The measurement uncertainty of the MAFM is U95=5nm+2×10-4l (l——the measured length between two arbitrary points in the measurement range in μm);

A new inequality of quadratic form of the three variables referring to two arbitrary points of the interior of a triangle is established,then application of this result is discussed,and finally three conjectures to be solved are put forward.

Meanwhile, the number of the interference fringes can be multiplied by the principle of phase multiplication, or the phase different between two arbitrary points in the field can be directly calculated, so that the measuring precision can be greatly improved.

By using the basic inequality of the triangle and its theorem, the unified proof of two inequality for two arbitrary points inside the triangle is given, two new inequalities are obtained, and two open problems are posed.

We study a generalized geometry of the triangle, based on the idea of letting two arbitrary points play the role that the centroid and the orthocenter play classically.

The problem of Arnold's diffusion consists in finding conditions on h which guarantee the existence of orbits Q of with connecting two arbitrary points of frequency space.

The shortest Manhattan path for two arbitrary points in and is outside the module, and thus it is a concave module.

The shortest Manhattan path for two arbitrary points in the separate submodules must be outside the module.

Compute another transformation which maps two arbitrary points of the conic to the line at infinity.

The relation of the moment of momentum between two arbitrary points is demonstrated in this article,from which theorem of momen of momentum about absolute motion of the system of parlicles to the center of moment of motion is inferred as well in the same article.In addition,the theorem of moment of momentum about relative motion of the system of particles to the center of moment of motion is proveb at the same time.Meanwhile,it is pointed out that the methob using the theorem of moment of momentum about...

The relation of the moment of momentum between two arbitrary points is demonstrated in this article,from which theorem of momen of momentum about absolute motion of the system of parlicles to the center of moment of motion is inferred as well in the same article.In addition,the theorem of moment of momentum about relative motion of the system of particles to the center of moment of motion is proveb at the same time.Meanwhile,it is pointed out that the methob using the theorem of moment of momentum about absolute motion of the system of particles to the center of moment of motion to solve some questions concerned is also suitable for common problems.

A non-iteration algorithm for solving multidimensional non-linear travel-time inversion is proposed. Because the ray paths depend on the velocity of the medium, it is a nonlinear problem to invert the velocity in a region from the travel-time data measured on its boundary. We illustrate the algorithm in terms of seismic longitudinal wave travel-time inversion. In the algorithm it is assumed that the angular coverage of the ray is complete, the image region has limited boundary, the velocity outside the image...

A non-iteration algorithm for solving multidimensional non-linear travel-time inversion is proposed. Because the ray paths depend on the velocity of the medium, it is a nonlinear problem to invert the velocity in a region from the travel-time data measured on its boundary. We illustrate the algorithm in terms of seismic longitudinal wave travel-time inversion. In the algorithm it is assumed that the angular coverage of the ray is complete, the image region has limited boundary, the velocity outside the image region is known. The travel-time between two arbitrary points on the boundary is given. It is also assumed that the velocity in the image region can be expressed as a linear combination of local base functions. In the first step, the velocities in the pixels adjacent to the boundary are inverted. Based on the fact that there must be a ray travelling through the only one pixel adjacent to the boundary, the data of this ray can be used to invert the velocity of the pixel which lead to solve an nonlinear equation with one unknown. In this way, the velocity of all the pixel closed to the boundary can be worked out. After that, we trace all of other rays from two terminals of the ray to the new boundary of unknown region and give two new terminals and travel-time of ray travelling through reduced unknown region. The ray directions at every two terminals in the first step can be determined by differentiating the travel-time of the neighbour ray from the Benndorf relation. Then the ray tracing is a initial value problem and can be calculated easily. The ray direction at two new terminals and traveling-time can be given in the ray tracing. Then the problem left is all the same as last step, but the unknown region is reduced and the algorithm used in the first step is repeated. In this way, the velocity of image region is inverted from outer layer to inner layer by layer stripping until the velocity of all the region is worked out.From the algorithm above, it can be seen that the inversion procedure can be finished in limited steps, no convergence problem exists. All the rays need to be traced only once. In every step, only equations of one unknown need to be solved, thus large linear algebraic equations are avoided. The computer time and space can be saved. When there are low velocity region in the image region or there is not complete angular coverage as in the cross-well seismic tomography, the algorithm will meet difficulties as all the algorithm for ray-travel-time tomography. Our algorithm, however, can shed a new light on the situation. The number of unknowns, in this case, can still be reduced in spite not to single ones by the algorithm.

狝 metrological atomic force microscope (MAFM) firstly has been developed in the world. As this MAFM is self-calibrated and quantized with a 3-D miniature-size laser interferometer system (laser beam is transmitted by an optical fibre from the laser to the interferometer) in the middle-sized measuring range of the nanometer measurement, the results of the MAFM will be directly traced to meter definition. Same results are also faced on scanning tunnel microscope (STM) in the field of nanometer technique and measurement....

狝 metrological atomic force microscope (MAFM) firstly has been developed in the world. As this MAFM is self-calibrated and quantized with a 3-D miniature-size laser interferometer system (laser beam is transmitted by an optical fibre from the laser to the interferometer) in the middle-sized measuring range of the nanometer measurement, the results of the MAFM will be directly traced to meter definition. Same results are also faced on scanning tunnel microscope (STM) in the field of nanometer technique and measurement. The quantized MAFM and MSTM will contribute exact explain and analysis to nanometer surface physics. The theoretic analysis for using the AFM as a metrological instrument, the eliminating and complementary methods of the instrument errors, and the experiments and the results are published in the paper. The measurement uncertainty of the MAFM is U95=5nm+2×10-4l (l——the measured length between two arbitrary points in the measurement range in μm); The measurement uncertainty along z axis is U95=(1.1～1.2nm)+2×10-4h(h——the measured height in μm).