Bilateral Hardy-type inequalities

This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.Comment: 40 pages, 2 figures; Acta Math. Sin. Eng. Ser. 201

A Nonabelian (1,0)(1,0) Tensor Multiplet Theory in 6D

We construct a general nonabelian (1,0) tensor multiplet theory in six dimensions. The gauge field of this (1,0) theory is non-dynamical, and the theory contains a continuous parameter $b$. When $b=1/2$, the (1,0) theory possesses an extra discrete symmetry enhancing the supersymmetry to (2,0), and the theory turns out to be identical to the (2,0) theory of Lambert and Papageorgakis (LP). Upon dimension reduction, we obtain a general ${\cal N}=1$ supersymmetric Yang-Mills theory in five dimensions. The applications of the theories to D4 and M5-branes are briefly discussed.Comment: 18 pages, published in JHEP. minor changes, references adde

Isospectral operators

For a large class of integral operators or second order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of $h$-transform (duality). This provides us a simple way to extend the known knowledge on the spectrum (or the estimation of the principal eigenvalue) from a smaller class of operators to a much larger one. In particular, an open problem about the positivity of the principal eigenvalue for birth--death processes is solved in the paper.Comment: Corrections are made in this versio

Symplectic Three-Algebra Unifying N=5,6 Superconformal Chern-Simons-Matter Theories

We define a 3-algebra with structure constants being symmetric in the first two indices. We also introduce an invariant anti-symmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The general N=5 superconformal Chern-Simons-matter (CSM) theory with SO(5) R-symmetry in three dimensions is constructed by using this algebraic structure. We demonstrate that the supersymmetry can be enhanced to N=6 if the sympelctic 3-algebra and the fields are decomposed in a proper fashion. By specifying the 3-brackets, some presently known N=5, 6 superconformal theories are described in terms of this unified 3-algebraic framework. These include the N=5, Sp(2N) X O(M) CSM theory with SO(5) R-symmetry , the N=6, Sp(2N) X U(1) CSM theory with SU(4) R-symmetry, as well as the ABJM theory as a special case of U(M) X U(N) theory with SU(4) R-symmetry.Comment: 31 pages, minor changes, final results remain the sam