Richard Ehrenborg conjectured that in a bipartite graph G with parts X and Y, the number of spanning trees is at most the product of the vertex degrees divided by |X|⋅|Y|. We make two main contributions. First, using techniques from spectral graph theory, we show that the conjecture holds for sufficiently dense graphs containing a cut vertex of degree 2. Second, using electrical network analysis, we show that the conjecture holds under the operation of removing an edge whose endpoints have sufficiently large degrees. Our other results are combinatorial proofs that the conjecture holds for graphs having |X| ≤ 2, for even cycles, and under the operation of connecting two graphs by a new edge. We also make two new conjectures based on empirical data, each of which is stronger than Ehrenborg's conjecture.

In 2005, Alon, et al. proved that tournaments arising from majority voting scenarios have minimum dominating sets that are bounded by a constant that depends only on the notion of what is meant by a majority. Moreover, they proved that when a majority means that Candidate A beats Candidate B when Candidate A is ranked above Candidate B by at least two out of three voters, the tournament used to model this voting scenario has a minimum dominating set of size at most three. This result gives 2-majority tournaments some significance among all tournaments and motivates us to investigate when a given tournament can be considered a 2-majority tournament. In this thesis, we prove, among other things, that the presence of an upset path in a tournament allows us to conclude the tournament is realizable as a 2-majority tournament.

A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.

This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the coefficients of the binomial expansion. Furthermore, combinatorics plays an increasingly important role in mathematics, starting when students enter high school and continuing on into their college years. Students become familiar with the traditional arguments based on classical arithmetic and algebra, however, methods of combinatorics can vary greatly. In combinatorics, perhaps the most important concept revolves around the coefficients of the binomial expansion, studying and proving their properties, and conveying a greater insight into mathematics.

Training paraprofessionals who work with children with Autism Spectrum Disorder and other related developmental disabilities can be a challenge due to limited resources, time, and money. Alternative ways to train paraprofessionals on a larger scale is needed. Interactive computerized training—a self-paced program that incorporates audio narration, video models, interactive activities, and competency checks—is one potential training method. Interactive computerized training has been successful at training college students and special education teachers to implement discrete trial instruction but their effectiveness in training paraprofessionals is unknown. The purpose of this study was to extend the literature on interactive computerized trainings to evaluate its utility to teach six paraprofessionals to implement discrete trial instruction. Errorless learning procedures are recommended during discrete trial instruction to minimize student errors and promote quicker skill acquisition. A secondary purpose was to evaluate the effectiveness of the interactive computerized training to teach paraprofessionals to implement an errorless learning procedure. Following the training, all participants increased their fidelity of implementation of discrete trial instruction, at varying levels, with a student in their classroom. One participant reached the performance criterion of 90% or higher fidelity following ICT alone and two participants required performance feedback. Three participants required live coaching to increase their fidelity of DTI components to 80%. All feedback was delivered from a distance. Fidelity remained high to untrained instructional programs and at 2-week follow up. Potential limitations and future research related to training paraprofessionals are discussed.

Individuals with disabilities often have a limited tact repertoire. This study compared the acquisition and generalization of tacts taught using different stimulus modes within discrete trial training (DTT) with children who have a limited tacting repertoire. The three stimulus modes that were compared were videos, pictures, and 3D objects. This research replicated Gómez’s (2015) methodology with a participant pool with lower tacting skills. In addition, this study assessed for generalization of the acquired tacts to a novel exemplar. In this study, tact training required fewer sessions when the picture and 3D object were used as stimulus modes. These results were then replicated across stimulus sets with Abraham and Alex. Finally, in this study greater generalization was observed for the tacts trained with a 3D object.

