Runoff variations in the Luanhe River Basin during 1956-2002

The decrease of runoff in the Luanhe river basin, which caused water crisis in Tian-jin for several times, was investigated using discharge data covering the period 1956-2002. The data from the differential integral curves of the annual runoff indicate that the decreasing point began in 1979 in the six sub-basins. The decrease of runoff in the Luanhe river basin resulted from the combination of climate effects and human activities, in which the latter plays an important role. This can be illustrated by noting that after 1979 the runoff generated by similar precipitation decreased under the condition that the total precipitation did not decrease in the entire basin. As a result, the annual runoff of the Luanhe river basin after 1979 decreased by about 6.46×10^8 m^3 each year. To analyze the runoff characteristics, it is inadequate to seek the runoff trends only and the identification of cyclical component of the runoff as accurate as possible is necessary. From the natural annual runoff discharge time series, we can see the annual runoff fluctuates around the long-term average. Analyzed by VRL （Variable Record Length） method, the main periods of 3, 5-6, 7, 9, 16-20 and 37-39 years were found. The last decade causing water crisis was the driest period in the history, and this condition will last several years from trend analysis and power spectrum analysis. So finding new water sources is urgent to solve water crisis in Tianjin city, and the South-North Water Transfer is a feasible option.

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic （MHD） equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.

A Unified Algorithm for Finding the Intersection Curve of Surfaces

In this papaert an INTEGRAL CURVE ALGORITHM is preselited, which turns the intersection curve of surfaces into the form of integral one and then uses 'PREDICTORCORRECTOR' technique to evaluate the intersection of surfaces.No matter how the surfaces are defined, the method always deaIs with the intersection curves in the same wny. To find a point on the curve one need only to calculate the JACOBI determinants of 'PREDICTOR point' and 'CORRECToR point' while the second order precision is guatanteed. Thus, not only is the problem of finding the intersection of surfaces resolved, but also the algorithms for generating both plane curve and space curve are unified.

PHASE-PORTRAIT OF A CERTAIN INTEGRABLE QUADRATIC DIFFERENTIAL SYSTEM WITH CENTER

We draw and analyse the phase-portrait of the integrable quadratic differential system (4) below, this result will be used in a latter paper.

The Cartesian analytical solutions for the Ndimensional compressible Navier-Stokes equations with density-dependent viscosity

In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.