Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets

Supertranslations to all orders

The transformation laws of the general linear superfield and chiral superfields under N=1 supertranslations are tabulated to all orders in the supertranslation parameters.Comment: 14 page

Lessons Learned in a Math Excel Workshop: The Importance of Maintaining High Cognitive Demands

Uri Treisman\u27s Emerging Scholars Workshop model has been implemented on many college campuses over the last twenty years. The Treisman model is based on groups of students meeting regularly in a social atmosphere to work collaboratively in solving challenging mathematics problems related to their introductory coursework. Emerging Scholars Programs (or Math Excel as it is called in many settings, including ours) have been particularly successful in increasing the academic success and participation of underrepresented groups in mathematics. The primary responsibilities of a workshop leader include the design of a session’s worksheet, as well as the facilitation of students\u27 problem solving efforts during the workshop session itself. In this paper, we discuss a mathematical tasks framework proposed by researchers in the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project that may be especially helpful to workshop leaders in making a successful implementation of Math Excel. This framework emphasizes the notion of the cognitive demand of a mathematical task. The level of cognitive demand is not a static attribute and may well change as students undertake a task in a classroom setting. QUASAR researchers noted how the initially high demands of a task may not be maintained in the classroom, and how teachers\u27 actions may lower the demands and consequently limit learning opportunities for students. Although the QUASAR project involved middle school mathematics instruction, we believe that this mathematical tasks framework can provide valuable lessons for Math Excel workshop leaders, and it suggests how critically important both the choice of problem tasks and the workshop leaders’ facilitation of student work can be. In this paper, we review the mathematical tasks framework and illustrate its application to scenarios actually encountered in our Math Excel workshops