The Microscope (Allen, R. M.)

Frederick M. Allen, 1916

Multiple-end Solutions to the Allen-cahn Equation in R

We construct new class of entire solutions of the Allen-Cahn equation ∆u+(1−u2)u = 0, in R2(∼ C). Given k ≥ 1, we find a family of solutions whose zero level sets are, away from a compact set, asymptotic to 2k straight lines (which we call the ”ends”). These solutions have the property that there exists θ0 < θ1 < . . . < θ2k = θ0 + 2π such that limr→+∞ u(re iθ) = (−1)j uniformly in θ on compact...

Two-end solutions to the Allen-Cahn equation in R

has been studied for several decades and is an important nonlinear PDE due to the fact that it lies at the interface of several different mathematical fields. The famous De Giorgi conjecture states that any entire solution to (1) in R which is monotone in one direction should be one dimensional, at least for n ≤ 8. The conjecture was proved in dimension n = 2 by Ghoussoub-Gui ([13]) and dimensi...

MACRODYSTROPHIA LIPOMATOSA: CT AND M R FINDINGS

Macrodystrophia lipomatosa is a rare congenital form of localized gigantism characterized by an increase in all mesenchymal elements, particularly fibroadipose tissue. MR imaging can be used to examine the soft tissue and detect the fibrous and fatty hypertrophy that characterizes this condition and also diagnose an associated fibrolipomatous hamartoma of the median or plantar nerve. We des...

Solvable Lie algebras with $N(R_n,m,r)$ nilradical

In this paper, we classify the indecomposable non-nilpotent solvable Lie algebras with $N(R_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $N(R_n,m,r)$.We also prove that these solvable Lie algebras are complete and unique, up to isomorphism.