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Στην παρούσα εργασία θα παρουσιαστούν διάφορα προβλήματα, με την επίλυσή τους, τα οποία ανήκουν στο πεδίο των Διακριτών Μαθηματικών. Πιο συγκεκριμένα στο πρώτο κεφάλαιο θα δούμε διάφορα προβλήματα αναδρομής όπου κάποια από αυτά θα είναι παραλλαγές γνωστών προβλημάτων (π.χ.: Ο ΠΥΡΓΟΣ ΤΟΥ HANOI) των οποίων οι (γνωστές) λύσεις θα μας βοηθάνε για να αποδεικνύουμε κάθε φορά αυτό που θέλουμε. Στο δεύτερο κεφάλαιο θα ασχοληθούμε με προβλήματα και ασκήσεις αθροισμάτων, τα οποία είναι παντού στα μαθηματικά, γι'αυτό χρησιμοποιούμε βασικά εργαλεία για να τα λύσουμε, αναπτύσσοντας γενικές τεχνικές έτσι ώστε η διαδικασία να είναι φιλική προς τον αναγνώστη. Στη συνέχεια στο τρίτο κεφάλαιο θα συναντήσουμε ασκήσεις οι οποίες αφορούν τις ακέραιες συναρτήσεις. Οι ακέραιοι αριθμοί αποτελούν τη ραχοκοκαλιά των Διακριτών Μαθηματικών, και εμείς συχνά χρειάζεται να μετατρέπουμε κλάσματα ή αυθαίρετους πραγματικούς αριθμούς σε ακέραιους. Ο στόχος λοιπόν αυτού του κεφαλαίου είναι να αποκτήσουμε οικειότητα και άνεση με τέτοιου είδους μετατροπές μέσα από τις ασκήσεις και να μάθουμε μερικές από τις αξιοσημείωτες ιδιότητες τους. Μέσα από τις ασκήσεις του τέταρτου κεφαλαίου γίνεται μια εισαγωγή στη θεωρία αριθμών ένα σημαντικό κλάδο των μαθηματικών που ασχολείται με τις ιδιότητες των ακεραίων. Τέλος στο κεφάλαιο 5 θα συναντήσουμε ασκήσεις και προβλήματα τα οποία βασίζονται στη μελέτη των διωνυμικών συντελεστών οι οποίοι είναι πολύ σημαντικοί στις εφαρμογές και επίσης πιo εύκολοι να τους χειριστούμε σε σύγκριση με άλλες ποσότητες των προηγούμενων κεφαλαίων. / -

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Thesis (M.Sc.Eng.) PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This thesis considers the efficient Hessian computation in inverse problems with specific application to the elastography inverse problem. Inverse problems use measurements of observable parameters to infer information about model parameters, and tend to be ill-posed. They are typically formulated and solved as regularized constrained optimization problems, whose solutions best fit the measured data. Approaching the same inverse problem from a probabilistic Bayesian perspective produces the same optimal point called the maximum a posterior (MAP) estimate of the parameter distribution, but also produces a posterior probability distribution of the parameter estimate, from which a measure of the solution's uncertainty may be obtained. This probability distribution is a very high dimensional function with which it can be difficult to work. For example, in a modest application with N = 104 optimization variables, representing this function with just three values in each direction requires 3^10000 U+2248 10^5000 variables, which far exceeds the number of atoms in the universe. The uncertainty of the MAP estimate describes the shape of the probability distribution and to leading order may be parameterized by the covariance. Directly calculating the Hessian and hence the covariance, requires O(N) solutions of the constraint equations. Given the size of the problems of interest (N = O(10^4 - 10^6)), this is impractical. Instead, an accurate approximation of the Hessian can be assembled using a Krylov basis. The ill-posed nature of inverse problems suggests that its Hessian has low rank and therefore can be approximated with relatively few Krylov vectors. This thesis proposes a method to calculate this Krylov basis in the process of determining the MAP estimate of the parameter distribution. Using the Krylov space based conjugate gradient (CG) method, the MAP estimate is computed. Minor modifications to the algorithm permit storage of the Krylov approximation of the Hessian. As the accuracy of the Hessian approximation is directly related to the Krylov basis, long term orthogonality amongst the basis vectors is maintained via full reorthogonalization. Upon reaching the MAP estimate, the method produces a low rank approximation of the Hessian that can be used to compute the covariance. / 2031-01-01

The past three decades have seen a fast expanding interest in thermal problems exhibiting a change of phase, more commonly known as Stefan problems. With the rapid advance in computer technology the use and expansion of numerical simulation schemes has been responsible for large advances in this field. The increasing size of computers has led to more sophisticated and complex numerical solutions becoming feasible from a computational point of view. On the other hand, part of this interest has developed from industrial quarters where a knowledge of the location of a melting/freezing boundary may be of critical importance for certain processes. Much experimental work has been completed in this field. However, it is still useful to be able to obtain quick, accurate numerical solutions to such problems and it is with this in mind that this thesis is presented. Ideas from both of the above areas of interest are treated. In the first case a simple to program and computationally efficient numerical scheme is proposed for solving one dimensional Stefan problems and its merits are discussed in relation to several of the more common existing solution schemes. This scheme is then modified to cater for a two dimensional problem which crudely imitates a possible heating configuration in some industrial processes. The problem, with its attendant difficulties, is first approximated by a 'test' problem which is constructed so as to admit an analytic solution. This allows assessment of the numerical procedure in two dimensions. In the course of this work a pseudo-analytic solution was obtained for the original two dimensional problem. Finally, in collaboration with the British Gas Corporation, a complex industrial freezing problem is discussed concerning the flow of liquid through an enclosed channel. Some simplifying assumptions are proposed to reduce the problem to a form for which a relatively simple numerical scheme may be adopted. Several simulations are completed to examine the effect of varying physical parameters on the solution and in particular to test for situations of blockage or steady-state.

