On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an -regular solution, a first-order error bound in the norm is shown and used to derive a second-order error bound in the norm. For the cubic Schrödinger equation with an -regular solution, first-order convergence in the norm is used to obtain second-order convergence in the norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and -conditional stability for error propagation, where for the Schrödinger-Poisson system and for the cubic Schrödinger equation.

     

New expansions of numerical eigenvalues for by nonconforming elements

The paper explores new expansions of the eigenvalues for in with Dirichlet boundary conditions by the bilinear element (denoted ) and three nonconforming elements, the rotated bilinear element (denoted ), the extension of (denoted ) and Wilson's elements. The expansions indicate that and provide upper bounds of the eigenvalues, and that and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the convergence rate can be obtained, where is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

     

The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data

This paper considers the stability and convergence results for the Euler implicit/explicit scheme applied to the spatially discretized two-dimensional (2D) time-dependent Navier-Stokes equations. A Galerkin finite element spatial discretization is assumed, and the temporal treatment is implicit/explict scheme, which is implicit for the linear terms and explicit for the nonlinear term. Here the stability condition depends on the smoothness of the initial data , i.e., the time step condition is in the case of , in the case of and in the case of for mesh size and some positive constant . We provide the -stability of the scheme under the stability condition with and obtain the optimal error estimate of the numerical velocity and the optimal error estimate of the numerical pressure under the stability condition with .

     

Equal moments division of a set

Let be the minimal positive integer , for which there exists a splitting of the set into  subsets, , , ..., , whose first moments are equal. Similarly, let be the maximal positive integer , such that there exists a splitting of into subsets whose first moments are equal. For , these functions were investigated by several authors, and the values of and have been found for and , respectively. In this paper, we deal with the problem for any prime . We demonstrate our methods by finding for any and for .

     

Rationality problem of three-dimensional purely monomial group actions: the last case

A -automorphism of the rational function field is called purely monomial if sends every variable to a monic Laurent monomial in the variables . Let be a finite subgroup of purely monomial -automorphisms of . The rationality problem of the -action is the problem of whether the -fixed field is -rational, i.e., purely transcendental over , or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.

     

Identifying minimal and dominant solutions for Kummer recursions

We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions and , where (not both equal to 0). The results are obtained by applying Perron's theorem, together with uniform asymptotic estimates derived by T. M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of , and , with .

     

The hyperdeterminant and triangulations of the 4-cube

The hyperdeterminant of format is a polynomial of degree in unknowns which has terms. We compute the Newton polytope of this polynomial and the secondary polytope of the -cube. The regular triangulations of the -cube are classified into -equivalence classes, one for each vertex of the Newton polytope. The -cube has coarsest regular subdivisions, one for each facet of the secondary polytope, but only of them come from the hyperdeterminant.

     

Weak coupling of solutions of first-order least-squares method

A theoretical analysis of a first-order least-squares finite element method for second-order self-adjoint elliptic problems is presented. We investigate the coupling effect of the approximate solutions for the primary function and for the flux . We prove that the accuracy of the approximate solution for the primary function is weakly affected by the flux . That is, the bound for is dependent on , but only through the best approximation for multiplied by a factor of meshsize . Similarly, we provide that the bound for is dependent on , but only through the best approximation for multiplied by a factor of the meshsize . This weak coupling is not true for the non-selfadjoint case. We provide the numerical experiment supporting the theorems in this paper.

     

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any and any non-exceptional modulus , we prove that, for large enough ( ), the interval contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

     

On the distinctness of modular reductions of maximal length sequences modulo odd prime powers

We discuss the distinctness problem of the reductions modulo of maximal length sequences modulo powers of an odd prime , where the integer has a prime factor different from . For any two different maximal length sequences generated by the same polynomial, we prove that their reductions modulo are distinct. In other words, the reduction modulo of a maximal length sequence is proved to contain all the information of the original sequence.