The solution of initial value problems (IVPs) provides the evolution of dynamic system state history for given initial conditions. Solving boundary value problems (BVPs) requires finding the system behavior where elements of the states are defined at different times. This dissertation presents a unified framework that applies modified Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs. Existing methods for solving IVPs and BVPs have not been very successful in exploiting parallel computation architectures. One important reason is that most of the integration methods implemented on parallel machines are only modified versions of forward integration approaches, which are typically poorly suited for parallel computation. The proposed MCPI methods are inherently parallel algorithms. Using Chebyshev polynomials, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. Combining Chebyshev polynomials with Picard iteration, MCPI methods iteratively refine estimates of the solutions until the iteration converges. The developed vector-matrix form makes MCPI methods computationally efficient. The power of MCPI methods for solving IVPs is illustrated through a small perturbation from the sinusoid motion problem and satellite motion propagation problems. Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, prior to parallel implementation. Modifying the algorithm to do double integration for second order systems, and using orthogonal polynomials to approximate position states lead to additional speedups. Finally, introducing perturbation motions relative to a reference motion results in further speedups. The advantages of using MCPI methods to solve BVPs are demonstrated by addressing the classical Lambert’s problem and an optimal trajectory design problem. MCPI methods generate solutions that satisfy both dynamic equation constraints and boundary conditions with high accuracy. Although the convergence of MCPI methods in solving BVPs is not guaranteed, using the proposed nonlinear transformations, linearization approach, or correction control methods enlarge the convergence domain. Parallel realization of MCPI methods is implemented using a graphics card that provides a parallel computation architecture. The benefit from the parallel implementation is demonstrated using several example problems. Larger speedups are achieved when either force functions become more complicated or higher order polynomials are used to approximate the solutions.

Based on the importance of environmental ethics in both causing and solving many of the environmental problems, the present study aimed to (1) Examine moral reasoning patterns (i.e. ecocentric, anthropocentric, non-environmental) of pre-service science teachers toward local and non-local environmental problems, (2) Investigate the effects of gender and grade level on moral reasoning patterns, and (3) Explain the factors that may have led to the observed differences in participants&rsquo / moral reasoning patterns. Throughout 2008-2009 Fall and Spring semesters, environmental cases regarding local and non-local environmental problems were distributed to a convenience sample of 120 pre-service science teachers who were enrolled in Middle East Technical University and moral decision-making interviews (MDMIs) were carried out with a sub-sample of 16 pre-service science teachers. In accordance with the purpose of the study, descriptive statistics, paired-samples t-tests, and multivariate analysis of variance (MANOVA) tests as well as qualitative analysis of the interviews were utilized. Analyses demonstrated that participants of the study mostly exhibited ecocentric moral reasoning for both local and non-local environmental problems, and their ecocentric and anthropocentric concerns showed statistically significant difference with regard to problems&rsquo / locality. Moreover, while gender did not have a statistically significant effect on participants&rsquo / moral reasoning patterns, grade level did have a statistically significant effect. Finally, analysis of the interviews revealed sixteen factors effective in participants&rsquo / environmental concerns and their moral reasoning regarding environmental issues.

Most of reports deal with bounded simply-connected domains; only a few involve in exterior and annular problems (Chen et al. [3], Katsuroda[10] and Ushijima and Chibu [30]). For exterior problems of Laplace's equations, there exist two kinds of infinity conditions, (1) |u|≤C and (2) u=O( ln r), which must be complied with by the fundamental solutions chosen. For u=O(ln r), the traditional fundamental solutions can be used. However, for |u|≤C, new fundamental solutions are explored, with a brief error analysis. Numerical experiments are carried out to verify the theoretical analysis made. Numerical experiments are also provided for annular domains, to show that the method of fundamental solutions (MFS) is inferior to the method of particular solutions (MPS), in both accuracy and stability. MFS and MPS are classified into the Trefftz method (TM) using fundamental solutions (FS) and particular solutions (PS), respectively. The remarkable advantage of MFS over MPS is the uniform $ln|overline{PQ_i}|$, to lead to simple algorithms and programming, thus to save a great deal of human power. Hence, we may reach the engineering requirements by much less efforts and a little payment. Besides, the crack singularity in unbounded domain is also studied. A combination of both PS and FS is also employed, called combination of MFS. The numerical results of MPS and combination of MFS are coincident with each other. The study in this thesis may greatly extend the application of MFS from bounded simply-connected domains to other more complicated domains.

