On resolving singularities

Let V be an irreducible affine algebraic variety over a field k of characteristic zero, and let (f_0,...,f_m) be a sequence of elements of the coordinate ring. There is probably no elementary condition on the f_i and their derivatives which determines whether the blowup of V along (f_0,...,f_m) is nonsingular. The result is that there indeed is such an elementary condition, involving the first and second derivatives of the $f_i,$ provided we admit certain singular blowups, all of which can be resolved by an additional Nash blowup. There is is a particular explicit sequence of ideals R=J_0, J_1, J_2,... \subset R so that V_i=Bl_{J_i}V is the i'th Nash blowup of V, with J_i|J_{i+1} for all i. Applying our earlier paper, V_i is nonsingular if and only if the ideal class of J_{i+1} divides some power of the ideal class of J_i. The present paper brings things down to earth considerably: such a divisibility of ideal classes implies that for some N\ge r+2 J_i^{N-r-2}J_{i+1}^{r+3}=J_i^NJ_{i+2}. Yet note that this identity in turn implies J_{i+2} is a divisor of some power of J_{i+1}. Thus although $V_i$ may fail to be nonsingular, when the identity holds the {\it next} variety V_{i+1} must be nonsingular. Thus the Nash question is equivalent to the assertion that the identity above holds for some sufficiently large i and N

Lloyd Morgan\u27s interpretation of emergent evolution

This thesis represents research in the general field of emergent evolution. The works of Lloyd Morgan and s. Alexander were studied in particular and the works of others in allied fields were used in a supplementary capacity. It maintains that Morgan offers a unique personal interpretation or the universe. Its purpose is to give an understanding of Morgan\u27e scheme and to call attention to certain problems within the limits of his theory

Letter from John Moody McCaleb to V. M. Metcalfe

Letter from John Moody McCaleb to V. M. Metcalfe dated 31 January 1936. McCaleb addresses Metcalfe as Uncle Minor. The correspondence is handwritten

Lloyd Morgan\u27s interpretation of emergent evolution

This thesis represents research in the general field of emergent evolution. The works of Lloyd Morgan and s. Alexander were studied in particular and the works of others in allied fields were used in a supplementary capacity. It maintains that Morgan offers a unique personal interpretation or the universe. Its purpose is to give an understanding of Morgan\u27e scheme and to call attention to certain problems within the limits of his theory

A study of mathematics course sequence and student performance in mathematics in a block scheduled high school

The 1983 study of the condition of American Education, A Nation at Risk, brought a significant amount of attention to the way in which schools provided students with the skills necessary to be productive citizens. Many of the recommendations contained in A Nation at Risk focused on the perceived poor performance of American students when compared to their counterparts worldwide. The year after publication of A Nation at Risk, the National Commission on Time and Learning published Prisoners of Time, a report that called for the establishment of high academic standards for students and a major restructuring of the school day to provide more time for focus on providing students with additional time to focus on content area subjects. The additional time, it was proposed, would help our students to become more competitive with their peers internationally. These two powerful documents provided the fuel necessary to ignite efforts to restructure the school day, particularly in high schools. While there have been several school restructuring initiatives, block scheduling has received much attention. Block scheduling radically alters the way in which schools organize the instructional day for students and staff. The most prominent form of block scheduling is the 4 x 4 block, which results in ninety-minute classes four times per day for each of two semesters. Thus, students take eight courses during the school year, four during each semester. Under this model, students complete a year of work in half the time. One consequence of block scheduling is that students may experience significant gaps in sequential courses, particularly in mathematics and foreign language. The extent to which this gap affects student achievement in mathematics was the focus of this study