Consider the Sturm-Liouville system : 8 > > > > > < > > > > > : − y00 + q(x)y = y y(0) cos + y0(0) sin = 0 y(1) cos + y0(1) sin = 0 , where q 2 L 1 (0, 1) and , 2 [0, £¾). Let 0 < x(n)1 < x(n)2 < ... < x(n)n − 1 < 1 be the nodal points of n-th eigenfunction in (0,1). The inverse nodal problem involves the determination of the parameters (q, , ) in the system by the knowledge of the nodal points . This problem was first proposed and studied by McLaughlin. Hald-McLaughlin gave a reconstruc- tion formula of q(x) when q 2 C 1 . In 1999, Law-Shen-Yang improved a result of X. F. Yang to show that the same formula converges to q pointwisely for a.e. x 2 (0, 1), when q 2 L 1 . We found that there are some mistakes in the proof of the asymptotic formulas for sn and l(n)j in Law-Shen-Yang¡¦s paper. So, in this thesis, we correct the mistakes and prove the reconstruction formula for q 2 L 1 again. Fortunately, the mistakes do not affect this result.Furthermore, we show that this reconstruction formula converges to q in L 1 (0, 1) . Our method is similar to that in the proof of pointwise convergence.

This dissertation consists of two parts. In the first part we show that for 1 k 1, a complex manifold M of dimension at least k in the boundary of a smooth bounded pseudoconvex domain in Cn is an obstruction to compactness of the @- Neumann operator on (p, q)-forms for 0 p k n, provided that at some point of M, the Levi form of b has the maximal possible rank n &#8722; 1 &#8722; dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction to compactness of the @-Neumann operator on (p, 1)-forms, provided that at some point of the disc, the Levi form has only one vanishing eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one vanishing eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. In the second part we obtain a weaker and quantified version of McNeal’s Property ( eP) which still implies the existence of a Stein neighborhood basis. Then we give some applications on domains in C2 with a defining function that is plurisubharmonic on the boundary.

Students need to build on their own understanding when problem solving. Mathematics reform is moving away from skill and drill types of activities and encouraging students to develop their own approaches to problem solving. The National Council of Teachers of Mathematics emphasizes the importance of representation by including it as a process standard in Principles and Standards for School Mathematics (2000) as a means for students to develop mathematically powerful conceptualization. Students use representation to make sense of and communicate mathematical concepts. This study considers the way fourth grade students view and solve division problems and whether problem type affected the choice of strategy. This study also looked at factors that affect students' score performance. Students in extant classrooms were observed in their regular mathematics instructional settings. Data were collected and quantified from pretests and posttests using questions formatted like students see on the state assessment. The results indicate that students moved from pre-algorithmic strategies to algorithmic strategies between pretest and posttest administration. The results also indicate that problem type did not predict students' choice of strategy and did not have an affect on the students' ability to arrive at a correct solution to the problem. This study found that the students' choice of strategy did play a significant role in their quest for correct solutions. The implication is that when students are able to make sense of the problem and choose an appropriate strategy, they are able to successfully solve division problems.

Universiẗat, Diss.--Paderborn, 1985.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 121-122). Also available on the Internet.

Thesis (M.A.) - Simon Fraser University, 2005. / Theses (Faculty of Education) / Simon Fraser University. Also issued in digital format and available on the World Wide Web.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1908. / Cover title. Vita.

Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 141-148). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 178-184).

Thesis (Ph. D.)--Washington State University, May 2009. / Title from PDF title page (viewed on Feb. 12. day, 2010). "Department of Physics and Astronomy." Includes bibliographical references (p. 186-199).