Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht

Clinical Decision-Making: Developing a 4 C Model Using Graph Theoretic Approach

The purpose of this paper is to propose a graph-theoretic mathematical model to measure how conducive the environment of a hospital is for decision-making. We propose a 4-C model, developed from four interacting factors: confidence, complexity, capability, and customer. In this graph-theoretic model, abstract information regarding the system is represented by the directed edges of a graph (or digraph), which together depict how one factor affects another. The digraph yields a matrix model useful for computer processing. The net effect of different factors and their interdependencies on the hospital's decision-making environment is quantified and a single numerical index is generated. This paper categorizes all the major factors that influence clinical decision-making and attempts to provide a tool to study and measure their interactions with each other. Each factor and each interaction among factors are to be quantified by healthcare experts according to their best judgment of the magnitude of its effect in a local hospital environment.A hospital case study is used to demonstrate how the 4-C model works. The graph-theoretic approach allows for the inclusion of new factors and generation of alternative environments by a combination of both qualitative and quantitative modeling. The 4-C model can be used to create both a database and a simple numerical scale that help a hospital set customized guidelines, ranging from patient admittance procedures to diagnostic and treatment processes, according to its specific situation. Implementing this methodology systematically can allow a hospital to identify factors that will lead to improved decision-making as well as identifying operational factors that present roadblocks

The significant others of subjective norm - A scientometric study of subjective norm in IS top-journals over two decades

The role of grounded approaches has been advocated for long in IS research. However, the inherent subjectivity of such approaches and the apparent lack of a basis to validate or even replicate such research has often been the subject of debate among IS researchers. As a result, many IS researchers tend to fall back on variance-theoretic approaches to conceptualize, design and operationalize their research. In this paper, we show how a grounded approach, interpretive structural modeling (ISM), can be used to qualitatively elicit individual cognitive structures. Further, we show how it can be applied to derive the shared aspects of such a structure across many individuals. We use the well-known technology acceptance model (TAM) to demonstrate the utility of our approach. We conclude the paper by discussing the strengths and weaknesses of this approach

Educating the Masses: Human Attitudes Affecting Reef Health

About two thirds of the world’s population lives within 60 km of a coastline, however many members of the public do not recognize the importance of the nearby coral reef ecosystems. Although reef degradation is currently occurring at alarming rates, there is hope for the future of coral reef health. It is proven that humans have a large effect on the current health of coral reefs. This thesis aims to determine how marine scientists and educators can best influence the general public to affect behaviors to improve reef health. To do this involves taking analyzing the many effects of human attitudes and behaviors on reefs. First, we study major approaches used by contemporary marine scientists to educate the broader public about marine ecology and reef degradation issues in particular. We will identify those initiatives and methods which show the most promise for altering human behaviors which threaten reef health. Secondly, we aim to define “populations of educational interest” by examining census data and other literature, which lead educators to determine important audiences that need to be educated. Thirdly, we will determine which behaviors and attitudes will have the largest effects on reef health. Based on current research, a digraph (directed graph) was created to model the influences that different attitudes have on different measures of reef health. The digraph model was then translated into a mathematical model which simulates a pulse process to show the effect of changes in this model. Three scenarios were developed. Influencing the general public to decrease their emission of greenhouse gasses would have positive effects on fish density, coral cover, reef framework and diversity. Human population itself was an important factor affecting reef health, and with changes in attitudes, if population decreases, reef health could be improved. Also, increasing education to influence the effect that boaters and divers have on physical damage would positively impact all reef health indicators. Information gained from the model, as well as the information gained from determining the “populations of interest” and furthering current educational outreach has the potential to allow educators better framing of future reef programs as well as alter aspects of current programs in order to obtain maximum results in behavior and attitude change, resulting in positive effects on reef health for the future

Interdependence and dynamics of essential services in an extensive risk context: a case study in Montserrat, West Indies

The essential services that support urban living are complex and interdependent, and their disruption in disasters directly affects society. Yet there are few empirical studies to inform our understanding of the vulnerabilities and resilience of complex infrastructure systems in disasters. This research takes a systems thinking approach to explore the dynamic behaviour of a network of essential services, in the presence and absence of volcanic ashfall hazards in Montserrat, West Indies. Adopting a case study methodology and qualitative methods to gather empirical data, we centre the study on the healthcare system and its interconnected network of essential services. We identify different types of relationship between sectors and develop a new interdependence classification system for analysis. Relationships are further categorised by hazard conditions, for use in extensive risk contexts. During heightened volcanic activity, relationships between systems transform in both number and type: connections increase across the network by 41%, and adapt to increase cooperation and information sharing. Interconnections add capacities to the network, increasing the resilience of prioritised sectors. This in-depth and context-specific approach provides a new methodology for studying the dynamics of infrastructure interdependence in an extensive risk context, and can be adapted for use in other hazard contexts

Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure

Moduli of quantum Riemannian geometries on <= 4 points

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for $\le 9$ points based on the known atlas of regular graphs. We also discuss aspects of the quantum theory defined by functional integration.Comment: 34 pages ams-latex, 4 figure

Investigating Abstract Algebra Students' Representational Fluency and Example-Based Intuitions

The quotient group concept is a difficult for many students getting started in abstract algebra (Dubinsky et al., 1994; Melhuish, Lew, Hicks, and Kandasamy, 2020). The first study in this thesis explores an undergraduate, a first-year graduate, and second-year graduate students' representational fluency as they work on a "collapsing structure", quotient, task across multiple registers: Cayley tables, group presentations, Cayley digraphs to Schreier coset digraphs, and formal-symbolic mappings. The second study characterizes the (partial) make-up of two graduate learners' example-based intuitions related to orbit-stabilizer relationships induced by group actions. The (partial) make-up of a learner's intuition as a quantifiable object was defined in this thesis as a point viewed in R17, 12 variable values collected with a new prototype instrument, The Non-Creative versus Creative Forms of Intuition Survey (NCCFIS), 2 values for confidence in truth value, and 3 additional variables: error to non-error type, unique versus common, and network thinking. The revised Fuzzy C-Means Clustering Algorithm (FCM) by Bezdek et al. (1981) was used to classify the (partial) make-up of learners' reported intuitions into fuzzy sets based on attribute similarity