Unitarity constraints on chiral perturbative amplitudes

Low lying scalar resonances emerge as a necessary part to adjust chiral perturbation theory to experimental data once unitarity constraint is taken into consideration. I review recent progress made in this direction in a model independent approach. Also I briefly review studies on the odd physical properties of these low lying scalar resonances, including in the $\gamma\gamma\to\pi^+\pi^-, \pi^0\pi^0$ processes.Comment: Talk given at: International Workshop on Effective Field Theories: from the pion to the upsilon, February 2-6 2009, Valencia, Spai

Boundary expansions and convergence for complex Monge-Ampere equations

We study boundary expansions of solutions of complex Monge-Ampere equations and discuss the convergence of such expansions. We prove a global conver- gence result under that assumption that the entire boundary is analytic. If a portion of the boundary is assumed to be analytic, the expansions may not converge locally

Boundary expansions for minimal graphs in the hyperbolic space

We study expansions near the boundary of solutions to the Dirichlet problem for minimal graphs in the hyperbolic space and characterize the remainders of the expansion by multiple integrals. With such a characterization, we establish optimal asymptotic expansions of solutions with boundary values of finite regularity and demonstrate a slight loss of regularity for nonlocal coefficients

On the Local Isometric Embedding in R^3 of Surfaces with Gaussian Curvature of Mixed Sign

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings exist.Comment: Comm. Anal. Geom., to appear, 47 page

Smooth local solutions to weingarten equations and σk\sigma_k-equations

In this paper, we study the existence of smooth local solutions to Weingarten equations and $\sigma_k$-equations. We will prove that, for $2 \leq k \leq n$, the Weingarten equations and the $\sigma_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associate linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately

Scalar resonance at 750 GeV as composite of heavy vector-like fermions

We study a model of scalars which includes both the SM Higgs and a scalar singlet as composites of heavy vector-like fermions. The vector-like fermions are bounded by the super-strong four-fermion interactions. The scalar singlet decays to SM vector bosons through loop of heavy vector-like fermions. We show that the surprisingly large production cross section of di-photon events at 750 GeV resonance and the odd decay properties can all be explained. This model serves as a good model for both SM Higgs and a scalar resonance at 750 GeV.Comment: 12 pages, no figure, references updated, version for publicatio