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Efficient mixing can be achieved in flows where turbulence is absent, if the trajectories of passively advected particles in the flow are chaotic. The chaotic nature of particle trajectories results in exponential stretching of material lines in the flow. Thus the interface along which diffusion occurs is stretched exponentially leading to efficient mixing. It has been demonstrated recently that regions in flow fields that exhibit poor mixing and non-chaotic particle trajectories can have an important bearing on the overall dynamics and transport of the entire domain. The space-time trajectories of physical stirrers or elliptic points in two dimensional flows can be classified according to braid groups. One can predict a lower bound on the topological entropy (i.e. exponential rate of stretching of material lines) of flows (h_f) by applying the Thurston-Nielsen classification theorems to these braids. This gives a reduced order model for the dynamics of transport of the entire flow field using just a few points. Recent work has shown that this methodology can be used to estimate a lower bound on h_f using the braids formed by Almost Cyclic Sets (ACS) in certain periodic Stokes' flows. These ACS are closely related to Almost Invariant Sets (AIS) which are identified using a probabilistic set oriented method that makes use of the descritised Perron-Frobenius operator of the flow map. This work extends this approach to flows at non-zero Reynolds numbers, which take into account the effects of inertia. The role of Finite Time Coherent Structures (FTCS) in the dynamics of flow fields is also investigated. Unlike ACS, the FTCS approach is more general as it can be applied to aperiodic flow fields. Further, the relationship between mixing efficiency and the topological entropy of flow fields at non-zero Reynolds numbers is also studied. / Ph. D.

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<p>In this thesis, we study several applications of self-affine sets to wavelet theory. Five major topics are considered here: wavelet sets (scaling sets), multiwavelet sets (generalized scaling sets), self-affine tiles, integral self-affine multi-tiles, self-affine sets. We divide the thesis into six chapters to discuss these topics. In Chapter 1, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup(K+d_2)$, where $B=A^t$ and $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. We show that it must be a constant in dimension $n=1$ or $2$ and it is bounded by $2\lvert K\rvert$ for any $n$. This result shows that all $A$-dilation self-affine scaling sets must be $A$-dilation MRA scaling sets in dimensions one and two. There exist results on the connection between the theory of wavelets and the theory of integral self-affine tiles and in particular, on the construction of wavelet bases using integral self-affine tiles. However, there are many non-integral self-affine tiles which can also yield wavelet basis. In Chapter 2 and Chapter 3, we give a complete characterization of all two dimensional $A$-dilation scaling sets $K$ such that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup (K+d_2)$ for some $d_1, d_2\in\mathbb{R}^2$, where $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. In Chapter 2, we deal with a particular case where $0\in\{d_1,d_2\}$, i.e. a self-affine tile $K$ satisfies $BK=K\bigcup (K+d)$ for some $d\in\mathbb{R}^2$. Chapter 3 is devoted to the general case with $d_1, d_2\in\mathbb{R}^2$. Moreover, we give a sufficient condition for a self-affine tile, possibly non-integral, to be an MRA scaling set in Chapter 3. Gabardo and Yu first considered using integral self-affine tiles in the Fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain self-similarity properties. In Chapter 4, we generalize their results to the integral self-affine multi-tiles setting. We characterize some analytic properties of integral self-affine multi-tiles under certain conditions. We also consider the problem of constructing (multi)wavelet sets using integral self-affine multi-tiles. Suppose that a measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with an $n\times n$ integral expansive matrix $B$. To our knowledge, no one considered how to represent an integral self-affine $\mathbb{Z}^n$-tiling set as the disjoint union of prototiles. In Chapter 5, we provide an algorithm to decompose $K$ into disjoint pieces $K_j$ which satisfy $K=\displaystyle\bigcup K_j$ such that the collection of the sets $K_j$ is an integral self-affine collection associated with matrix $B$ and the number of pieces $K_j$ is minimal. Using this algorithm, we can determine whether a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with any given $n\times n$ integral expansive matrix $B$. Furthermore, the minimal decomposition we provide is unique. Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In Chapter 6, we not only consider the problem how to compute the Lebesgue measure of self-affine sets $K(B,\mathcal{D})$, but also consider the Hausdorff measure for those with zero Lebesgue measure under the assumption that $K(B,\mathcal{D})$ is a self-similar set. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $\mu=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}\delta_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $\mu$.</p> / Doctor of Science (PhD)

