A note on computing environments and network reliability

Interest has increased during recent years in using microcomputers for implementation of network reliability algorithms. There are indications that some implementations suffer severe restrictions imposed by such a computing environment, whereas some other results indicate otherwise. This brief note attempts to clarify the apparently contradictory conclusions that would be naturally drawn from recent papers that treat this subject.     

Output probability from a general combinational network

In this paper, we present the method which calculates the probability of the output of a general combinational network being 1, when the probabilities is given for each input being 1. Also we present the method which derives the output probability expression in terms of a given set of input probabilities.The method is based on a basic probability axiom and Binary Decision Diagrams. A Boolean function for a combinational network is transformed into a Binary Decision Diagram and the output probability expression is derived from it. Therefore a derivation of the output probability for a combinational network can be straightforward. Some examples are showed that the method using Binary Decision Diagrams is simple and efficient. We can see a variation of the output probability for a combinational network according to the given set of input probabilities.     

Contractibility of Efficient Frontier of Simply Shaded Sets

In finite dimensional Euclidean space, we prove the contractibility of the efficient frontier of simply shaded sets. This work extends the result of Peleg [7], which confirms the contractibility of the efficient frontier in the convex case.     

Chaotic resonance: A simulation

Stochastic resonance is a statistical phenomenon that has been observed in periodically modulated, noise-driven, bistable systems. The characteristic signatures of the effect include an increase in the signal-to-noise of the output as noise is added to the system, and exponentially decreasing peaks in the probability density as a function of residence times in one state. Presented are the results of a numerical simulation where these same signatures were observed by adding achaotic driving term instead of a white noise term. Although the probability distributions of the noise and chaos inputs were significantly different, the stochastic and chaotic resonances were equal within the experimental error.     

Geometric Knowledge of Middle School Students in a Reform-based Mathematics Curriculum

National and international studies have found U.S. elementary students to be weak in their understandings and applications of geometric concepts. The University of Chicago School Mathematics Project's (UCMSP) Everyday Mathematics Program is one of the current reform-based elementary curricula incorporating geometry throughout the K-6 curriculum, with an emphasis on hands-on and problem-solving activities. In this study, the geometric knowledge of fifth and sixth graders using the UCSMP curriculum is compared to the knowledge of students using more traditional curricula. Because UCSMP students had been in the program since kindergarten, this research attempts to measure the longitudinal effects of such an approach. Along with an overall score, a subset of test items was used to assign each student a van Hiele level for geometric thinking, as well as a reasoning score. On all measures, UCSMP students substantially outperformed their counterparts, and nearly all differences were significant. Aspects of the UCSMP curriculum and the van Hiele model for learning geometry are discussed relative to these results.     

A System for Three-Dimensional Interactive Simulation of Hand Biomechanics

A real-time interactive graphics simulation of the mechanics of the human thumb has been developed. The simulation utilizes a realistic data structure for the bones of the thumb which may be expanded to include the hand and forearm. This is coupled to a model of the kinematics of the joints in a concurrent processing arrangement in which the dynamic graphical transformations for scaling, translation, rotation, perspective, and clipping are all performed on a special-purpose display processor. The model and the control functions are distributed between this processor and a separate general-purpose superminicomputer. The resulting system presents the user with a realistic simulation of the movements of the thumb in normal and impaired states. The user may choose from a menu of options, including an interactive tendon transfer simulation for the current hand being simulated. He may control (interactively and in real time) the view, observation position, skeletal motion, and parameters for use in the model of joint mechanics. This type of computer modeling, utilizing a realistic three-dimensional data structure, models of musculoskeletal kinematics, and interactive programming, shows great potential for bringing mathematical modeling into useful clinical, research, and educational applications.