Periodic solutions of sublinear impulsive differential equations

In this paper, we consider sublinear second order differential equations with impulsive effects. Basing on the Poincar\'{e}-Bohl fixed point theorem, we first will prove the existence of harmonic solutions. The existence of subharmonic solutions is also obtained by a new twist fixed point theorem recently established by Qian etc in 2015 (\cite{Qian15})

Boundedness of solutions in impulsive Duffing equations with polynomial potentials and C1C^{1} time dependent coefficients

In this paper, we are concerned with the impulsive Duffing equation $$x''+x^{2n+1}+\sum_{i=0}^{2n}x^{i}p_{i}(t)=0,\ t\neq t_{j},$$ with impulsive effects $x(t_{j}+)=x(t_{j}-),\ x'(t_{j}+)=-x'(t_{j}-),\ j=\pm1,\pm2,\cdots$, where the time dependent coefficients $p_i(t)\in C^1(\mathbb{S}^1)\ (n+1\leq i\leq 2n)$ and $p_i(t)\in C^0(\mathbb{S}^1)\ (0\leq i\leq n)$ with $\mathbb{S}^1=\mathbb{R}/\mathbb{Z}$. If impulsive times are 1-periodic and $t_{2}-t_{1}\neq\frac{1}{2}$ for $0< t_{1}<t_{2}<1$, basing on a so-called large twist theorem recently established by X. Li, B. Liu and Y. Sun in \cite{XLi}, we find large invariant curves diffeomorphism to circles surrounding the origin and going to infinity, which confines the solutions in its interior and therefore leads to the boundedness of these solutions. Meanwhile, it turns out that the solutions starting at $t=0$ on the invariant curves are quasiperiodic.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1705.0272

Periodic solutions of semilinear Duffing equations with impulsive effects

In this paper we are concerned with the existence of periodic solutions for semilinear Duffing equations with impulsive effects. Firstly for the autonomous one, basing on Poincar\'{e}-Birkhoff twist theorem, we prove the existence of infinitely many periodic solutions. Secondly, as for the nonautonomous case, the impulse brings us great challenges for the study, and there are only finitely many periodic solutions, which is quite different from the corresponding equation without impulses. Here, taking the autonomous one as an auxiliary equation, we find the relation between these two equations and then obtain the result also by Poincar\'{e}-Birkhoff twist theorem

On some Liouville type Theorems for the stationary MHD and Hall-MHD equations

We prove several Liouville type results for the stationary MHD and Hall-MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces or satisfying a priori decay assumption, must be zero.Comment: Our proof is incomplete. We need to check i

Dynamic Detection of False Data Injection Attack in Smart Grid using Deep Learning

Modern advances in sensor, computing, and communication technologies enable various smart grid applications. The heavy dependence on communication technology has highlighted the vulnerability of the electricity grid to false data injection (FDI) attacks that can bypass bad data detection mechanisms. Existing mitigation in the power system either focus on redundant measurements or protect a set of basic measurements. These methods make specific assumptions about FDI attacks, which are often restrictive and inadequate to deal with modern cyber threats. In the proposed approach, a deep learning based framework is used to detect injected data measurement. Our time-series anomaly detector adopts a Convolutional Neural Network (CNN) and a Long Short Term Memory (LSTM) network. To effectively estimate system variables, our approach observes both data measurements and network level features to jointly learn system states. The proposed system is tested on IEEE 39-bus system. Experimental analysis shows that the deep learning algorithm can identify anomalies which cannot be detected by traditional state estimation bad data detection

Optimal Puncturing of Polar Codes With a Fixed Information Set

For a given polar code construction, the existing literature on puncturing for polar codes focuses in finding the optimal puncturing pattern, and then re-selecting the information set. This paper devotes itself to find the optimal puncturing pattern when the information set is fixed. Puncturing the coded bits corresponding to the worst quality bit channels, called the worst quality puncturing (WQP), is proposed, which is analyzed to minimize the bit channel quality loss at the punctured positions. Simulation results show that WQP outperforms the best existing puncturing schemes when the information set is fixed.Comment: Polar codes, puncture, quasi-uniform puncturing,worst quality puncturin

A Comparison Study of Credit Card Fraud Detection: Supervised versus Unsupervised

Credit card has become popular mode of payment for both online and offline purchase, which leads to increasing daily fraud transactions. An Efficient fraud detection methodology is therefore essential to maintain the reliability of the payment system. In this study, we perform a comparison study of credit card fraud detection by using various supervised and unsupervised approaches. Specifically, 6 supervised classification models, i.e., Logistic Regression (LR), K-Nearest Neighbors (KNN), Support Vector Machines (SVM), Decision Tree (DT), Random Forest (RF), Extreme Gradient Boosting (XGB), as well as 4 unsupervised anomaly detection models, i.e., One-Class SVM (OCSVM), Auto-Encoder (AE), Restricted Boltzmann Machine (RBM), and Generative Adversarial Networks (GAN), are explored in this study. We train all these models on a public credit card transaction dataset from Kaggle website, which contains 492 frauds out of 284,807 transactions. The labels of the transactions are used for supervised learning models only. The performance of each model is evaluated through 5-fold cross validation in terms of Area Under the Receiver Operating Curves (AUROC). Within supervised approaches, XGB and RF obtain the best performance with AUROC = 0.989 and AUROC = 0.988, respectively. While for unsupervised approaches, RBM achieves the best performance with AUROC = 0.961, followed by GAN with AUROC = 0.954. The experimental results show that supervised models perform slightly better than unsupervised models in this study. Anyway, unsupervised approaches are still promising for credit card fraud transaction detection due to the insufficient annotation and the data imbalance issue in real-world applications

Symmetric properties for Choquard equations involving fully nonlinear nonlocal operator

In this paper, the positive solutions to Choquard equation involving fully nonlinear nonlocal operator are shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. The key ingredients are to obtain the "narrow region principle" and "decay at infinity" for the corresponding problems. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities

Estimates for L{\phi}-Lipschitz and L{\phi}-BMO Norms of Differential Forms

In this paper, we define the L{\phi}-Lipschitz norm and L{\phi}-BMO norm of differential forms using Young functions, and prove the comparison theorems for the homotopy operator T on differential forms with L{\phi}-Lipschitz and L{\phi}-BMO norms. As applications, we give the L{\phi}-BMO norm estimate for conjugate A-harmonic tensors and the weighted L{\phi}-Lipschitz norm estimate for the homotopy operator T.Comment: 14 page

Genus Two Stable Maps, Local Equations and Modular Resolutions

We geometrically describe a canonical sequence of modular blowups of the relative Picard stack of the Artin stack of pre-stable genus two curves. The final blowup stack locally diagonalizes certain tautological derived objects. This implies a resolution of the primary component of the moduli space of genus two stable maps to projective space and meanwhile makes the whole moduli space admit only normal crossing singularities. Our approach should extend to higher genera.Comment: 81 pages, 7 figure