Consider (1) -yn1+ q1y1 = (λr11 + µr12)y1 on [0, 1] y’1(0) = cot α1 and = y’1(1) = a1λ + b1 y1(0) y1(1) c1λ+d1 (2) - yn2 + q2y2 = (λr21 + µr22)y2 on [0, 1] y’2(0) = cot α2 and = y’2(1) = a2µ + b2 y2(0) y2(1) c2µ + d2 subject to certain definiteness conditions; where qi and rij are continuous real valued functions on [0, 1], the angle αi is in [0, π) and ai, bi, ci, di are real numbers with δi = aidi − bici > 0 and ci = 0 for I, j = 1,2. Under the Uniform Left Definite condition we have proved an asymptotic theorem and an oscillation theorem. Analysis of (1) and (2) subject to the Uniform Ellipticity condition focus on the location of eigenvalues, perturbation theory and the local analysis of eigenvalues. We also gave a bound for the number of nonreal eigenvalues. We also have studied the system T1(x1) = (λA11 + µA12)(x1) and T2(x2) = (λA21 + µA22)(x2) where Aij (j =1, 2) and Ti are linear operators acting on finite dimensional Hilbert spaces Hi (i = 1, 2). For a pair of commutative operators Γ = (Γ0, Γ1) constructed from Aij and Ti on the Hilbert space tensor product H1 ⊗ H2, we can associate a natural Koszul complex namely Dºr-(λ,μ) D1 r-(λ,μ) 0 H H ø H H 0 We have constructed a basis for the Koszul quotient space N(D1Г−(λ,µ))/R(D0Г−( λ,µ)) in terms of the root basis of (Г0, Г1). (For equations pl refer the PDF file)

The importance of mathematics instruction including "real life" contexts relevant to students’ lives and experiences is widely acknowledged (Common Core State Standards Initiative, 2010; National Council of Teachers of Mathematics, 2000; 2006; 2009), however questions about why contextualized mathematics is beneficial and how different types of contextualization impact problem solving have yet to be fully addressed by research. Common justifications for contextualized mathematics include the idea that relevant contexts may help students to apply what they learn in school to out-of-school situations, and that relevant contexts may scaffold learning by providing a bridge between what students understand and the content they are trying to learn. The present study investigates these justifications, as well as students' beliefs and problem-solving methods, using story problems on linear functions. A situated cognition theoretical framework (Greeno, 2006) is used to interpret student behavior in the complex, social system of "school mathematics." In a series of interviews, students from a low-performing urban school were presented with algebra problems. Some problems were personalized to the ways in which they described using mathematics in their everyday lives, while others were normal story problems, story problems with equations, or abstract symbolic equations. Results showed that students rarely explicitly used situational knowledge when solving story problems, had consistent issues with verbal interpretation of stories, and engaged in non-coordinative reasoning where they bypassed the intermediate step of understanding the given situation before trying to solve the problem. After completing most of Algebra I, students still had considerable difficulty with symbolic representations, and struggled to coordinate formal and informal mathematical reasoning. Problems with the same mathematical structure with different amounts of verbal and symbolic support elicited different strategies from students, with personalized problems having high response rates and high use of informal strategies. This suggests that students can use sophisticated, situation-based reasoning on contextualized problems, and that different problem framings may scaffold learning. However, results also demonstrated that the culture of schooling, and story problems as an artifact of this culture, undermines many of the justifications for contextualizing mathematics, and that students need more authentic ways to develop their mathematical reasoning. / text

Η παρούσα διπλωματική αναφέρεται σε ελλειπτικά προβλήματα συνοριακών συνθηκών σε κανονικά πολύγωνα, εστιάζοντας στην αρκετά γενική εξίσωση Helmholtz. Θα εφαρμοσθούν οι γνωστές υπολογιστικές μέθοδοι επίλυσης ελλειπτικών προβλημάτων (όπως η παρεμβολή με τμηματικά κυβικά πολυώνυμα) και θα αναπτυχθούν και μέθοδοι που προκύπτουν από νέες μαθηματικές αναλύσεις του προβλήματος. / In this work we deal with elliptic boundary value problems which are defined in regular polygons. The numerical results presented in the defence are derived using well established methods, such as the finite differemces and the 2d collocation, as well as a new method introduced recently which appears to yield nice results.

