregressive processes, and (4) introduction of a statistical software package for multitaper spectral analysis. We present four results. (1) We introduce and demonstrate the feasibility of a level-of-change estimator. (2) We present a spectral analysis and coherence study of the Burgundy grape harvest date series that includes locating a change-point. (3) We present a study showing an advantage using multitaper spectral estimates when calculating autocorrelation coefficients. And (4) we introduce an R software package, available on the CRAN, to perform multitaper spectral estimation.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2014-10-15 00:52:05.8422014-10-15T04:00:00Z