Using a version of Elliot Eisner's connoisseurship, this paper examines the current Manitoba Framework document for high school mathematics, finds it wanting, and proposes a new way of considering curriculum, using high school mathematics as an example space. Drawing inspiration from the writing of John Dewey, the model makes use of complexity theory as described by Brent Davis and Elaine Simmt (2003) to contribute to curriculum theorizing. The Manifold & Intention Model uses major themes, termed Manifolds, as organizing devices for creating and using curricula. The underlying social, mathematical and educational assumptions surrounding the curriculum are opened to scrutiny in the Intention. The content that appears in a curriculum must meet the Content Evaluation Criteria and in the Common Content and Local Content, specific learning outcomes are eschewed, replaced by exemplars. The Manifold & Intention Model is a social and generative way to create curriculum.

The notion of communication complexity was introduced by Yao in his seminal paper [Yao79]. In [BFS86], Babai Frankl and Simon developed a rich structure of communication complexity classes to understand the relationships between various models of communication complexity. This made it apparent that communication complexity was a self-contained mini-world within complexity theory. In this thesis, we study the place of regular languages within this mini-world. In particular, we are interested in the non-deterministic communication complexity of regular languages. / We show that a regular language has either O(1) or O(log n) non-deterministic complexity. We obtain several linear lower bound results which cover a wide range of regular languages having linear non-deterministic complexity. These lower bound results also imply a result in semigroup theory: we obtain sufficient conditions for not being in the positive variety Pol(Com). / To obtain our results, we use algebraic techniques. In the study of regular languages, the algebraic point of view pioneered by Eilenberg ([Eil74]) has led to many interesting results. Viewing a semigroup as a computational device that recognizes languages has proven to be prolific from both semigroup theory and formal languages perspectives. In this thesis, we provide further instances of such mutualism.

In this dissertation, I examine trajectories of cultural evolution among complexhunter-gatherers and middle range societies. Broadly, I consider the theoretical issuesrelated to these two areas of study and how we should conceptualize the study of socioculturalevolution in societies organized at this scale. I apply these ideas to the study ofthe prehistoric hunter-gatherers who occupied Sapelo Island, Georgia, U. S. A.Specifically, I examine the Archaic period (4200 – 3000 B. P.) occupation of the SapeloShell Ring complex, located on the western side of the island. In particular, I study issuesof sedentism, settlement aggregation, mound construction, and the emergence of socialinequality as they relate to shell rings in the southeastern United States, as well as otherareas of the world. One of the central problems for studying these sites is whether shellrings form by gradual accumulation or by intentional construction and the concomitantsocial formations associated with these two different behaviors. Using geophysicalsurvey, artifact distributions, and radiocarbon dating, I examine the use and nature ofspace at the site as well as site formation processes. I present the results of both thegrowth band analysis on clams and the isotopic analysis on clams and oysters from thesite in order to address season of occupation. In addition to this new data, a reanalysis ofprevious excavations is presented. Combined, these data lend important insights intodifferent dimensions of socio-political complexity. Specifically, these data suggest thatthe Sapelo Shell Ring complex population was relatively large for its time. It addition, itseems that at least some portion of the population occupied the site year-round. Despiteit large population size and reduced mobility the occupants of the site maintained at leastsome degree of egalitarian social relations.

Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer's βhierarchy, and Cai and Chen's guess-and-check model.

This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems. A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs. The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case. Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest. In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.

Image similarity or distortion assessment is fundamental to a broad range of applications throughout the field of image processing and machine vision. These include image restoration, denoising, coding, communication, interpolation, registration, fusion, classification and retrieval, as well as object detection, recognition, and tracking. Many existing image similarity measures have been proposed to work with specific types of image distortions (e.g., JPEG compression). There are also methods such as the structural similarity (SSIM) index that are applicable to a wider range of applications. However, even these "general-purpose" methods offer limited scopes in their applications. For example, SSIM does not apply or work properly when significant geometric changes exist between the two images being compared. The theory of Kolmogorov complexity provides solid groundwork for a generic information distance metric between any objects that minorizes all metrics in the class. The Normalized Information Distance (NID) metric provides a more useful framework. While appealing, the challenge lies in the implementation, mainly due to the non-computable nature of Kolmogorov complexity. To overcome this, a Normalized Compression Distance (NCD) measure was proposed, which is an effective approximation of NID and has found successful applications in the fields of bioinformatics, pattern recognition, and natural language processing. Nevertheless, the application of NID for image similarity and distortion analysis is still in its early stage. Several authors have applied the NID framework and the NCD algorithm to image clustering, image distinguishability, content-based image retrieval and video classification problems, but most reporting only moderate success. Moreover, due to their focuses on ! specific applications, the generic property of NID was not fully exploited. In this work, we aim for developing practical solutions for image distortion analysis based on the information distance framework. In particular, we propose two practical approaches to approximate NID for image similarity and distortion analysis. In the first approach, the shortest program that converts one image to another is found from a list of available transformations and a generic image similarity measure is built on computing the length of this shortest program as an approximation of the conditional Kolmogorov complexity in NID. In the second method, the complexity of the objects is approximated using Shannon entropy. Specifically we transform the reference and distorted images into wavelet domain and assume local independence among image subbands. Inspired by the Visual Information Fidelity (VIF) approach, the Gaussian Scale Mixture (GSM) model is adopted for Natural Scene Statistics (NSS) of the images to simplify the entropy computation. When applying image information distance framework in real-world applications, we find information distance measures often lead to useful features in many image processing applications. In particular, we develop a photo retouching distortion measure based on training a Gaussian kernel Support Vector Regression (SVR) model using information theoretic features extracted from a database of original and edited images. It is shown that the proposed measure is well correlated with subjective ranking of the images. Moreover, we propose a tone mapping operator parameter selection scheme for High Dynamic Range (HDR) images. The scheme attempts to find tone mapping parameters that minimize the NID of the HDR image and the resulting Low Dynamic Range (LDR) image, and thereby minimize the information loss in HDR to LDR tone mapping. The resulting images created by minimizing NID exhibit enhanced image quality.

