Wavelet transformation on a finite interval

Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

Interpretation of mullite HREM images using the Potential-exchange-method

Multi-slice image contrast calculations using the periodic continuation theorem (supercell-model) and a new approximation developed by Hamid Rahman (1989) called the “Potential exchange method” show good agreement in the electron microscope image contrast simulation for mullite. The method can be used to calculate a non-periodic potential and is most accurate when the defects (light atom vacancies) are distributed on (100), (010) or (001) planes and accompanied with cation shifts within the unit cell.Mullite AlThe average structure of mullite (Fig. 1a,The structure analysis shows that the vacancies occur on the tetrahedral linking positions (OX-ray (

Multislice calculations for quasi-periodic and non-periodic objects

Computer simulations of electron diffraction patterns and images are an essential step in the process of structure and/or defect elucidation. So far most programs are designed to deal specifically with crystals, requiring frequently the space group as imput parameter. In such programs the deviations from perfect periodicity are dealt with by means of “periodic continuation”.However, for many applications involving amorphous materials, quasiperiodic materials or simply crystals with defects (including finite shape effects) it is convenient to have an algorithm capable of handling non-periodicity. Our program “HeGo” is an implementation of the well known multislice equations in which no periodicity assumption is made whatsoever. The salient features of our implementation are: 1) We made Gaussian fits to the atomic scattering factors for electrons covering the whole periodic table and the ranges [0-2]Å−1 and [2-6]Å−1.

A Comparison of Calculated and Experimental Hrtem Images for Twinned Boron Carbide

ABSTRACTHigh resolution TEM images of boron carbide (B13C2) have been recorded and compared with images calculated using the multislice method as implemented by M. A. O'Keefe in the SHRLI programs. Images calculated for the [010] zone, using machine parameters for the JEOL 2000FX AEM operating at 200 keV, indicate that for the structure model of Will et al., the optimum defocus image can be interpreted such that white spots correspond to B12 icosahedra for thin specimens and to low density channels through the structure adjacent to the direct inter-icosahedral bonds for specimens of intermediate thickness (-40 > t > -100 nm). With this information, and from the symmetry observed in the TEM images, it is likely that the (101) twin plane passes through the center of icosahedron located at the origin. This model was tested using the method of periodic continuation. Resulting images compare favorably with experimental images, thus supporting the structural model. The introduction of a (101) twin plane through the origin creates distortions to the icosahedral linkages as well as to the intra-icosahedral bonding. This increases the inequivalence of adjacent icosahedral sites along the twin plane, and thereby increases the likelihood of bipolaron hopping.

A multislice approach to the rheed and rem simulation

The multislice method of calculating dynamical diffraction is based on the physical optics theory of Cowley and Moodie, which has been shown is equivalent to the quantum theory, back scattering being neglected. This method is capable of the calculation of RHEED diffraction amplitudes and the REM image simulation provided that the surface structure and surface potential have been properly modeled.The procedure for RHEED and REM calculations used her,e is basically the same as the conventional multislice calculation. The periodic continuation assumption has been employed to include the desired model surface (Fig. 1). A large dimension normal to the surface must be used so that there is no interference between these extended unit cells. The choice of the specimen thickness depends on the competition between the requirements for short computing time and larger model surface required to simulate the real equilibrium diffraction processes.