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For many applications, such as cryptography and digital communications, binary sequences with certain specific properties are required. These include a balance of 0's and 1's in a period, ideal runs frequencies, good auto- and cross-correlation spectra, and high linear complexity. Perfect Linear Complexity Profile sequences (PLCPs) have the linear complexity of all subsequences (starting with the first bit) equal to half the length of the subsequence (this is the expected value for a random sequence). We investigate the density - proportion of ones - of finite length PLCPs, both in general and for specific examples. We gain results on the average, maximal and minimal densities, as well as their limits as the length tends to infinity. We also study whether the PLCP property is preserved under various decimations. PLCPs are characterised by a simple linear recurrence modulo 2. We look at similar "nearly" perfect profiles and characterise sequences with these profiles in terms of similar recurrences. Also sequences with a PLCP up to a point and then constant complexity are characterised in terms of the convergents in the continued fraction expansion of the generating function of PLCPs, and we look briefly at their corresponding periods. Sequences with bounded jumps in their linear complexity are discussed and a method of generating them is suggested. The interleaving of shifts of a sequence with out-of-phase auto-correlation equal to -1 and balance, in a specific order, seems to be a fundamental method of generating longer sequences with this auto-correlation property. It is shown that two pairs of families of these sequences, derived in different ways, are in fact equivalent. The analysis highlights the general method mentioned above, and so provides examples of families of sequences with 2-valued auto-correlation by changing the ingredients in the interleaving pattern. We also look at the cross-correlation of sequences with this interleaved structure.

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j, k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j, k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)-mixed graphs) and k−edge-coloured graphs ((0, k)−mixed graphs). A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. An m−colouring of a (j, k)−mixed graph is a homomorphism from that graph to a target with m vertices. The (j, k)−chromatic number of a (j, k)−mixed graph is the least m such that an m−colouring exists. When (j, k) = (0, 1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring. Similarly, when (j, k) = (1, 0) and (j, k) = (0, k) these definitions are consistent with the usual definitions of homomorphism and colouring for oriented graphs and k−edge-coloured graphs, respectively. In this thesis we study the (j, k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1, 0)−chromatic number, more commonly called the oriented chromatic number, and the (0, k)−chromatic number. In examining oriented graphs, we provide improvements to the upper and lower bounds for the oriented chromatic number of the families of oriented graphs with maximum degree 3 and 4. We generalise the work of Sherk and MacGillivray on the 2−dipath chromatic number, to consider colourings in which vertices at the ends of iii a directed path of length at most k must receive different colours. We examine the implications of the work of Smolikova on simple colourings to study of the oriented chromatic number of the family of oriented planar graphs. In examining k−edge-coloured graphs we provide improvements to the upper and lower bounds for the family of 2−edge-coloured graphs with maximum degree 3. In doing so, we define the alternating 2−path chromatic number of k−edge-coloured graphs, a parameter similar in spirit to the 2−dipath chromatic number for oriented graphs. We also consider a notion of simple colouring for k−edge-coloured graphs, and show that the methods employed by Smolikova ́ for simple colourings of oriented graphs may be adapted to k−edge-coloured graphs. In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs. / Graduate

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This dissertation addresses the role of ritual practices in the emergence of complex forms of social organization during the Late Archaic Period of the Central Andes (ca. 3000 1500 B.C.). This theme is approached through description and analysis of ritual architecture remains recovered in excavations at the site of Cerro Lampay, located in the Fortaleza Valley, within the North Central Coast of Peru.The emergence of social complexity is approached from the perspective of Practice Theory, noting the relevance of ritual practices in the generation, reproduction, and/or transformation of social conditions of existence.Following these theoretical principles, archaeological information is analyzed through a methodological frame built to understand the performative aspects of ritual and its material manifestations. A particular emphasis is put on the analysis of architectural remains, which are analyzed from proxemics and space syntax perspectives, in order to define the patterns of human interaction produced during the conduct of ritual.The inference of behavioral patterns conducted within construction events and ritual performances have allowed me to propose a scenario of a community with emergent leaders and a dual organization, which was responsible for the building, use, and closure of the architectural compounds found at Cerro Lampay. Ritual practices such as conspicuous consumption and feasting played a key role in the development of social dynamics and might have been a significant power source for the emergent leaderships.

Despite the acceptance of best practice as a standard for quality health care delivery, the exact nature of what constitutes best practice does not share universal definition or application. The purpose of this dissertation is to propose an integrative conceptual representation of nursing best practice from the philosophical perspective of complexity science.A five-step concept clarification approach was used to identify the concept, systematize observations and descriptions, develop an operational definition, construct a model, and formulate hypotheses. An expert panel explored preliminary validity of the definition and model.Purposive recruitment of clinicians and scholars was conducted for expert panel membership. The expert panel rated the strength of the model dimensions of adequacy, representative-ness, consistency, clarity, simplicity, generalness, accessibility, importance, and relevance, as well as interest in development and application. Narrative data from open-ended questions was incorporated into model refinement.Clinician properties and context properties emerged as two principle domains of interdependent influence. Key dynamic processes included critical thinking by which clinicians operationalize properties into practice choices, and informative reflection by which the organization monitors and improves performance through information flow and learning. All aspects of the conceptual model, with the exception of consistency of relationships, were ultimately rated as strengths by the expert panel. Relationships among constructs were identified as complex, diverse, and difficult to isolate. Expert perception was that clinician and context properties most likely equally influence nursing best practice, but that context properties may have greater influence than clinician properties over time.This model incorporates a full range of interdependence across clinician and context domains of influence. This model requires further operationalization of constructs prior to formal validity testing. The application of complexity science introduces challenges to research and measurement in the study of complex adaptive systems. The model presented in this dissertation provides a perspective from which a better understanding of health care system interdependencies may arise.

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Entropy measures have been widely used to quantify the complexity of theoretical and experimental dynamical systems. In this thesis, two novel entropy measures are developed based on using coarse quantization to classify and compare dynamical features within a time series; quantized dynamical entropy (QDE) and a quantized approximation of sample entropy (QASE). Following this, comprehensive guidelines for the quantification of complexity are presented based on a detailed investigation of the performance characteristics of the two developed measures and three existing measures; permutation entropy, sample entropy and fuzzy entropy. The sensitivity of the considered entropy measures to changes in dynamics was assessed using the case study of characterizing bipedal walking gait dynamics. Based on the analysis conducted, it was found that sample entropy and fuzzy entropy, while computationally inefficient, provide the best overall performance. In instances where computational efficiency is vital, QDE and QASE serve as viable alternatives to existing methods.