Bakalauro darbe analizuojamos tėvų emigraciją patyrusių vaikų problemos mokykloje. Šioms problemoms atskleisti buvo iškelti tokie uždaviniai: išanalizuoti mokslinę literatūrą, siekiant atskleisti tėvų emigraciją patyrusių vaikų psichosocialines problemas, išsiaiškinti tėvų emigraciją patyrusių vaikų psichosocialinių problemų ypatumus, atskleisti tėvų emigraciją patyrusių vaikų adaptacijos ypatumus bei mokykloje jiems kylančias problemas, parengti socialinės - edukacinės pagalbos pasiūlymų paketą. Siekiant kuo išsamiau atskleisti tyrimo uždavinius, pasirinkti kiekybinis ir kokybinis tyrimo metodai. Tyrime iš viso dalyvavo 108 mokiniai, kurių bent vienas iš tėvų yra išvykęs į užsienį, iš Šiaulių bei Pakruojo miestų bendrojo lavinimo mokyklų. Jų amžius – 12-18 metų. 102 mokiniai dalyvavo anketinėje apklausoje, 6 mokiniai interviu. Tyrimų metu stengtasi nustatyti tėvų emigraciją patyrusių vaikų būdingiausias emocijas, savivertės ypatumus, tarpusavio santykius su tėvais, pedagogais, bendraamžiais, giminaičiais, asmeninės adaptacijos aspektus po tėvų išvykimo bei vyraujančias problemas mokykloje. Tyrimu nustatyta, kad tėvų emigraciją patyrusiems 12 – 18 metų vaikams būdingiausios emocijos: ilgesys, nerimas, kaltė, kurios įtakoja pyktį, agresyvumą bei stresą. Nors emigrantų vaikai patiria daug neigiamų emocijų, tačiau save vertina gerai, megzdami socialinius santykius jaučiasi pilnaverčiais žmonėmis, savimi pasitiki. Tėvų emigracijos... [toliau žr. visą tekstą] / This Bachelor’s thesis analyses problems at school of children, who experienced parents’ emigration. In order to reveal these issues the following tasks were raised: to analyse scientific literature, with the aim to reveal psycho-social problems of children who experienced parents’ emigration, to find out the assessment of psychological condition of children who experienced parents’ emigration, to reveal adaptation peculiarities and problems that children who experienced parents’ emigration face at school, to prepare the package of social-educational aid proposals. Quantitative and qualitative research methods have been chosen so as to reveal the goals of the research in as many details. The research involved 108 students of Šiauliai and Pakruojis town general education schools, with at least one of the parents being abroad. Age of students was 12-18 years old. 102 students participated in a questionnaire survey, 6 students were interviewed. The aim of the researches was to determine the most typical emotions, self-esteem characteristics and relationships with parents, teachers, peers, relatives, aspects of personal adaptation after parents’ departure and prevailing problems at school of children who experienced parents' emigration. Research results identified the following emotions characteristic to children, aged 12-18, who experienced parents' emigration: longing, anxiety, guilt that altogether cause anger, aggression and stress... [to full text]

This study aimed at revealing the mediational role of mothers&rsquo / psychological adjustment between mothers&rsquo / parental attitudes and their preschool children&rsquo / s behavioral problems relationship. The participants were 204 married mothers, 64 fathers, 195 chief teachers and 25 assistant teachers of preschool children aged between 1&frac12 / - 5 years-old, who are attending nurseries in different neighborhoods of Ankara. Prior to the main analyses, the reliability analyses of Child Behavior Check List - 1&frac12 / - 5 (CBCL &ndash / 1&frac12 / - 5), and Caregiver-Teacher Report Form (C-TRF) were conducted. These analyses yielded many significant coefficients, though the strength of the correlations varied from low to high. Following the reliability analyses, Hierarchical Regression Analyses were performed to test the mediational role of mothers&rsquo / psychological adjustment between their parental attitudes and children&rsquo / s internalizing, externalizing, and total behavior problems relationships. For these analyses, the variances accounted for by children&rsquo / s and mothers&rsquo / demographic characteristics were controlled. Results revealed that, &lsquo / Rejection of the Home Making Role&rsquo / measure of PARI and Trait Anxiety measure explained the largest amount of variances while predicting the children&rsquo / s behavioral problems. According to hierarchical regression analyses, rejecting attitudes of mothers significantly associated with children&rsquo / s internalizing, externalizing, and total behavior problems. Moreover, mothers&rsquo / rejecting attitudes significantly associated with their trait anxiety which is in turn associated with children&rsquo / s behavior problems. However, after controlling for the effect of trait anxiety, the relationship between mothers&rsquo / rejecting attitudes and their children&rsquo / s behavior problems was disappeared. Therefore, mothers&rsquo / trait anxiety was found to mediate their rejecting attitudes and preschool children&rsquo / s internalizing, externalizing, and total behavior problems. Results were discussed by referring the relevant literature.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 101-103).

Thesis (M.A.T.S.)--University of Texas at El Paso, 2008. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.