Law of large numbers for branching symmetric Hunt processes with measure-valued branching rates

We establish weak and strong law of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state-dependent. Our work is motivated by recent work on strong law of large numbers for branching symmetric Markov processes by Chen-Shiozawa [J. Funct. Anal., 250, 374--399, 2007] and for branching diffusions by Engl\"ander-Harris-Kyprianou [Ann. Inst. Henri Poincar\'e Probab. Stat., 46, 279--298, 2010]. Our results can be applied to some interesting examples that are covered by neither of these papers

Strong law of large numbers for supercritical superprocesses under second moment condition

Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$\psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)}(e^{-\lambda y}-1+\lambda y)n(x,dy), \quad x\in E, \quad\lambda> 0,$$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$\sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty.$$ Put $T_tf(x)=\mathbb{P}_{\delta_x}$. Let $\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\hat{\phi}_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $\lambda_0$. Under some conditions on the spatial motion and the $\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$\lim_{t\rightarrow\infty}e^{-\lambda_0 t}< f, X_t> = W_\infty\int_E\hat{\phi}_0(y)f(y)m(dy),\quad \mathbb{P}_{\mu}{-a.s.},$$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$

Comparative validation of the D. melanogaster modENCODE transcriptome annotation

Accurate gene model annotation of reference genomes is critical for making them useful. The modENCODE project has improved the D. melanogaster genome annotation by using deep and diverse high-throughput data. Since transcriptional activity that has been evolutionarily conserved is likely to have an advantageous function, we have performed large-scale interspecific comparisons to increase confidence in predicted annotations. To support comparative genomics, we filled in divergence gaps in the Drosophila phylogeny by generating draft genomes for eight new species. For comparative transcriptome analysis, we generated mRNA expression profiles on 81 samples from multiple tissues and developmental stages of 15 Drosophila species, and we performed cap analysis of gene expression in D. melanogaster and D. pseudoobscura. We also describe conservation of four distinct core promoter structures composed of combinations of elements at three positions. Overall, each type of genomic feature shows a characteristic divergence rate relative to neutral models, highlighting the value of multispecies alignment in annotating a target genome that should prove useful in the annotation of other high priority genomes, especially human and other mammalian genomes that are rich in noncoding sequences. We report that the vast majority of elements in the annotation are evolutionarily conserved, indicating that the annotation will be an important springboard for functional genetic testing by the Drosophila community

Bis{2-[(E)-benzyl­imino­meth­yl]-4-methyl­phenolato-κ2 N,O}nickel(II)

In the title complex, [Ni(C15H14NO)2], the NiII atom is located on an inversion centre and is coordinated by two O and two N atoms from two symmetry-related bidentate Schiff base ligands in a slightly distorted square-planar geometry. The phenyl and benzene rings in the ligand mol­ecule form a dihedral angle of 72.79 (8)°