Maximal Estimates of Schrödinger Equations on Rearrangement Invariant Sobolev Spaces

We establish the maximal estimates for the solutions of some initial value problems on rearrangement-invariant quasi-Banach function spaces. Our result covers the cases for which the initial value problem is given by the Schrödinger equation.     

Bivariate composite vector valued rational interpolation

In this paper we point out that bivariate vector valued rational interpolants (BVRI) have much to do with the vector-grid to be interpolated. When a vector-grid is well-defined, one can directly design an algorithm to compute the BVRI. However, the algorithm no longer works if a vector-grid is ill-defined. Taking the policy of ``divide and conquer', we define a kind of bivariate composite vector valued rational interpolant and establish the corresponding algorithm. A numerical example shows our algorithm still works even if a vector-grid is ill-defined.

     

Crashworthiness analysis and optimization of fourier varying section tubes

Thin-walled structures are widely used as energy absorption devices for their proven advantages on lightweight and crashworthiness. However, a majority of studies have being focus on exploring separately the crashworthiness of the thin-walled structure with a specific geometric section, such as circular, square, hexagon, octagon etc., and little research has investigated the relationship of crashworthiness among thin-walled structures with different sections systematically. This paper utilizes Fourier series expansion to generate a series of novel sectional configurations, namely Fourier varying sectional tubes (FVSTs), to look into their advantages of crashworthiness, thereby developing some FVSTs with highest possible energy absorption capacity. Based on the validated finite element (FE) models, parametric analysis is conducted to investigate the effects of cross-sectional configuration, perimeter and thickness of FVSTs on collapse mode and energy absorption. The results showed that the collapse modes of FVSTs are fairly sensitive to cross-sectional configuration, perimeter and wall thickness. Of these FVSTs generated, the highest specific energy absorption (SEA) increases 77.54% by increasing perimeter and 69.73% by decreasing wall thickness. Finally, a discrete optimization based on the orthogonal arrays is conducted to obtain the optimal FVST for maximizing SEA under the constraint of the initial peak crushing force (IPCF). The optimized FVSTs are of superior crashworthiness and great potential as an energy absorber.     

基于三次样条插值的艾滋病治疗模型

基于数学建模提供的艾滋病治疗方案中的实验数据,依据病人的初始CD4状态,将实验数据分类筛选,采用一维三次样条插值法描绘各类病人服药时间与病情的关系图形,进而由图形确定最佳治疗终止时间;并建立了CD4与时间的函数关系模型,借助于性价比函数进行疗效、价格的综合比较分析,此模型解法对最佳治疗方案的选择提供了有力依据.     

H-bases for polynomial interpolation and system solving

The H-basis concept allows, similarly to the Gröbner basis concept, a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H-bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are, under mild conditions, already H-bases. We apply H-bases to the solution of polynomial systems by the eigenmethod and to multivariate interpolation.     

Determination of an acceptable level of spectral data compression by Discrete Wavelet Transforms

Today, due to the ever increasing amount of data generated by analytical instruments, good compression methods are desired to keep computation time acceptable. The lower the volume and noise content of data, the easier it becomes to investigate and interpret the modeling results. Discrete Wavelet Transform (DWT) is an effective data compression and noise suppression tool. Compression can be performed at different levels, in each, the size of signal part of the data reduces to half the size. This work includes an approach for determining an acceptable level of compression of data where the aim is to achieve minimal loss of information and no significant change in the structure of data, which could mean, e.g. no loss in predictive ability or the effective rank of the data-set. The method is based on estimation of the Singular Values (SVs) from a data matrix and the Singular Values at each level of compression followed by the application of Median Absolute Deviation (MAD) of the correlation between original SVs and compression SVs as a simple statistical test for the determination of the optimum level of compression. We illustrate the method using FT-Raman data from aqueous solutions of three sugars (glucose, trehalose and sucrose) and NMR data from mixtures of three alcohols. A sudden change in prediction error sum of square plots from Partial Least Squares (PLS) modeling confirms the results from MAD statistics.     

Nonnegative Solutions to Systems with Symmetric Circulant Matrix

For a system with symmetric circulant matrix, conditions on the right-hand side vector which ensure the positivity of the solution to the system are found. As an application of the results obtained, the problem of positive spline interpolation of positive functions on uniform grids is studied.     

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata, which describe the discrete and exactly causal unitary evolution of a lattice of quantum systems, have been recently considered as a fundamental approach to quantum field theory and a linear automaton for the Dirac equation in one dimension has been derived. In the linear case a quantum cellular automaton is isomorphic to a quantum walk and its evolution is conveniently formulated in terms of transition matrices. The semigroup structure of the matrices leads to a new kind of discrete path-integral, different from the well known Feynman checkerboard one, that is solved analytically in terms of Jacobi polynomials of the arbitrary mass parameter.     

On the Size of the Plasma Region

AMS (MOS): 35J60, 35B40, 35J67