The understanding of maternal factors associated with child feeding problems is limited due to a lack of research which has examined a comprehensive range of maternal factors and the existing literature focussing on a narrow range of ideas about the wider familial context in which feeding problems occur. The broad aim of this thesis is to investigate maternal parenting of children with and without clinical feeding problems to provide insights into the wider context in which feeding problems occur. In study 1, thematic analysis of interviews with 10 mothers of children with, and 10 mothers of children without, clinical feeding problems revealed that mothers of children with clinical feeding problems appeared to have less maternal self-efficacy for managing parenting challenges than mothers in the non-clinical group. A template analysis found that these perceptions seemed to be informed by four theoretical sources of self-efficacy: mastery experiences, verbal persuasion, vicarious experience and physiological state. In study 2, 278 mothers of children with and without clinical feeding problems completed existing self-report measures of maternal self-efficacy. It was found that lower levels of maternal self-efficacy for establishing structure and routine around instrumental child care tasks and for implementing discipline and setting limits for the child was predictive of problematic child feeding behaviour and maladaptive maternal responses to child feeding problems. In study 3, 215 mothers of children with and without feeding problems completed self-report measures of the theoretical components of self-efficacy (mastery experiences, verbal persuasion, vicarious experience, physiological state). Physiological state was found to be the strongest predictor of maternal self-efficacy for establishing structure and routines as well as for providing discipline and setting limits for the child. Maternal self-efficacy mediated the relationship between physiological state and problematic child feeding behaviour and the relationship between parenting stress and maladaptive maternal responses. The final study was a pilot study using autophotography. This study explored parenting dimensions and tasks which contributed towards perceptions of maternal self-efficacy in 13 mothers of children without clinical feeding problems. Findings suggested that child feeding was an especially difficult and complex task for mothers to manage, with many mothers reporting perceptions of low efficacy for managing mealtimes. This appeared to be due to the large number of health related concerns and the worry mothers felt when children did not eat a healthful diet. Overall, results of this thesis suggest that mothers who lack confidence in their ability to manage child behaviour report more problematic child feeding behaviours and use more maladaptive strategies for managing feeding problems. Findings of the studies in this thesis suggest that mothers of children with feeding problems, and who are experiencing high levels of stress, may be especially vulnerable to lower levels of maternal self-efficacy which may exacerbate difficulties. Increasing maternal self-efficacy for providing structure, routines and discipline in mothers experiencing feeding problems in their children may improve outcomes for those affected.

After analysing the major theoretical contributions to health and development, the author develops a dialectical materialist approach. Health, both physical and mental, is defined as a movement of energy that is simultaneously conserved and expanded. It is reality in movement. It is the vital ingredient of labour power and the capacity to create. Fundamental to humanity's health, both on a personal and public level, is productive activity. The relations and forces of production are the main determinants of public health. The relative power of the oppressed and the contemporary means of production affect both the type and spread of disease. In the specific historic Investigation of health and development in the Maltese Islands, the prevalence of contemporary diseases is appraised with reference to the balance of forces between nations, classes and the sexes. The particular significance of imperialism, merchant capital and religion is discussed. Examination of three major diseases, cholera, undulant fever and cancer, between 1837 and 1987, is the empirical basis of the thesis. The transition from high mortality rates to high morbidity rates in the past forty years reflects Malta's late and uneven development. Health policy to overcome disease is limited because health and disease are manifestations of the mode of production. Health in developing countries is placed in the dialectic of imperialism and development, chauvinism and development and, essentially, the dialectic of capitalism and development.

The diploma thesis thematically deals with the adaptation of exercises for teaching arithmetic and algebra at primary school. The aim of the thesis was to create a collection of methodically processed exercises for the second grade students. In seven parts, the collection focuses on working with fraction from introduction of it to particular operations with fractions. The thesis has following structure: the first part is dealing with theoretic adaptation of literary sources on the given topic. The second part contains practical orientation with a motivational example, an explanatory example, exercises for practising, an exercise to test the curriculum, an exercise for the quick-witted and an exercise for home preparation in every thematic unit.

College students' drinking patterns have been a cause for concern for a number of years. The present study evaluated the relative effectiveness of two brief interventions aimed at reducing alcohol consumption among heavy-drinking students. The first intervention delivered personalised feedback about students' alcohol use and other alcohol-related information. The other one delivered nonpersonalised feedback. It was hypothesised that the personalised feedback would be more successful than nonpersonalised feedback in motivating heavy-drinking students to reduce their alcohol consumption. The study began with a large-scale, screening survey of students' alcohol use. The survey first identified heavy-drinking students, who then completed a baseline assessment comprising questionnaire measures related to personality, motivation, reasons for drinking, high-risk drinking situations, and alcohol-related problems. Following the baseline assessment, the heavy-drinking students (n= 111) were randomly assigned to either one of three groups; personalised feedback, nonpersonalised feedback, or a non-intervention control group. Students (n= 110) in all three groups were followed-up 12 weeks after the interventions had been delivered. The results showed that personalised alcohol-related feedback produced the greatest increase in students' readiness to change their excessive drinking. However, there was no evidence for an effect of intervention on students' actual consumption. At baseline it was found that as students' alcohol-related problems increased there were also increases in (a) the amount of alcohol that they consumed, (b) the negative-affect situations in which they drank, and (c) their maladaptive motivational patterns. In fact, each of the latter three variables contributed uniquely to the variance in alcohol-related problems. The results were discussed from the perspective of a motivational model of alcohol use (Cox & Klinger, 1988). It was concluded that the findings of the present study have important implications for future brief interventions among students.