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.

Dietary Intakes and Periodontal Outcomes After Sanative Therapy

Diet has an important role in the maintenance of oral health, but the relationship between diet and clinical outcomes following sanative therapy (ST) has not been investigated. Due to their antioxidant and anti-inflammatory properties, we hypothesized that periodontal patients with higher intakes of vitamin C, vitamin D, eicosapentaenoic acid (EPA) or docosahexaenoic acid (DHA) would have greater reductions in probing depth (PD) after ST. Patients completed the Block food frequency questionnaire, a supplement use questionnaire and had their serum 25-hydroxyvitamin D measured. There were no significant associations between intakes of vitamin C, vitamin D, EPA, DHA and PD. There were, however, negative associations between intakes of linoleic acid, α- linolenic acid or total vegetable intake and PD, as well as a positive association between the total omega-6/omega-3 ratio and PD (p < 0.05). Therefore, dietary intakes of essential fatty acids and vegetables may be important modulators of periodontal outcomes following ST.