A versatile approach to calculus and numerical methods

Traditionally the calculus is the study of the symbolic algorithms for differentiation and integration, the relationship between them, and their use in solving problems. Only at the end of the course, when all else fails, are numerical methods introduced, such as the Newton-Raphson method of solving equations, or Simpson’s rule for calculating areas. The problem with such an approach is that it often produces students who are very well versed in the algorithms and can solve the most fiendish of symbolic problems, yet have no understanding of the meaning of what they are doing. Given the arrival of computer software which can carry out these algorithms mechanically, the question arises as to what parts of calculus need to be studied in the curriculum of the future. It is my contention that such a study can use the computer technology to produce a far more versatile approach to the subject, in which the numerical and graphical representations may be used from the outset to produce insights into the fundamental meanings, in which a wider understanding of the processes of change and growth will be possible than the narrow band of problems that can be solved by traditional symbolic methods of the calculus

Transition Delay in Hypervelocity Boundary Layers By Means of CO₂/Acoustic Instability Interaction

The potential for hypervelocity boundary layer stabilization was investigated using the concept of damping Mack’s second mode disturbances by vibrational relaxation of carbon dioxide (CO₂) within the boundary layer. Experiments were carried out in the Caltech T5 hypervelocity shock tunnel and the Caltech Mach 4 Ludwieg tube. The tests used 5-degree half-angle cones (at zero angle of attack) equipped near the front of the cone with an injector consisting of either discrete holes or a porous section. Gaseous CO₂, argon (Ar) and air were injected into the boundary layer and the effect on boundary layer stability was evaluated by optical visualization, heat flux measurements and numerical simulation. In T5, tests were carried out with CO₂ in the free stream as well as injection. Injection experiments in T5 were inconclusive; however, experiments with mixtures of air/CO₂ in the free stream demonstrated a clear stabilizing effect, limiting the predicted amplification N-factors to be less than 13. During the testing activities in T5, significant improvements were made in experimental technique and data analysis. Testing in the Ludwieg tube enabled optical visualization and the identification of a shear-layer like instability downstream of the injector. Experiments showed and numerical simulation confirmed that injection has a destabilizing influence beyond a critical level of injection mass flow rate. A modified injection geometry was tested in the Ludwieg tube and we demonstrated that it was possible to cancel the shock wave created by injection under carefully selected conditions. However, computations indicate and experiments demonstrate that shear-layer like flow downstream of the porous wall injector is unstable and can transition to turbulence while the injected gas is mixing with the free stream. We conclude that the idea of using vibrational relaxation to delay boundary layer transition is a sound concept but there are significant practical issues to be resolved to minimize the flow disturbance associated with introducing the vibrationally-active gas into the boundary layer

An Immersive Telepresence System using RGB-D Sensors and Head Mounted Display

We present a tele-immersive system that enables people to interact with each other in a virtual world using body gestures in addition to verbal communication. Beyond the obvious applications, including general online conversations and gaming, we hypothesize that our proposed system would be particularly beneficial to education by offering rich visual contents and interactivity. One distinct feature is the integration of egocentric pose recognition that allows participants to use their gestures to demonstrate and manipulate virtual objects simultaneously. This functionality enables the instructor to ef- fectively and efficiently explain and illustrate complex concepts or sophisticated problems in an intuitive manner. The highly interactive and flexible environment can capture and sustain more student attention than the traditional classroom setting and, thus, delivers a compelling experience to the students. Our main focus here is to investigate possible solutions for the system design and implementation and devise strategies for fast, efficient computation suitable for visual data processing and network transmission. We describe the technique and experiments in details and provide quantitative performance results, demonstrating our system can be run comfortably and reliably for different application scenarios. Our preliminary results are promising and demonstrate the potential for more compelling directions in cyberlearning.Comment: IEEE International Symposium on Multimedia 201

Time-varying Learning and Content Analytics via Sparse Factor Analysis

We propose SPARFA-Trace, a new machine learning-based framework for time-varying learning and content analytics for education applications. We develop a novel message passing-based, blind, approximate Kalman filter for sparse factor analysis (SPARFA), that jointly (i) traces learner concept knowledge over time, (ii) analyzes learner concept knowledge state transitions (induced by interacting with learning resources, such as textbook sections, lecture videos, etc, or the forgetting effect), and (iii) estimates the content organization and intrinsic difficulty of the assessment questions. These quantities are estimated solely from binary-valued (correct/incorrect) graded learner response data and a summary of the specific actions each learner performs (e.g., answering a question or studying a learning resource) at each time instance. Experimental results on two online course datasets demonstrate that SPARFA-Trace is capable of tracing each learner's concept knowledge evolution over time, as well as analyzing the quality and content organization of learning resources, the question-concept associations, and the question intrinsic difficulties. Moreover, we show that SPARFA-Trace achieves comparable or better performance in predicting unobserved learner responses than existing collaborative filtering and knowledge tracing approaches for personalized education