Codes on Graphs: Observability, Controllability, and Local Reducibility

Original manuscript: August 30, 2012This paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility. Trimness and properness are dual properties of constraint codes. A linear or group realization with a constraint code that is not both trim and proper is locally reducible. A linear or group realization on a finite cycle-free graph is minimal if and only if every local constraint code is trim and proper. A realization is called observable if there is a one-to-one correspondence between codewords and configurations, and controllable if it has independent constraints. A linear or group realization is observable if and only if its dual is controllable. A simple counting test for controllability is given. An unobservable or uncontrollable realization is locally reducible. Parity-check realizations are controllable if and only if they have independent parity checks. In an uncontrollable tail-biting trellis realization, the behavior partitions into disconnected sub-behaviors, but this property does not hold for nontrellis realizations. On a general graph, the support of an unobservable configuration is a generalized cycle

Google distance analysis of the MIT curriculum

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaf 50).The MIT curriculum has traditionally been organized into Departments and Schools, or groups of Departments. This influences degree titles and their requirements, subject listings seen by students and professors, and more. But subjects, Departments and Schools are not systematically analyzed or organized based on the knowledge they cover. The system built mined a subset of Institute subjects, taken from OpenCourseWare, for key topics, and performed Google searches on those terms. By composing keyword search result vectors for each subject, scores can be calculated between all pairs of subjects. These scores were used by an MDS layout algorithm and a Hierarchical Clustering algorithm to suggest two new organizations each for Department 6 (E.E. and C.S.) and for the MIT Departments into Schools. Convex hulls in MDS graphs of Department 6, colored based on classes a student has taken, are also used to predict new departments the student will take.by Casey Dugan.M.Eng