Every graph G = (V, E) has a dominating set S ⊆ V(G) such that any vertex not in S is adjacent to a vertex in S. We define a paired-dominating set S to be a dominating set S = {v1, v2,..., v2t-1, v2t} where M = {v1v2, v3v4, ..., v2t-1v2t} is a perfect matching in 〈S〉, the subgraph induced by S. The domination number of a graph G is the smallest cardinality of any dominating set of G, and the paired-domination number is the smallest cardinality of any paired-dominating set. Determining the domination number for grid graphs is a well-known open problem in graph theory. Not surprisingly, determining the paired-domination number for grid graphs is also a difficult problem. In this thesis, we survey past research in domination, paired-domination and grid graphs to obtain background for our study of paired-domination in grid graphs. We determine the paired-domination number for grid graphs Gr,c where r ∈ {2,3}, for infinite dimensional grid graphs, and for the complement of a grid graph.

The purpose of this thesis is to describe and document parentage statements and paired stela found on Maya monuments or portable objects. While the existence of parentage statements, within the Maya text, has been known since the late 1970s, no thesis or major research project has ever been published pertaining to this class of glyphs within the Maya script. In 1977 Christopher Jones is credited with the discovery of parentage statements in Maya hieroglyphic texts (Jones 1977). His discovery was followed by a detailed analysis of parentage statements by Floyd Lounsbury, Peter Mathews, and Linda Schele (1977). They introduced most of the known parentage statements and detailed their use in the hieroglyphic texts. While their work, which introduced parentage statements, in general, is cited regularly as the all-encompassing source of information about parentage statements, it remains unpublished. This thesis provides a source for all new and previous work done on parentage statements and insight into possibly why parentage statements were used by the Classic Maya. An analysis of 253 archaeological sites containing a total of 2473 glyphic monuments was conducted as part of this thesis. Justin Kerr's online vase and photo collections were also checked for parentage statements on portable objects and ceramic vessels. Parentage statements were found on 225 monuments and 70 portable objects bringing the total to 295 different monuments with parentage statements. Each parentage statement is identified and transcribed phonetically. The usage of each parentage statement and its variants are then mapped through time. Also introduced in this thesis is the death of a parent glyph commonly known as the Winged Capped Ajaw Death Phrase. An analysis of its usage demonstrates that it refers to the death of a parent. Paired stelae were analyzed as another possible form of parentage statements. This thesis tests three hypotheses on why parentage statements were used. They included ancestor worship, political legitimacy, and heir designation. The weaknesses and strengths of each hypothesis are demonstrated through case studies. Lastly, the appendices include source material for each site and monument researched as part of this thesis.

<p> The study seeks to derive a spatial preference model for urban places, based on a farm population's spatial choices of urban places for retail expenditure. Specifically, the study tests for the similarity in households' preference orderings of urban places, and finds a high degree of similarity. This is achieved using a model with only two simple variables, namely town population and distance to town. Tests indicate no major variable is omitted in the model. The information on households' preference orderings enables the aggregate preference order to be defined. </p> <p> Tests are inconclusive as to whether households also assign worths to urban places which indicate their awareness of the amount by which different places are preferred. </p> <p> The second part of the analysis seeks to determine if different types of households have different spatial preferences. Differences are revealed in the value different types attach to the locational convenience of places. Farm households with members working off the farm reveal a lower than average preference for convenient shopping places. Smaller, more affluent households display a stronger preference for convenient locations and a lower preference for large towns than less affluent households with young children. </p> / Thesis / Doctor of Philosophy (PhD)

