Subjects: Mathematics >> Applied Mathematics submitted time 2022-11-09
Abstract： In recent years, mixed integer linear programming (MILP, in short) is widely used to search differential characteristics and linear approximations with high probability and gradually becomes a powerful tool of automated cryptanalysis in symmetric ciphers. A key problem in the MILP method is how to fully characterize a set $S subseteq {0,1 }^n$ with as few linear integer inequalities $L$ as possible, which is called a full linear integer inequality characterization (FLIIC, in short) problem. In this work we establish a complete theory to solve the best solution of a FLIIC problem. We start from plain sets which can be characterized by exactly one linear integer inequality, and give their essential properties, including type, sparsity, degeneration, order, minimal and maximal element, norm and its bound, etc. Based on these essential properties, we further provide an efficient algorithm of solving a FLIIC problem with $S$, which can produce all minimal plain closures of $S$ and output a best FLIIC theoretically. As examples, we give their best solutions for differential properties of some common S-boxes used in block ciphers.
Peer Review Status:
Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2021-11-29
Abstract： In the filed of machine learning and mathematical optimization, it is a challenge to mathematically explain optimality of loss function for deep learning. Loss function is high-dimensional, non-convex, and non-smooth. It was, however, observed that gradient descent could reach zero training loss of this highly non-convex function. Loss landscape analysis is critical to reveal reasons why deep networks are easily optimizable. We reviewed the advance on loss landscape analysis, such as landscape features (number and spatial distribution of local minima, connectivity between global optima, and global optimality of critical points), convergence of gradient descent, and visualization of loss landscape. This survey aimed to promote interpretable and reliable deep learning in critical applications. "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2020-02-18
Abstract： " " "Novel Coronavirus Pneumonia (NCP, or alternatively 2019-nCoV), initially blown up in Wuhan in December of 2019, has been quickly spread all over China, and even other countries of the world, which has produced an important effect to the agricultural and industrial activities, and daily life. It is expected that a well-known recognition is essential to the effective prevention of the disease. Based on the daily announced numbers of the infective people from the National and Hubei provincial Health commissions, a logistic model is applied in this paper for data fitting, in order to provide some scientific information for the effective prevention and controlling of the disease. Using the parameters obtained from the data simulation, a susceptible-infected (SI) model is used to forecast the future trend of the NCP. Our work indicated that the epidemic will last at least two additional weeks in Hubei, but should come to an apex in one week in other areas of China. "
Peer Review Status:Awaiting Review
Subjects: Physics >> Electromagnetism, Optics, Acoustics, Heat Transfer, Classical Mechanics, and Fluid Dynamics Subjects: Information Science and Systems Science >> Basic Disciplines of Information Science and Systems Science Subjects: Mathematics >> Mathematics （General） submitted time 2017-11-26
Abstract： The infrared imaging grayscale variation caused by the influence of atmosphere on infrared radiation transmission is a problem that infrared target tracking application needs to cope with. The object of this paper is to model the law of infrared imaging grayscale variation in Lie group, which is important to design an efficient and robust target tracking algorithm. This paper firstly analyzes the infrared radiation transmission model, and then derives the brightness model of infrared imaging by considering the mechanism of infrared imaging. Furthermore, it is theoretically proved that the infrared imaging grayscale variation caused by the atmosphere obeys to the Lie group structure, and a non-Euclidean mathematical representation of the infrared imaging grayscale variation is proposed. Finally, according to the infrared imaging grayscale variation model, the field experimental data collected under different environments are fitted, and the regression analysis results demonstrate the correctness of the model, which validates the rationality of the Lie group representation of the infrared imaging grayscale variation.
Peer Review Status:Awaiting Review
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