On the Clifford-Fourier transform

For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) := e^{\frac{i \pi}{2} \Gamma_{y}}e^{-i }$, replacing the kernel $e^{i }$ of the ordinary Fourier transform, where $\Gamma_{y} := - \sum_{j<k} e_{j}e_{k} (y_{j} \partial_{y_{k}} - y_{k}\partial_{y_{j}})$. An explicit formula of $K(x,y)$ is derived, which can be further simplified to a finite sum of Bessel functions when $m$ is even. The closed formula of the kernel allows us to study the Clifford-Fourier transform and prove the inversion formula, for which a generalized translation operator and a convolution are defined and used.Comment: Some small changes, 30 pages, accepted for publication in IMR

Low dimensional cohomology of general conformal algebras gcNgc_N

We compute the low dimensional cohomologies $\tilde H^q(gc_N,C)$, H^q(gc_N,\C) of the infinite rank general Lie conformal algebras $gc_N$ with trivial coefficients for $q\le3, N=1$ or $q\le2, N\ge2$. We also prove that the cohomology of $gc_N$ with coefficients in its natural module is trivial, i.e., H^*(gc_N,\C[\ptl]^N)=0; thus partially solve an open problem of Bakalov-Kac-Voronov in [{\it Comm. Math. Phys.,} {\bf200} (1999), 561-598].Comment: 18 page

Making sense o fhormone-mediated defense networking: from rice to Arabidopsis

Phytohormones are not only essential for plant growth and development but also play central roles in triggering the plant immune signaling network. Historically, research aimed at elucidating the defense-associated role of hormones has tended to focus on the use of experimentally tractable dicot plants such as Arabidopsis thaliana. Emerging from these studies is a picture whereby complex crosstalk and induced hormonal changes mold plant health and disease, with outcomes largely dependent on the lifestyle and infection strategy of invading pathogens. However, recent studies in monocot plants are starting to provide additional important insights into the immune-regulatory roles of hormones, often revealing unique complexities. In this review, we address the latest discoveries dealing with hormone-mediated immunity in rice, one of the most important food crops and an excellent model for molecular genetic studies in monocots. Moreover, we highlight interactions between hormone signaling, rice defense and pathogen virulence, and discuss the differences and similarities with findings in Arabidopsis. Finally, we present a model for hormone defense networking in rice and describe how detailed knowledge of hormone crosstalk mechanisms can be used for engineering durable rice disease resistance

Edge States and Quantum Hall Effect in Graphene under a Modulated Magnetic Field

Graphene properties can be manipulated by a periodic potential. Based on the tight-binding model, we study graphene under a one-dimensional (1D) modulated magnetic field which contains both a uniform and a staggered component. New chiral current-carrying edge states are generated at the interfaces where the staggered component changes direction. These edge states lead to an unusual integer quantum Hall effect (QHE) in graphene, which can be observed experimentally by a standard four-terminal Hall measurement. When Zeeman spin splitting is considered, a novel state is predicted where the electron edge currents with opposite polarization propagate in the opposite directions at one sample boundary, whereas propagate in the same directions at the other sample boundary.Comment: 5 pages, 4 figure

An updated catalog of M31 globular-like clusters: UBVRI photometry, ages, and masses

We present an updated UBVRI photometric catalog containing 970 objects in the field of M31, selected from the Revised Bologna Catalog (RBC v.4.0), including 965, 967, 965, 953, and 827 sources in the individual UBVRI bands, respectively, of which 205, 123, 14, 126, and 109 objects do not have previously published photometry. Photometry is performed using archival images from the Local Group Galaxies Survey, which covers 2.2 deg^2 along the major axis of M31. We focus on 445 confirmed `globular-like' clusters and candidates, comprising typical globular and young massive clusters. The ages and masses of these objects are derived by comparison of their observed spectral-energy distributions with simple stellar population synthesis. Approximately half of the clusters are younger than 2 Gyr, suggesting that there has been significant recent active star formation in M31, which is consistent with previous results. We note that clusters in the halo (r_ projected>30kpc) are composed of two different components, older clusters with ages >10 Gyr and younger clusters with ages around 1 Gyr. The spatial distributions show that the young clusters (<2 Gyr) are spatially coincident with the galaxy's disk, including the `10 kpc ring,' the `outer ring,' and the halo of M31, while the old clusters (> 2 Gyr) are spatially correlated with the bulge and halo. We also estimate the masses of the 445 confirmed clusters and candidates in M31 and find that our estimates agree well with previously published values. We find that none of the young disk clusters can survive the inevitable encounters with giant molecular clouds in the galaxy's disk and that they will eventually disrupt on timescales of a few Gyr. Specifically, young disk clusters with a mass of 10^4 M_\odot are expected to dissolve within 3.0 Gyr and will, thus, not evolve to become globular clusters.Comment: 35 pages, 20 figures and 5 tables, accepted for publication in Ap