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Pain is a complex, ubiquitous phenomenon that can be debilitating and costly. Although it is well known that some individuals can easily tolerate pain while others are more intolerant to pain, little is known of the neurophysiological bases of these differences. Because differences in sensory information processing may underlie variability in tolerance to pain and because measures of sensory gating are used to explore differences in sensory information processing, sensory gating among college students (N = 14) who are tolerant or intolerant to pain was investigated. This investigation explored the hypothesis that those who were more tolerant to pain would evidence greater sensory gating. Pain tolerance was first determined using a cold pressor task. Sensory gating was then determined by the amount of attenuation of the amplitude of a second painful, electrical, somatosensory stimulus (S2) in relation to the amplitude of an identical first stimulus (S1) in a paired-stimulus evoked potential (EP ) paradigm. The results obtained showed the intolerant group exhibiting greater physiological reactivity than the tolerant group, indicating that the tolerant group attained greater sensory gating than the intolerant group. / Ph. D.

I. A Comparison of the Effectiveness of Tournaments. A paired-comparison experiment involving t treatments is analogous to a tournament with t players. A balanced experiment, in which every possible pair occurs once per replication, is the counterpart of a round robin tournament. When the objective is to pick the best treatment, the balanced design may prove to be more expensive than necessary. The knock-out tournament has been suggested as an alternative requiring fewer units of each treatment per replication. In this paper round robin, replicated knock-out, and double elimination tournaments are investigated for their effectiveness in selecting the best one or tour players. Effectiveness is gauged in terms of the two criteria (a) the probability that the best player wins and (b) the expected number of games. For general values of the parameters involved, expressions are derived for the evaluation of the criteria. Comparisons are made on the basis of series of assigned parameter values. Possibilities for the extension of the study are briefly discussed. II. Ties in Paired-Comparison Experiments. In making paired comparisons a judge frequently is unable to express a real preference in a number of the pairs he judges. In spite of this, some or the methods in current use do not permit the judge to declare a tie. In other methods tied observations are either ignored or divided equally or randomly between the tied members. It appears that there is a need, at least in the estimation of response-scale values, for a method which takes tied observations into account. In the Thurstone-Mosteller method the standardized distribution of the difference of two stimulus responses is normal with unit variance and mean equal to the difference or the two mean stimulus responses. In prohibiting ties the assumption is in effect made that all differences, however small, are perceptible to the judge. In this paper the assumption is made that a tie will occur whenever the difference between the judge's responses to the two stimuli lie below a certain threshold, i.e. if the difference lies between -t and t the judge will declare a tie. The parameter t and the mean stimulus responses are estimated by least squares. To overcome a difficulty presented by correlated data, an angular response law is postulated for the response-scale differences. In the resulting transformed data non-homogeneity of variances is encountered. In effecting a weighted solution, weights are first determined by using a preliminary unweighted analysis, and an iterative procedure is proposed. Large-sample variances and covariances of the estimates are obtained. A test of the validity of the model is described. A computational procedure is set up, and exemplified through application to experimental data. / Ph. D.

There is a transitional period between preschool and first grade during which children develop from reliance on instructions to self-sufficient, spontaneous retrieval. Past research has revealed that retrieval instructions are vital to preschoolers' retention of paired-associates. Preschool and kindergarten children were presented with a mixed-list of 20 paired-associate pictures to learn. Ten of the 20 pairs depicted 2 nouns as interacting in an uncommon or bizarre (funny) manner with one another whereas the other 10 pictures depicted component nouns as normally (nonfunnily) interacting. Type and timing of instructions to learn the paired-associates were manipulated. Funny-group subjects were provided with encoding instructions highlighting the silliness quality of the pictures. Nonfunny-group subjects were provided with encoding instructions which concentrated on the interactive aspects of the pictures. A control group was told to remember the pictures “really hard.” For the former two instructional groups, half of each group served as a control during test-time. These control groups were told to try really hard to remember the pictures. The experimental halves of these groups were given elaborative retrieval instructions at test-time, dependent upon their r e encoding instructions. Reference to the funny pictures was provided for half the Funny group whereas reference to the normal interaction was given to half the Nonfunny group. This study revealed that elaborative instructions do not enhance memory for paired-associates over non-elaborative or control instructions. These preschoolers did not exhibit retrieval deficiencies though control retrieval instructions may have provided sufficient information to enhance memory. As well, non bizarre (nonfunny) interactive pictures were remembered better than bizarre (funny) interactive pictures. / M.S.