This thesis examines the historical array of 'social' and 'technical' factors that have shaped the development and evolution of Korean national Coded Character Set (CCS) standards. CCS standards refer to a layer of compatibility standards which specify rules for digital representation of textual data at the most basic level of Information and Communication Technologies (ICTs). The effective and efficient operation of information processing, storage, and exchange is thus dependent on the employment of technically sound, economically viable, and culturally adequate CCS standards at national, regional and international levels. Historicaily, the CCS standards had emerged around the cultural presumptions and practices of the US and Western Europe due to the economic and tcchnical dominance of the region from the formative stage of ICTs development. As the need for global information infrastructure and multilingual information processing has been growing, however, the international CCS standards regime has evolved (from 1SO 646, to ISO 2022, and JSO/IEC 10646-1) to incorporate various national scripts around the world, and the issues have arisen over the adequate representation of these scripts within the international standards regime. For example, the incorporation of East Asian scripts, such as Chinese, Japanese, and Korean, presents a formidable challenge with their exceptionally large repertoire. In particular, the design and implementation of Korean national CCS standards, normally a exclusive domain of experts and bureaucrats, had caused a series of heated public controversies during the 1980s and 1990s. Despite the intensity of disputes and the breadth of participation in the controversies on Korean national CCS standards, the standardisation process had not been subject to a detailed socio-economic analysis, the lack of which allowed deterministic and simplistic speculations to appear, implying technological rationality, economic imperatives or corporate strategics alone have guided the CCS standards along a linear development path with increasingly larger and more powerful standards replacing previous ones. Drawing on the social shaping of technology perspective, the case study examines the evolution of Korean national CCS standards, focusing both on the process in which a standards emerges as a result of network building activities and alliance formation of various actors, and on the changes in immediate and broader contexts around the standardisation which directly and indirectly affect the interests alliances and evolution of standards. Contrary to the deterministic and simplistic perspectives, the case study suggests a structured but also dynamic social shaping process of the Korean national CCS standards. Four major themes forwarded in the case study are as below: First, the case identified a received view on the Korean controversy which can be characterised as 'technological fix on cultural problem' in a sense that technical challenge experienced in Korean character encoding was a product of distinctive local culture and the problem was fixed by the steady advances in the information technology. Without denying the importance of the state of technological capabilities, however, the case shows that social choices had been made both in international and national standards and had critical roles in shaping various controversy and whole national standards sdting process. Second, the case study identifies two contrasting modes of standardisation, 'technicisation' and 'politicisation,' and examines how the fluctuation between them has affected the development of Korean national standards setting process. In the discourse of technicised standardisation, technological knowledge is accepted as neutral, asocial 'hard fact.' Accordingly, the social choices are obscured and the standardisation process is to be dominated by the negotiation among disinterested experts over the relative technical merits of standards. Under the politicised mode of standardisation,' the political nature of CCS standards design - conflicting values and incongruent ascription of technological properties as a result - is brought forward. The standardisation is characterised by the formation of and competition among interests alliances. The outcome of standardislltion seems to be dependent on which mode is dominant as well as who prevails in each mode. Third, the case study raises a question about the relationship between the standards and interests embedded in them. The ascription of certain technological properties to standards and the interests alliances built around them proved unstable and dynamic. Both of them seem to be influenced by network building activities of llctors and their backdrop, a specific configuration of economic, social, cultural, political and technological factors, enabling and constraining the activities of actors involved in the standardisation process. As the makeup of the configuration changes for various reasons, - for example, globalised software market, social movements, surge of nationalism, political democratisation, advances in related technological field - the meanings attached by the actors to the standard also shifts, and the interests alliances based on them are unmade and replaced by new ones, producing a series of character set standards. Fourth, the study also draws attention to the complexity involved in the nlltional standardisation process and the challenge faced by the social research into those intricate social shaping process. The standardisation process involved many actors at different levels and across various geographical locales. Also there had been recurrent but unpredictable changes in the relationship among the actors and between the actors and artefacts. A recent trend in the social shaping approach - a call for a decentralised concept of actor and the transforming terrain of innovation - was found hrlpful to meet the challenge. In particular, the concept of 'development arena' was found useful to meet the challenge and to understand the case in balance between the actions of different 'modes of performance' and the contexts of varying 'configurations' of heterogeneous clements.

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published_or_final_version / Computer Science / Master / Master of Philosophy

This thesis considers problems concerning Latin squares and sets of mutually quasiorthogonal Latin squares (MQOLS). We show how MQOLS are related to a number of other designs and establish bounds on Nq(n), the maximum number of Latin squares of order n in a mutually quasi-orthogonal set. We report the number of quasi-complete mappings admitted by each group of order 15 or less, and explain the surprising result that each of the non-cyclic groups of order 8 possesses exactly 384 complete mappings. For each group G of order 15 or less, we identify the sizes of all maximal sets of mutually orthogonal orthomorphisms. We also identify a number of new maximal sets for larger groups. We present a method to determine all proper, maximal sets of MQOLS of order n and carry this out for n < 6. Also we present a search for 3 MQOLS of order 10, which, whilst not identifying such a set, led to the identification of all resolutions of each (10,3, 2)-balanced incomplete block design. We give a construction for MQOLS based on groups, and use this to determine new sets of 2n - 1 MQOLS of order 2n based on two infinite classes of group. Existence results for MQOLS based on groups are also extended. Two constructions for (n x n)/k semi-Latin squares are given, one of which provides some new A-, D- and E-optimal examples with k > n which out-perform the existing A-, D- and E-optimal examples in the E'-criteria. Finally we consider the problem of determining invertible directed terraces of each non-abelian group of order < 21, and in so doing construct the first doubly balanced bipartite tournament of odd order.

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