Thesis--M. Phil., University of Hong Kong, 1980.

Not available / text

published_or_final_version / Mathematics / Master / Master of Philosophy

In this work we study the following problem on constrained approximation. Let f be a continuous mapping defined on a bounded domain with piecewise smooth boundary in R^n and taking values in R^n. What are necessary and sufficient conditions for f to be uniformly approximable by C^1-smooth mappings with nonnegative Jacobian? When the dimension is equal to one, this is just approximation by monotone smooth functions. Hence, the necessary and sufficient condition is: the function is monotone. On the other hand, for higher dimensions the description is not as clear. We give a simple necessary condition in terms of the topological degree of continuous mapping. We also give some sufficient conditions for dimension 2. It also turns out that if the dimension is greater than one, then there exist real-analytic mappings with nonnegative Jacobian that cannot be approximated by smooth mappings with positive Jacobian. In our study of the above mentioned question we use topological degree theory, Schoenflies-type extension theorems, and Stoilow's topological characterization of complex analytic functions.

No description available.

In this work we study the following problem on constrained approximation. Let f be a continuous mapping defined on a bounded domain with piecewise smooth boundary in R^n and taking values in R^n. What are necessary and sufficient conditions for f to be uniformly approximable by C^1-smooth mappings with nonnegative Jacobian? When the dimension is equal to one, this is just approximation by monotone smooth functions. Hence, the necessary and sufficient condition is: the function is monotone. On the other hand, for higher dimensions the description is not as clear. We give a simple necessary condition in terms of the topological degree of continuous mapping. We also give some sufficient conditions for dimension 2. It also turns out that if the dimension is greater than one, then there exist real-analytic mappings with nonnegative Jacobian that cannot be approximated by smooth mappings with positive Jacobian. In our study of the above mentioned question we use topological degree theory, Schoenflies-type extension theorems, and Stoilow's topological characterization of complex analytic functions.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 127-132). Also available via the Internet.

Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 62-64). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Hong Kong, 1973. / Also available in print.

Thesis (Ph.D.)--Computing, Georgia Institute of Technology, 2008. / Committee Chair: Vazirani, Vijay; Committee Member: Cook, William; Committee Member: Kalai, Adam; Committee Member: Tetali, Prasad; Committee Member: Thomas, Robin