New methods for testing hypotheses in paired-comparison experiments are presented in this dissertation. The methods are developed on the basis of a very general mathematical model and they are, in general, quite easy to employ. Two tests of the null hypothesis that all treatments have equal stimuli, against its general alternative, are proposed. One test is for the case in which it is assumed prior to the experiment that no interaction will take place between repetitions and preference probabilities (the probabilities of the possible comparison preferences). The other test is for the case in which the above assumption cannot be made. The number of times a treatment is preferred is called its score. For the “no interaction" case, the test procedure is based on a test statistic that is a function D of the corrected sum of squares of the treatment scores. In the other case, the value of D is calculated for each group of homogeneous repetitions and then the values are summed to give the new test statistic. It is established that a X²-approximation may be used to determine the critical value of the test statistic for experiments outside the range of the tabled distributions. This test procedure is shown to be simpler than other approximate tests and, in general, at least as accurate with respect to errors of the first kind. It is shown that the two test methods discussed above may be extended to ranking experiments in balanced incomplete block designs with more than two treatments per block. To test the null hypothesis of no interaction between preference probabilities and repetitions, against its general alternative, a test method based on the theory of X² homogeneity tests is introduced. Means are presented for testing whether (1) a particular treatment is better than the average of the treatment stimuli; (2) two particular treatment stimuli are not equal; and (3) the treatment receiving the highest score is better than the average. The three test procedures are based essentially on the binomial distribution of the treatment scores under the null hypothesis. In each case, the test procedure is conservative. A procedure analogous to Tukey's test based on allowances is developed to test the null hypothesis of equal treatment stimuli and to separate the significantly different treatment scores when it rejects the null hypothesis. A method for judging contrasts of treatment scores similar to Scheffe's (1953) method for judging contrasts in the analysis of variance is proposed. The test method based on D, mentioned earlier, is used in place of the F-test employed in the Scheffe method. The use of paired-comparison experiments to test factorial effects is discussed and a test method based on orthogonal contrasts of the treatment scores is suggested. Because of correlations that arise, it is necessary to restrict this method to cases in which the only factors that are allowed to appear at more than two levels are those that will not interact with the other factors in the experiment. The test methods are illustrated through application on the data from two paired-comparison experiments. / Ph. D.

Yes / Romantic relationships are extremely important to people’s happiness and well-being, yet many people do not seek advice with relationship issues or may do so only once serious problems arise. Paired is a commercially available relationships app. Launched in October 2020, it currently has over 12,000 daily active users, predominantly in the US and UK. Public self-management of care (i.e. self-help) is target for technological investment, as digital health and well-being apps gain popularity. There are currently over 318,000 health apps available worldwide, with a further 200+ new health apps coming onto the market each day. Research has shown that mobile health (mHealth, i.e. health and well-being apps) can be effective in supporting behaviour change: helping us to adopt and maintain healthy behaviours. However, many health and well-being apps are not based on reliable research evidence, the only indication of an app’s quality deriving from ‘user reviews’. Paired is evidence-based. Focusing on the area of romantic relationships, it seeks to support and enhance couple relationships, before the point when professional help may be needed. Researchers at The Open University (OU) and the University of Brighton evaluated the effectiveness of Paired, using a mixed methods approach... We created the Quality of Relationship Index (QRI), a rigorously-developed measure of overall relationship quality. We found that quality of communication – the underpinning dimension of all relationship maintenance behaviour – was most strongly related to overall relationship quality... In developing the QRI, we explored change in different aspects of relationship quality, whilst using Paired... Integration of the findings from our analyses of different data sources, and the ‘dose-response’ effect that we consistently observed, together give us confidence that Paired is responsible for the improvements to relationship quality that its users enjoy. / The Open University

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