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PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
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We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
THE TWODIMENSIONAL LAZERMCKENNA CONJECTURE FOR AN EXPONENTIAL NONLINEARITY
, 2006
"... Abstract. We consider the problem of AmbrosettiProdi type ∆u + e u = sφ1 + h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded, smooth domain in R 2, φ1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h ∈ C 0,α ( ¯ Ω). We prove that given k ≥ 1 this problem has ..."
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Abstract. We consider the problem of AmbrosettiProdi type ∆u + e u = sφ1 + h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded, smooth domain in R 2, φ1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h ∈ C 0,α ( ¯ Ω). We prove that given k ≥ 1 this problem has at least k solutions for all sufficiently large s> 0, which answers affirmatively a conjecture by Lazer and McKenna [22] for this case. The solutions found exhibit multiple concentration behavior around maxima of φ1 as s → +∞.
Application of numerical continuation to compute all solutions of semilinear elliptic equations
"... (Communicated by A. Sommese) Abstract. We adapt numerical continuation methods to compute all solutions of finite difference discretizations of nonlinear boundary value problems involving the Laplacian in two dimensions. New solutions on finer meshes are obtained from solutions on coarser meshes usi ..."
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(Communicated by A. Sommese) Abstract. We adapt numerical continuation methods to compute all solutions of finite difference discretizations of nonlinear boundary value problems involving the Laplacian in two dimensions. New solutions on finer meshes are obtained from solutions on coarser meshes using a complex homotopy deformation. Two difficulties arise. First, the number of solutions typically grows with the number of mesh points and some form of filtering becomes necessary. Secondly, bifurcations may occur along homotopy paths of solutions and efficient methods to swap branches are developed when the mappings are analytic. For polynomial nonlinearities we generalize an earlier strategy for finding all solutions of twopoint boundary value problems in one dimension and then introduce exclusion algorithms to extend the method to general nonlinearities. 1