Transition Fronts in Time Heterogeneous and Random Media of Ignition Type

The current paper is devoted to the investigation of wave propagation phenomenon in reaction-diffusion equations with ignition type nonlinearity in time heterogeneous and random media. It is proven that such equations in time heterogeneous media admit transition fronts or generalized traveling wave solutions with time dependent profiles and that such equations in time random media admit generalized traveling wave solutions with random profiles. Important properties of generalized traveling wave solutions, including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts, are also obtained

Bounds of Riesz Transforms on LpL^p Spaces for Second Order Elliptic Operators

For any fixed $p>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transforms associated with second order elliptic operators with real, symmetric, bounded measurable coefficients.Comment: To appear in Annales de L'Institut Fourier (Grenoble

Cobordism invariants of the moduli space of stable pairs

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.Comment: 21 pages. Some typos fixed. To appear in J. London Math. So