Keywords:Stochastic differential equations of jump type Existence and uniqueness Invariant measuresThe purpose of this paper is twofold. Firstly, we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi-martingales. Secondly, we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent.

Semiconducting molecules of perylene diimide (PDI) with strong light absorption properties in the near-infrared region and good biocompatibility have received increasing attention in the field of theranostics, especially as photoacoustic (PA) imaging agents. Herein, we report a series of [Cu]-labeled PDI nanoparticles (NPs) of different sizes (30, 60, 100, and 200 nm) as dual positron emission tomography (PET) and PA imaging probes and photothermal therapy agents. The precise size control of the PDI NPs can be achieved by adjusting the initial concentration of PDI molecules in the self-assembly process, and the photophysical property of different sized PDI NPs was studied in detail. Furthermore, we systematically investigated the size-dependent accumulation of the PDI NPs in the lymphatic system after local administration and in tumors after intravenous injection by PA and PET imaging. The results revealed that 100 nm is the best size for differentiating popliteal and sciatic LNs since the interval is around 60 min for the NPs to migrate from popliteal LNs to sciatic LNs, which is an ideal time window to facilitate surgical sentinel LN biopsy and pathological examination. Furthermore, different migration times of the different-sized PDI NPs will provide more choices for surgeons to map the specific tumor relevant LNs. PDI NP theranostics can also be applied to imaging-guided cancer therapy. The NPs with a size of 60 nm appear to be the best for tumor imaging and photothermal cancer therapy due to the maximum tumor accumulation efficiency. Thus, our study not only presents organic PDI NP theranostics but also introduces different-sized NPs for multiple bioapplications.

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Yield is one of the most important traits for rapeseed (Brassica napus L.) breeding, but its genetic basis remains largely ambiguous. Association mapping has provided a robust approach to understand the genetic basis of complex agronomic traits in crops. In this study, a panel of 192 inbred lines of B. napus from all over the world was genotyped using 451 single-locus microsatellite markers and 740 amplified fragment length polymorphism markers. Six yield-related traits of these inbred lines were investigated in three consecutive years with three replications, and genome-wide association studies were conducted for these six traits. Using the model controlling both population structure and relative kinship (Q + K), a total of 43 associations (P < 0.001) were detected using the means of the six yield-related traits across 3 years, with two to fourteen markers associated with individual traits. Among these, 18 markers were repeatedly detected in at least 2 years, and 12 markers were located within or close to QTLs identified in previous studies. Six markers commonly associated with correlated traits. Conditional association analysis indicated that five of the associations between markers and correlated traits are caused by one QTL with pleiotropic effects, and the remaining association is caused by linked but independent QTLs. The combination of favorable alleles of multiple associated markers significantly enhances trait performance, illustrating a great potential of utilization of the associations in rapeseed breeding programs.