An elliptic curve is a pair $(E,O),$ where $E$ is a smooth projective curve of genus 1 and $O$ is a point of $E$, called the point at infinity. Every elliptic curve can be given by a Weierstrass equation $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Let $\mathbb{Q}$ be the set of rationals. $E$ is said to be defined over $\mathbb{Q}$ if the coefficients $a_i, i=1,2,3,4,6$ are rationals and $O$ is defined over $\mathbb{Q}$.

Let $E/ \mathbb{Q}$ be an elliptic curve and let $E(\mathbb{Q})_{tors}$ be the torsion group of points of $E$ defined over $\mathbb{Q}$. The theorem of Mazur asserts that $E (\mathbb{Q})_{tors}$ is one of the following 15 groups $$E(\mathbb{Q})_{tors}=\begin{cases} \mathbb{Z}/m\mathbb{Z}, & m=1,2,\ldots,10,12 \\ \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z}, & m=1,2,3,4.\end{cases}.$$ We say that an elliptic curve $E'/\mathbb{Q}$ is isogenous to the elliptic curve $E$ if there is an isogeny, i.e. a morphism $\phi:E\rightarrow E'$ such that $\phi(O)=O$ , where $O$ is the point at infinity.

We give an explicit model of all elliptic curves for which $E(\mathbb{Q})_{tors}$ is in the form $\mathbb{Z}/m\mathbb{Z}$ where $m$ = 9,10,12 or $\mathbb{Z}/ 2 \mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z} \ {\rm where} \ m=4$, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit model of all their isogenous curves with cyclic kernels consisting of rationals points.

Non-tensor product bivariate fractal interpolation functions defined on gridded rectangular domains are constructed. Linear spaces consisting of these functions are introduced. The relevant Lagrange interpolation problem is discussed. A negative result about the existence of affine fractal interpolation functions defined on such domains is obtained.

The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.

This paper discusses band-limited scaling function, especially on the interval band case and three interval bands case, its relationship with oversampling property and weakly translation invariance are also studied. At the end, we propose an open problem.

This paper is concerned with the second order elliptic problems with small periodic coefficients on a bounded domain with a curved boundary. A two-scale curved element method which couples linear elements and isoparametric elements is proposed. The error estimate is obtained over the given smooth domain. Furthermore, an additive Schwarz method is provided for the isoparametric element method.

The motion of surface waves under the effect of bottom is a very interesting and challenging phenomenon in the nature, we use boundary integral method to compute and analyze this problem. In the linear analysis, the linearized equations have bounded error increase under some compatible conditions. This contributes to the cancellation of instable Kelvin-Helmholtz terms. Under the effect of bottom, the existence of equations is hard to determine, but given some limitations it proves true. These limitations are that the swing of interfaces should be small enough, and the distance between surface and bottom should be large enough. In order to maintain the stability of computation, some compatible relationship must be satisfied like that of [5]. In the numerical examples, the simulation of standing waves and breaking waves are calculated. And in the case of shallow bottom, we found that the behavior of waves is rather singular.

The Laguerre Gauss-Radau interpolation is investigated. Some approximation results are obtained. As an example, the Laguerre pseudospectral scheme is constructed for the BBM equation. The stability and the convergence of proposed scheme are proved. The numerical results show the high accuracy of this approach.

For the first order nonstationary hyperbolic equation taking the piecewise linear discontinuous Galerkin solver, we prove that under the uniform rectangular partition, such a discontinuous solver, after postprocessing, can have two and a half approximative order which is half order higher than the optimal estimate by Lesaint and Raviart under the rectangular partition.

A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters invoved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.