Recent interest in identifying and measuring health outcomes represents an advance in our understanding of how health care for individuals should be evaluated. However, the concept of health outcomes has mainly focussed on improvements in health status. Non-health outcomes of health care may also be important to patients. In this thesis, four tasks were undertaken with the aim of identifying non-health outcomes and establishing the extent of their relevance and importance to patients. First, the illness experience literature was reviewed to identify potential non-health outcomes. Seven categories of non-health outcomes were identified: information, being treated with dignity, being able to trust the health care provider, having distress recognised and supported, participating in decision making, legitimation and reassurance. Second, to gain an in-depth understanding of these concepts, topic-specific literature was reviewed and synthesised. Third, in order to confirm how relevant and important the concepts were to patients, a qualitative study was conducted with each of two different groups of health service users. Broadly, patients considered that all the non-health concepts were relevant, although the extent to which they were important varied. Fourth, to test the relative importance of the seven concepts, a Stated Preference Discrete Choice experiment in the context of general practice was conducted. This study showed that most people thought their GP demonstrated behaviour likely to result in the production of non-health outcomes. The results showed that although all the non-health outcomes were, to some extent, preferred by respondents, trust was most important, followed by legitimation and recognition of and support for emotional distress. Once again, these results point to the importance of context in the evaluation of health care from the patient's perspective. While still being perceived as positive aspects of health care, the provision of information and acting autonomously or participating in decisions about their health care were the non-health outcomes considered least important by patients

"The Department of Electrical and Computer Engineering, the University of Newcastle." Includes index. Bibliography: p. [137]-143.

<p>The delivery of health care services has been under pressure due to limited funding and increasing demand. This has highlighted the need to increase not only the effectiveness but also the efficiency of health care delivery. Discrete-event simulation has been suggested as an analysis tool in health care management to support the planning of health care resources.</p><p>The overall purpose of this thesis is to investigate the possibilities and requirements for using discrete-event simulation in analyzing and planning the use of hospital resources. This is achieved by three case studies that focus on improvements in patient flow of emergency patients that require a radiology examination, intensive care unit capacity planning and operating room allocation strategies, respectively.</p><p>The first case investigates the current stage of digitization and process orientation in hospital care as a prerequisite for efficient process simulation and analysis. The study reveals an emergency-radiology patient flow process that is not very well measured and uncovers disparate information systems storing incompatible and fragmented data. These results indicate that the current degree of process orientation and the current IT infrastructure does not enable efficient use of quantitative process analysis and management tools like simulation.</p><p>In the second case the possibilities to develop generic hospital unit simulation models by building and validating a generic intensive care unit (ICU) model are explored. The results show that some of the modeling approaches described in literature cannot replicate the actual behavior observed in all studied ICUs. It is important to identify patient groups for different admission priorities, to account for over-utilizations in the model logic, and to discover and properly model dependencies in the input data. The research shows that it is possible to develop a generic ICU simulation model that could realistically describe the performance of different real ICUs in terms of occupancy, coverage and transfers.</p><p>The value of simulation modeling in health care management is examined in the third case through the development and use of a simulation model for optimal resource allocation and patient flow in a hospital operating department. The goal of the simulation modeling in this case was to identify bottlenecks in the patient flow and to try different alternatives for allocation of operating room capacity in order to increase the utilization of operating room resources. The final model was used to evaluate four different proposed changes to operating room time allocation.</p>

<p>“In many interesting papers on discrete velocity models (DVMs), authors postulate from the beginning that the finite velocity space with "good" properties is given and only after this step they study the Discrete Boltzmann Equation. Contrary to this approach, our aim is not to study the equation, but to discuss all possible choices of finite phase spaces (sets) satisfying this type of "good" restrictions. Due to the velocity discretization it is well known that it is possible to have DVMs with "spurious" summational invariants (conservation laws which are not linear combinations of physical invariants). Our purpose is to give a method for constructing normal models (without spurious invariants) and to classify all normal plane models with small number of velocities (which usually appear in applications). On the first step we describe DKMs as algebraic systems. We introduce for this an abstract discrete model (ADM) which is defined by a matrix of reactions (the same as for the concrete model). This matrix contains as rows all vectors of reactions describing the "jump" from a pre-reaction state to a new reaction state. The conservation laws corresponding to the many-particle system are uniquely determined by the ADM and do not depend on the concrete realization. We find the restrictions on ADM and then we give a general method of constructing concrete normal models (using the results on ADMs). Having the general algorithm, we consider in more detail, the particular cases of models with mass and momentum conservation (inelastic lattice gases with pair collisions) and models with mass, momentum and energy conservation (elastic lattice gases with pair collisions).”</p>