A Polynomial time Algorithm for Hamilton Cycle with maximum Degree 3

Based on the famous Rotation-Extension technique, by creating the new concepts and methods: broad cycle, main segment, useful cut and insert, destroying edges for a main segment, main goal Hamilton cycle, depth-first search tree, we develop a polynomial time algorithm for a famous NPC: the Hamilton cycle problem. Thus we proved that NP=P. The key points of this paper are: 1) there are two ways to get a Hamilton cycle in exponential time: a full permutation of n vertices; or, chose n edges from all k edges, and check all possible combinations. The main problem is: how to avoid checking all combinations of n edges from all edges. My algorithm can avoid this. Lemma 1 and lemma 2 are very important. They are the foundation that we always can get a good branch in the depth-first search tree and can get a series of destroying edges (all are bad edges) for this good branch in polynomial time. The extraordinary insights are: destroying edges, a tree contains each main segment at most one time at the same time, and dynamic combinations. The difficult part is to understand how to construct a main segment's series of destroying edges by dynamic combinations (see the proof of lemma 4). The proof logic is: if there is at least on Hamilton cycle in the graph, we always can do useful cut and inserts until a Hamilton cycle is got. The times of useful cut and inserts are polynomial. So if at any step we cannot have a useful cut and insert, this means that there are no Hamilton cycles in the graph.Comment: 49 pages. This time, I add a detailed polynomial time algorithm and proof for 3S

Assembly history of subhalo populations in galactic and cluster sized dark haloes

We make use of two suits of ultra high resolution N-body simulations of individual dark matter haloes from the Phoenix and the Aquarius Projects to investigate systematics of assembly history of subhaloes in dark matter haloes differing by a factor of $1000$ in the halo mass. We have found that real progenitors which built up present day subhalo population are relatively more abundant for high mass haloes, in contrast to previous studies claiming a universal form independent of the host halo mass. That is mainly because of repeated counting of the 're-accreted' (progenitors passed through and were later re-accreted to the host more than once) and inclusion of the 'ejected' progenitor population(progenitors were accreted to the host in the past but no longer members at present day) in previous studies. The typical accretion time for all progenitors vary strongly with the host halo mass, which is typical about $z \sim 5$ for the galactic Aquarius and about $z \sim 3$ for the cluster sized Phoenix haloes. Once these progenitors start to orbit their parent haloes, they rapidly lose their original mass but not their identifiers, more than $55$ ($50$) percent of them survive to present day for the Phoenix(Aquarius) haloes. At given redshift, survival fraction of the accreted subhalo is independent of the parent halo mass, whilst the mass-loss of the subhalo is more efficient in high mass haloes. These systematics results in similarity and difference in the subhalo population in dark matter haloes of different masses at present day.Comment: 7 pages, 6 figures, moderate changes, accepted to MNRA

The E3 ubiquitin ligase c-IAP1 regulates PCSK9-mediated LDLR degradation: Linking the TNF-α pathway to cholesterol uptake

Proprotein convertase subtilisin/kexin type 9 (PCSK9), in addition to LDLR (low-density lipoprotein receptor) and APOB (apolipoprotein B), is one of three loci implicated in autosomal dominant hypercholesterolaemia (ADH)^1^. A number of PCSK9 gain-of-function mutations and loss-of-function mutations have been identified from families afflicted with ADH with hypercholesterolaemia or hypocholesterolaemia, respectively^1-4^. In humans, the main function of PCSK9 appears to be the post-transcriptional regulation of the number of cell-surface LDL receptors^5-7^. To date, only LDLR and its closest family members VLDLR and ApoER2 have been shown to bind with PCSK9^8,9^. To find new binding partners for PCSK9, we used a shotgun proteomic method to analyse the protein complex pulled down by immunoprecipitation against FLAG-tagged PCSK9 protein. Among 22 potential novel binding proteins identified, we found that the cellular inhibitor of apoptosis protein 1 (c-IAP1^10^) and the TNF receptor-associated factor 2 (TRAF2^11^) complex are regulated differently in different dominant PCSK9 mutations that occur naturally. Further immunoprecipitation analysis showed that c-IAP1 is a direct binding partner for PCSK9. One of the "gain-of-function" mutants, PCSK9-S127R, which has impaired autocatalytic activity, is defective in binding to c-IAP1. The other dominant mutation, PCSK9-D374Y^12^, which is 10-fold more potent in degrading the LDLR protein than wild-type PCSK9, can be significantly ubiquitinated by c-IAP1 in vitro. The ubiquitinated PCSK9-D374Y is unable to degrade LDLR, which is its main cause of hypercholesterolaemia in patients. These results indicate that there is a novel cholesterol uptake regulation pathway linking PCSK9/LDLR to the E3 ubiquitin ligase c-IAP1 in a TNF-[alpha] response pathway. This highlights the possibility of developing new treatments for human cardiovascular diseases through ubiquitin ligase-mediated ubiquitination of target proteins in cholesterol metabolism

Crop Yield Prediction Using Deep Neural Networks

Crop yield is a highly complex trait determined by multiple factors such as genotype, environment, and their interactions. Accurate yield prediction requires fundamental understanding of the functional relationship between yield and these interactive factors, and to reveal such relationship requires both comprehensive datasets and powerful algorithms. In the 2018 Syngenta Crop Challenge, Syngenta released several large datasets that recorded the genotype and yield performances of 2,267 maize hybrids planted in 2,247 locations between 2008 and 2016 and asked participants to predict the yield performance in 2017. As one of the winning teams, we designed a deep neural network (DNN) approach that took advantage of state-of-the-art modeling and solution techniques. Our model was found to have a superior prediction accuracy, with a root-mean-square-error (RMSE) being 12% of the average yield and 50% of the standard deviation for the validation dataset using predicted weather data. With perfect weather data, the RMSE would be reduced to 11% of the average yield and 46% of the standard deviation. We also performed feature selection based on the trained DNN model, which successfully decreased the dimension of the input space without significant drop in the prediction accuracy. Our computational results suggested that this model significantly outperformed other popular methods such as Lasso, shallow neural networks (SNN), and regression tree (RT). The results also revealed that environmental factors had a greater effect on the crop yield than genotype.Comment: 9 pages, Presented at 2018 INFORMS Conference on Business Analytics and Operations Research (Baltimore, MD, USA). One of the winning solutions to the 2018 Syngenta Crop Challeng

New bounds for circulant Johnson-Lindenstrauss embeddings

This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as an important class of structured random JL embeddings, are formed by randomizing the column signs of a circulant matrix generated by a random vector. With the help of recent decoupling techniques and matrix-valued Bernstein inequalities, we obtain a new bound $k=O(\epsilon^{-2}\log^{(1+\delta)} (n))$ for Gaussian circulant JL embeddings. Moreover, by using the Laplace transform technique (also called Bernstein's trick), we extend the result to subgaussian case. The bounds in this paper offer a small improvement over the current best bounds for Gaussian circulant JL embeddings for certain parameter regimes and are derived using more direct methods.Comment: 11 pages; accepted by Communications in Mathematical Science