Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.

A comparison of two computer-assisted remedial reading programs for adolescent unskilled readers : component reading skills and repeated reading

This study compared the relative effectiveness of two computerized remedial reading programs in improving the reading word recognition, rate, and comprehension of adolescent readers demonstrating significant and longstanding reading difficulties. One of the programs involved was Autoskill Component Reading Subskills Program, which provides instruction in isolated letters, syllables, and words, to a point of rapid automatic responding. This program also incorporates reading disability subtypes in its approach. The second program, Read It Again. Sam, delivers a repeated reading strategy. The study also examined the feasibility of using peer tutors in association with these two programs. Grade 9 students at a secondary vocational school who satisfied specific criteria with respect to cognitive and reading ability participated. Eighteen students were randomly assigned to three matched groups, based on prior screening on a battery of reading achievement tests. Two I I groups received training with one of the computer programs; the third group acted as a control and received the remedial reading program offered within the regular classroom. The groups met daily with a trained tutor for approximately 35 minutes, and were required to accumulate twenty hours of instruction. At the conclusion of the program, the pretest battery was repeated. No significant differences were found in the treatment effects of the two computer groups. Each of the two treatment groups was able to effect significantly improved reading word recognition and rate, relative to the control group. Comprehension gains were modest. The treatment groups demonstrated a significant gain, relative to the control group, on one of the three comprehension measures; only trends toward a gain were noted on the remaining two measures. The tutoring partnership appeared to be a viable alternative for the teacher seeking to provide individualized computerized remedial programs for adolescent unskilled readers. Both programs took advantage of computer technology in providing individualized drill and practice, instant feedback, and ongoing recordkeeping. With limited cautions, each of these programs was considered effective and practical for use with adolescent unskilled readers.