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.

As a psychotherapist, my work with thousands of clients over the last two decades has illuminated a theme––when there is a misalignment between one’s values and beliefs and one’s lived experience, dis-ease, discomfort, dissonance and self-alienation take hold. The corollary often being true: An authentic life––living closely aligned with one’s beliefs and values––promotes experiences of health and happiness. Psychological research supports the notion that well-being and happiness are products of living authentically (Brown, Ryan, Laguardia, & Rawsthorne, 2005; Kernis & Goldman, 2006; Wood, Linley, Maltby, Baliousis, & Joseph, 2008). This qualitative hermeneutic phenomenological study was a response to my curiosity about the practice of living an aligned life and answered the overarching question: “What is the experience of the conscious practice of authenticity?” with the sub-question: “How does the conscious practice of authenticity relate to transformative learning?” Relevant literature is reviewed in the areas of transformative learning, authenticity, and complexity thinking: particularly the quality of emergence represented in the methodological orientation of the hermeneutic circle. Data collection strategies including interviews, an experiential component of “in the moment” recording, researcher field notes in the form of a commonplace book, and a web-based forum, as well as close readings of relevant literature. Data interpretation was based on a six stage integrative and creative hermeneutic analysis (adapted from Ajjiwa & Higgs, 2007). The stages of interpretation were (1) immersion, (2) understanding, (3) abstraction, (4) synthesis and theme development, (5) illumination and illustration of the phenomenon, (6) integration and critique. The research findings introduced the terms querencia (home or source in a person) and querencetic living (living from one’s inner source or knowing). The findings indicate that querencetic living generates wellness, happiness and peace and is comprised of four parts: 1) knowing our querencia (values, beliefs, needs, feelings), 2) being attuned to our querencia, 3) trusting our knowing/not knowing/changing knowing and 4) enacting our querencia through voice and action. The nature of our querencia is emergent (ever-changing and fluid) where our relationship to our querencia (being attuned, trusting and enacting) is unbending. / Thesis (Ph.D, Education) -- Queen's University, 2014-05-29 11:15:55.339

Using a version of Elliot Eisner's connoisseurship, this paper examines the current Manitoba Framework document for high school mathematics, finds it wanting, and proposes a new way of considering curriculum, using high school mathematics as an example space. Drawing inspiration from the writing of John Dewey, the model makes use of complexity theory as described by Brent Davis and Elaine Simmt (2003) to contribute to curriculum theorizing. The Manifold & Intention Model uses major themes, termed Manifolds, as organizing devices for creating and using curricula. The underlying social, mathematical and educational assumptions surrounding the curriculum are opened to scrutiny in the Intention. The content that appears in a curriculum must meet the Content Evaluation Criteria and in the Common Content and Local Content, specific learning outcomes are eschewed, replaced by exemplars. The Manifold & Intention Model is a social and generative way to create curriculum.

The Greater Victoria School District has undergone a reconfiguration to include middle schools in their public education system. This reconfiguration has students from grades six to eight learning in a new setting. Cedar Hill Middle school is an example of one of the middle schools that is now in its eighth year of existence. The school is the centre of the learning community that is made up of students, staff and parents. As part of an ongoing review and reflection, this research used an Appreciative Inquiry methodology to try to identify the strengths found within the learning community. Through a dialogic interview process, a sample of staff, parents and most importantly students were interviewed to collect the best experiences that they had experienced as a part of the learning community. The data was open coded to identify best experiences, categorize them, and then identify the themes or relationships between the experiences of the different members of the learning community. These relationships were then used to create a set of powerful propositions that can be used to potentially guide the growth of the Cedar Hill learning community. Through this process two main themes emerged. The first was the importance of connections within the system and the second was the need for diversity within the system. This supported the assumption that the learning community was indeed a complex system and reinforced the idea that Appreciative Inquiry is a tool that can be used to support and develop complex systems. / Graduate