On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization

In this paper, we give a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which are proposed by Ye, Dai, Lam (1999) and Faug$\mu$ere, Perret (2006, 2008, 2009). We show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). We also propose a conjecture such that the decomposition for a set of polynomials can be computed from that of its homogenization with high probability. Finally, we prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space. Combining these results together, we prove that the algorithm can compute a degree proper decomposition for a set of randomly decomposable quartic polynomials with probability one when the base field is of characteristic zero, and with probability close to one when the base field is a finite field with sufficiently large number under the assumption that the conjeture is correct

Rational general solutions of algebraic ordinary differential equations

We give a necessary and sufficient condition for an alge-braic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to com-pute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants

Deep Learning for Video Object Segmentation:A Review

As one of the fundamental problems in the field of video understanding, video object segmentation aims at segmenting objects of interest throughout the given video sequence. Recently, with the advancements of deep learning techniques, deep neural networks have shown outstanding performance improvements in many computer vision applications, with video object segmentation being one of the most advocated and intensively investigated. In this paper, we present a systematic review of the deep learning-based video segmentation literature, highlighting the pros and cons of each category of approaches. Concretely, we start by introducing the definition, background concepts and basic ideas of algorithms in this field. Subsequently, we summarise the datasets for training and testing a video object segmentation algorithm, as well as common challenges and evaluation metrics. Next, previous works are grouped and reviewed based on how they extract and use spatial and temporal features, where their architectures, contributions and the differences among each other are elaborated. At last, the quantitative and qualitative results of several representative methods on a dataset with many remaining challenges are provided and analysed, followed by further discussions on future research directions. This article is expected to serve as a tutorial and source of reference for learners intended to quickly grasp the current progress in this research area and practitioners interested in applying the video object segmentation methods to their problems. A public website is built to collect and track the related works in this field: https://github.com/gaomingqi/VOS-Review

Erchen Decoction Prevents High-Fat Diet Induced Metabolic Disorders in C57BL/6 Mice

Erchen decoction (ECD) is a traditional Chinese medicine prescription, which is used in the treatment of obesity, hyperlipidemia, fatty liver, diabetes, hypertension, and other diseases caused by retention of phlegm dampness. In this study we investigated the potential mechanism of ECD, using metabolism-disabled mice induced by high-fat diet. Body weight and abdominal circumference were detected. OGTT was measured by means of collecting blood samples from the tail vein. Blood lipid levels and insulin were measured using biochemical assay kit. Real-time PCR was used to measure the CDKAL1 gene expression and western blot was used to measure the protein expression. Through the research, it was found that ECD showed markedly lower body weight and abdominal circumference than those in the HFD group. Consistently, we observed that ECD significantly improved glucose tolerance, promoted the secretion of insulin and decreased the level of TG, TC level. Meanwhile, we observed significantly increased CDKAL1 mRNA and protein level in the ECD group. Therefore, we speculate that the potential molecular mechanism of ECD is to promote the CDKAL1 expression, ameliorate islet cell function, and raise insulin levels to regulate the metabolic disorder