On fermionic representation of the framed topological vertex

The Gromov-Witten invariants of \mathbb{C}^3 with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al in the fermionic picture. We will propose a similar formula for the framed topological vertex and prove it in the case when there are one or two branes.Comment: An acknowledgement adde

Linear isometric invariants of bounded domains

We introduce two new conditions for bounded domains, namely $A^p$-completeness and boundary blow down type, and show that, for two bounded domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for some real number $p>0$ with $p\neq$ even integers, then $D_1$ and $D_2$ must be holomorphically equivalent, where for a domain $D$, $A^p(D)$ denotes the space of $L^p$ holomorphic functions on $D$.Comment: 14pages, comments welcome

Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds

We show that a singular Hermitian metric on a holomorphic vector bundle over a Stein manifold which is negative in the sense of Griffiths (resp. Nakano) can be approximated by a sequence of smooth Hermitian metrics with the same curvature negativity. We also show that a smooth Hermitian metric on a holomorphic vector bundle over a Stein manifold restricted to a submanifold which is negative in the sense of Griffiths (resp. Nakano) can be extended to the whole bundle with the same curvature negativity.Comment: 10pages. Comments welcome

AIM2 and Psoriasis

Psoriasis is a chronic inflammatory skin disease occurring worldwide, with multiple systemic complications, which seriously affect the quality of life and physical and mental health of patients. The pathogenesis of psoriasis is related to the environment, genetics, epigenetics, and dysregulation of immune cells such as T cells, dendritic cells (DCs), and nonimmune cells such as keratinocytes. Absent in melanoma 2 (AIM2), a susceptibility gene locus for psoriasis, has been strongly linked to the genetic and epigenetic aspects of psoriasis and increased in expression in psoriatic keratinocytes. AIM2 was found to be activated in an inflammasome-dependent way to release IL-1β and IL-18 to mediate inflammation, and to participate in immune regulation in psoriasis, or in an inflammasome-independent way by regulating the function of regulatory T(Treg) cells or programming cell death in keratinocytes as well as controlling the proliferative state of different cells. AIM2 may also play a role in the recurrence of psoriasis by trained immunity. In this review, we will elaborate on the characteristics of AIM2 and how AIM2 mediates the development of psoriasis

A new proof of Kiselman's minimum principle for plurisubharmonic functions

We give a new proof of Kiselman's minimum principle for plurisubharmonic functions, based on Ohsawa-Takegoshi extension theorem