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Full-Text Articles in Physical Sciences and Mathematics
Maximally Inflected Real Rational Curves, Viatcheslav Kharlamov, Frank Sottile
Maximally Inflected Real Rational Curves, Viatcheslav Kharlamov, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in the Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.
On The Well-Posedness Of The Wave Map Problem In High Dimensions, Andrea Nahmod, Atanas Stefanov, Karen Uhlenbeck
On The Well-Posedness Of The Wave Map Problem In High Dimensions, Andrea Nahmod, Atanas Stefanov, Karen Uhlenbeck
Mathematics and Statistics Department Faculty Publication Series
We construct a gauge theoretic change of variables for the wave map from R × Rn into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - n ≥ 4 - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4.
On Schrodinger Maps, Andrea R. Nahmod, Atanas Stefanov, Karen Uhlenbeck
On Schrodinger Maps, Andrea R. Nahmod, Atanas Stefanov, Karen Uhlenbeck
Mathematics and Statistics Department Faculty Publication Series
We study the question of well-posedness of the Cauchy problem for Schr¨odinger maps from R 1 ×R 2 to the sphere S 2 or to H2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schr¨odinger system of equations and then study this modified Schr¨odinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schr¨odinger map itself from it. In …