Favorite Downcrossing Sites of One-Dimensional Simple Random Walk

For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local time at time $n$ achieves the maximum among all sites. In this paper, we study the cardinality of the favorite downcrossing site set, and will show that with probability 1 there are only finitely many times at which there are at least four favorite downcrossing sites and three favorite downcrossing sites occurs infinitely often. Some related open questions will be introduced.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:2111.0068

Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by T\'{o}th and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369], and Ding and Shen [Ann. Probab. {\bf 46} (2018) 2545-2561], disproves a conjecture mentioned in Remark 1 on page 368 of T\'{o}th and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369].Comment: 23 page

Characterisation of the vitreous proteome in proliferative diabetic retinopathy

Background: Diabetes can lead to serious microvascular complications such as proliferative diabetic retinopathy (PDR), which results in severe vision loss. The diabetes-induced alterations in the vitreous protein composition in diabetic patients with PDR may be responsible for the presence of PDR. The vitreous humour can be utilised in a variety of studies aimed toward the discovery of new targets for the treatment or prevention of PDR and the identification of novel disease mechanisms. The aim of this study was to compare the protein profile of vitreous humour from diabetic patients with PDR with that of vitreous humour from normal human eyes donated for corneal transplant. Results: Vitreous humour from type 2 diabetic patients with PDR (n = 10) and from normal human eyes donated for corneal transplant (n = 10) were studied. The comparative proteomic analysis was performed using twodimensional fluorescence difference gel electrophoresis (2-D DIGE). Differentially produced proteins (abundance ratio> 2 or <-2, p < 0.01) were identified by matrix-assisted laser desorption ionisation time-of-flight mass spectrometry (MALDI-TOF MS) and MALDI-TOF tandem mass spectrometry. A total of 1242 protein spots were detected on the 2-D master gel of the samples, and 57 spots that exhibited statistically significant variations were successfully identified. The spots corresponded to peptide fragments of 29 proteins, including 8 proteins tha

A convenient tandem one-pot synthesis of donor-acceptor-type triphenylene 2,3-dicarboxylic esters from diarylacetylene

A tandem one-pot method for the direct synthesis of polysubstituted triphenylene 2,3-dicarboxylic esters with different substitution patterns was developed by enyne metathesis of diarylacetylene, followed by Diels–Alder, aromatization and a cyclization cascade