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Full-Text Articles in Physical Sciences and Mathematics
Branches Of Radial Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji
Branches Of Radial Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji
All HMC Faculty Publications and Research
We consider the radially symmetric solutions to the equation −Δu(x) = λƒ(u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω denotes the unit ball in RN (N > 1), centered at the origin and λ > 0. Here ƒ: R→R is assumed to be semipositone (ƒ(0) < 0), monotonically increasing, superlinear with subcritical growth on [0, ∞). We establish the structure of radial solution branches for the above problem. We also prove that if ƒ is convex and ƒ(t)/(tƒ'(t)−ƒ(t)) is a nondecreasing function then for each λ > 0 there exists at most one positive solution u such that (λ, u) belongs to the unbounded branch of positive solutions. Further when ƒ(t) = tp − k, k > 0 and 1 < p < (N + 2)/(N − 2), we prove that the set of positive solutions is connected. Our results are motivated by and extend the developments in [4].
Distortion And Evolution Of A Localized Vortex In An Irrotational Flow, Joseph F. Lingevitch, Andrew J. Bernoff
Distortion And Evolution Of A Localized Vortex In An Irrotational Flow, Joseph F. Lingevitch, Andrew J. Bernoff
All HMC Faculty Publications and Research
This paper examines the interaction of an axisymmetric vortex monopole, such as a Lamb vortex, with a background irrotational flow. At leading order, the monopole is advected with the background flow velocity at the center of vorticity. However, inhomogeneities of the flow will cause the monopole to distort. It is shown that a shear‐diffusion mechanism, familiar from the study of mixing of passive scalars, plays an important role in the evolution of the vorticity distribution. Through this mechanism, nonaxisymmetric vorticity perturbations which do not shift the center of vorticity are homogenized along streamlines on a Re1/3 time scale, much faster …
Descartes And Problem-Solving, Judith V. Grabiner
Descartes And Problem-Solving, Judith V. Grabiner
Pitzer Faculty Publications and Research
What can Descartes' Geometry teach us about problem solving?