ar X iv : 0 70 5 . 42 46 v 2 [ m at h . G R ] 3 1 M ay 2 00 7 The equation w ( x , y ) = u over free groups

ar X iv : 0 70 5 . 46 61 v 1 [ m at h . A G ] 3 1 M ay 2 00 7 ALGEBRAIC CYCLES AND MUMFORD - GRIFFITHS INVARIANTS

Let X be a projective algebraic manifold and let CH(X) be the Chow group of algebraic cycles of codimension r on X, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration {F }ν≥0 on CH(X)⊗Q due to the second author, we construct a space of arithmetic Hodge theoretic invariants ∇J(X) and corresponding map φ X : Gr F CH(X)⊗Q → ∇J(X), and determine conditions on X for whi...

ar X iv : 0 80 5 . 37 89 v 1 [ m at h . A P ] 2 4 M ay 2 00 8 The balance between diffusion and absorption in semilinear parabolic equations ∗

Let h : [0,∞) 7→ [0,∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m,q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = uk of ∂tu − ∆u m + h(t)u = 0 in Ω × (0, T ) satisfying uk(x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the asso...

ar X iv : 0 70 5 . 10 42 v 1 [ m at h . M G ] 8 M ay 2 00 7 NON - POSITIVE CURVATURE AND THE PTOLEMY INEQUALITY

We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.

ar X iv : 0 70 7 . 03 46 v 1 [ m at h - ph ] 3 J ul 2 00 7 THE ONE - DIMENSIONAL SCHRÖDINGER - NEWTON EQUATIONS

We prove an existence and uniqueness result for ground states of one-dimensional Schrödinger-Newton equations.

ar X iv : 0 70 5 . 31 41 v 1 [ as tr o - ph ] 2 2 M ay 2 00 7 The SARG Planet Search