## Search

Now showing items 1-10 of 91

#### Error-Tolerant Coding and the Genetic Code

(University of Waterloo, 2006)

The following thesis is a project in mathematical biology building upon the so-called "error minimization hypothesis" of the genetic code. After introducing the biological context of this hypothesis, I proceed to develop ...

#### Two- and Three-Dimensional Coding Schemes for Wavelet and Fractal-Wavelet Image Compression

(University of Waterloo, 2001)

This thesis presents two novel coding schemes and applications to both two- and three-dimensional image compression. Image compression can be viewed as methods of functional approximation under a constraint on the amount ...

#### Approximate Private Quantum Channels

(University of Waterloo, 2006)

This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for ε-randomizing maps,

**n**+ 2log(1/ε) +**c**bits required to ε-randomize an arbitrary**n**-qubit state by improving a scheme of Ambainis and Smith [5] based on small bias spaces [16, 3]. We show by a probabilistic argument that in fact the great majority of random schemes using slightly more than this many bits of key are also ε-randomizing. We provide the first known nontrivial lower bound for ε-randomizing maps, and develop several conditions on them which we hope may be useful in proving stronger lower bounds in the future....#### On the Role of Partition Inequalities in Classical Algorithms for Steiner Problems in Graphs

(University of Waterloo, 2006)

The Steiner tree problem is a classical, well-studied, $\mathcal{NP}$-hard optimization problem. Here we are given an undirected graph $G=(V,E)$, a subset $R$ of $V$ of terminals, and non-negative costs $c_e$ for all ...

#### Combinatorial Constructions for Transitive Factorizations in the Symmetric Group

(University of Waterloo, 2004)

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (σ<i>r</i>,. . . ,σ1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product ...

#### Numerical Stability in Linear Programming and Semidefinite Programming

(University of Waterloo, 2006)

We study numerical stability for interior-point methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the difficulties inherent in current methods and present robust algorithms. <br /><br /> We start with the error bound analysis of the search directions for the normal equation approach for LP. Our error analysis explains the surprising fact that the ill-conditioning is not a significant problem for the normal equation system. We also explain why most of the popular LP solvers have a default stop tolerance of only 10<sup>-8</sup> when the machine precision on a 32-bit computer is approximately 10<sup>-16</sup>. <br /><br /> We then propose a simple alternative approach for the normal equation based interior-point method. This approach has better numerical stability than the normal equation based method. Although, our approach is not competitive in terms of CPU time for the NETLIB problem set, we do obtain higher accuracy. In addition, we obtain significantly smaller CPU times compared to the normal equation based direct solver, when we solve well-conditioned, huge, and sparse problems by using our iterative based linear solver. Additional techniques discussed are: crossover; purification step; and no backtracking. <br /><br /> Finally, we present an algorithm to construct SDP problem instances with prescribed strict complementarity gaps. We then introduce two

**measures of strict complementarity gaps**. We empirically show that: (i) these measures can be evaluated accurately; (ii) the size of the strict complementarity gaps correlate well with the number of iteration for the SDPT3 solver, as well as with the local asymptotic convergence rate; and (iii) large strict complementarity gaps, coupled with the failure of Slater's condition, correlate well with loss of accuracy in the solutions. In addition, the numerical tests show that there is no correlation between the strict complementarity gaps and the geometrical measure used in [31], or with Renegar's condition number....#### Classification of Nilpotent Lie Algebras of Dimension 7 (over Algebraically Closed Field and R)

(University of Waterloo, 1998)

This thesis is concerned with the classification of 7-dimensional nilpotent Lie algebras. Skjelbred and Sund have published in 1977 their method of constructing all nilpotent Lie algebras of dimension <i>n</i> given those ...

#### Interior-Point Algorithms Based on Primal-Dual Entropy

(University of Waterloo, 2006)

We propose a family of search directions based on primal-dual entropy in the context of interior point methods for linear programming. This new family contains previously proposed search directions in the context of ...

#### Nonlinear Dimensionality Reduction with Side Information

(University of Waterloo, 2006)

In this thesis, I look at three problems with important applications in data processing. Incorporating side information, provided by the user or derived from data, is a main theme of each of these problems. <br /><br ...

#### Variational Spectral Analysis

(University of Waterloo, 2000)

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values ...