Summary: The aim of this paper is to propose a homomorphic encryption scheme based on identity. For the LWE difficult problem on lattice, the paper adds the identity information to the parameter settings when selecting. The outputs meet homomorphic addition that prove the correctness and security of the scheme under IND-ID-CPA. The scheme can resist the attacks from quantum computing, and the keys are shorter. It is easy to manage the keys.

Summary: To search for simple methods for conforming analytic function by harmonic function. The aforesaid methods respectively derives by using equivalent condition of Cauchy-Riemann equation and uniqueness theorem of analytic function. Herein are obtained two variable substitution methods which are more simple and convenient for conforming analytic function by harmonic function. Also are obtained simple methods for distinguishing analyticity of functions and converting analytic functions from binary real function form to complex function form.

Summary: The main purpose of this paper is to solve the Burgers-Huxley equation and obtain its exact solutions. The Burgers-Huxley equation is solved by the homogeneous balance method and is finally simplified by means of Mathematica software. Six explicit traveling wave solutions of Burgers-Huxley equation in different forms are obtained. The homogeneous balance method is an effective and universally applicable means to solve certain nonlinear partial differential equations.

Summary: The aim of this paper is to analyze the properties and the dependence of the maximum and minimum functions on the interval of definition. The continuities, monotonicity and differentiability of the maximum and minimum functions are investigated with analytical method. The maximum and minimum functions discussed herein are continuously dependent on their interval of definition, however, the differentiability of the kinds of functions is uncertain.

Summary: The aim of this paper is to propose a new random algorithm for solving the very large-scale linear least squares problems because some state-of-art methods for the least squares problems are not suitable for solving very large-scale problems since the computational complexity and the memory complexity of the methods are very high. The coefficients matrix of the very large-scale least squares problems is reduced by random sampling; then the resulted matrix is transformed with the fast Walsh-Hadamard to save the important information as soon as possible; at last, the reduced problem is solved by QR decomposition to obtain the approximated solution of the original problem. The computational complicity and the memory cost of the new algorithm are less than the related algorithms. The numerical experiments show that the new algorithm is efficient to save the training time with comparable approximated solution, and the new algorithm can solve larger size of problems with the same computing platform.