No description available.

No description available.

In this dissertation we study a pioneering model of analog computation called General Purpose Analog Computer (GPAC), introduced by Shannon in 1941. The GPAC is capable of manipulating real-valued data streams. Its power is characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address two limitations of this model. The first is its fundamental inability to reason about functions of more than one independent variable (the `time' variable). In particular, the Shannon GPAC cannot be used to specify solutions of partial differential equations. The second concerns the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. To overcome these limitations, we extend the class of data types by taking channels carrying information on a general complete metric space X; for example the class of continuous functions of one real variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and add two new modules: a differential module which computes spatial derivatives; and a continuous limit module which computes limits. We then build networks using X-stream channels and the abovementioned modules. This leads us to a framework in which the speci cations of such analog systems are given by fi xed points of certain operators on continuous data streams, as considered by Tucker and Zucker. We study the properties of these analog systems and their associated operators. We present a characterization which generalizes Shannon's results. We show that some non-differentially algebraic functions such as the gamma function are generable by our model. Finally, we attempt to relate our model of computation to the notion of tracking computability as studied by Tucker and Zucker. / Thesis / Doctor of Philosophy (PhD)

No description available.

No description available.

No description available.

No description available.

No description available.

Thesis (Ed.D.)--Boston University / 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. / The present study investigated undergraduate students' understanding of slope fields and equilibrium solutions as they solved problems in differential equations. The following questions were addressed: 1) Does performance on complex problems vary by context (mathematical, non-mathematical)? 2) When considering a complex problem in a mathematical and a non-mathematical context, are participants who answer the problem in one context correctly more likely to answer the corresponding problem in the other context correctly? 3) Does performance on simple problems predict performance on complex problems? A written test, Differential Equations Concept Assessment (DECA), was designed and administered to 91 participants drawn from three introductory differential equations courses. Of those participants, 13 were interviewed. DECA consists of four complex problems, two in mathematical contexts and two in non-mathematical contexts, and six simple problems that assess aspects of slope fields and equilibrium solutions. The data obtained from DECA and the interviews showed that participants performed significantly better on complex problems in non-mathematical contexts than on complex problems in mathematical contexts. There was a significant relationship found between performance on a problem in a mathematical context and performance on the isomorphic problem in the context of population growth, but a significant relationship was not found between a different pair of isomorphic problems, one in a mathematical context and the other in the context oflearning. However, for all the complex problems, participants illustrated a preference for algebraic rather than geometric methods, even when a geometric approach was a more efficient method of solution. Although performance on simple problems was not found to be a strong predictor of performance on complex problems, the simple problems proved to elicit difficulties participants had with aspects of slope fields and equilibrium solutions. For example, participants were found to overgeneralize the notion of equilibrium solution as being any straight line and as existing at all values where a differential equation equals zero. Participants were also found to identify slope fields as determining only equilibrium solutions. / 2999-01-01

by Chi-chung Lee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 45-47). / INTRODUCTION --- p.1 / Chapter CHAPTER 1 --- RADIAL SYMMETRY OF GROUND STATES --- p.7 / Chapter CHAPTER 2 --- EXISTENCE OF A GROUND STATE --- p.14 / Chapter CHAPTER 3 --- UNIQUENESS OF GROUND STATE I --- p.23 / Chapter CHAPTER 4 --- UNIQUENESS OF GROUND STATE II --- p.35 / REFERENCES --- p.45