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The double exponential formula is known to be very powerful for evaluation of various kinds of integrals, in particular integrals with end point singularities or integrals over the half infinite interval. It is also known that a weak point of this formula is the inefficiency when applied to a slowly decaying oscillatory integral over the half infinite interval such as I = ∫0∞f1(x) sin x dx, f1(x) is an algebraic function. In this paper we propose a new type of the double exponential formula which is quite efficient for evaluation of the integral mentioned above. It is based on such a transformation that makes the points of the formula after the transformation approach to the zeros of sin x double exponentially for large x.

This paper gives a survey of the results known to date about quadrature formulas obtained by variable transformation followed by an application of the trapezoidal rule with an equal mesh size. It has been shown that a formula obtained by an appropriate transformation is in general efficient and also robust against the end point singularity. Various kinds of useful transformations together with the asymptotic error behaviors of the resulting quadrature formulas are summarized. In particular special emphasis is placed on an asymptotically optimal formula called the double exponential formula, abbreviated as the DE-rule, which is characterized by the double exponential decrease of its weights in the neighborhood of the end points of the transformed interval of integration.

We study a numerical solution of the multi-dimensional time dependent Schrödinger equation using a split-operator technique for time stepping and a spectral approximation in the spatial coordinates. We are particularly interested in systems with near spherical symmetries. One expects these problems to be most efficiently computed in spherical coordinates as a coarse grain discretization should be sufficient in the angular directions. However, in this coordinate system the standard Fourier basis does not provide a good basis set in the radial direction. Here, we suggest an alternative basis set based on Chebyshev polynomials and a variable transformation.Furthermore, it is shown how the use of operator splitting produces a splitting error which introduces high frequency modes in the numerical solution in the case of the singular Coulomb potential. Incorporating the Coulomb potential into the radial Laplacian provides a much better splitting. Fortunately our new basis set allows this in some cases.Numerical experiments are presented which demonstrate the advantages and limitations of our technique. Details are demonstrated by 1D toy examples, while the superior efficiency is demonstrated by a 3D example.

In practice, especially in engineering, we often encounter the linear absolute-value objective-function problems with linear constraints, such as ∑| X i − C i |. Using powerful simplex algorithm, we derive a variable transformation method to solve it.

The objective of this research is to study what factors may affect the schooling rates of 15-17 years old in Taiwan province in 1990. After finding out some possible factors and collecting those data regarding those factors, we separate the data (by stratified random sampling) into two sets. One set is used to construct the model, and the other set shall be used to test the model. The main process to build a regression model is as follows. First, we shall use simple linear regression models to help us to see if each factor may have relation with the schooling rates. With the analysis of residuals and so on, we then make appropriate transformations on each of these factors. Finally, we use three common statistical regression techniques (including stepwise regression, forward selection, and backward elimination methods) to develop a suitable multiple regression model. It seems that, by our understanding of schooling rates in Taiwan, this model is not unreasonable. In addition, we verify the assumptions of the model by graphical methods and statistical tests. We also do the inferences about regression parameters. Furthermore, ye use the results of the analysis of variance and predictions of new observations to evaluate the prediction ability of the model. Finally, we use the most appropriate multiple regression model to give some suggestions to improve (or keep) the schooling rates of 15-17 years old.

The numerical treatment of nonlinear model fitting problems often can be simplified by manipulating the model equations. Algebraic manipulations, including nonlinear transformations of model parameters, do not change the numerical result of the adjustment. Therefore, such manipulations can be a powerful method to improve the performance of solution algorithms. Nonlinear transformations of the observations, on the other hand, do change the numerical results unless the normal equations are transformed accordingly. The latter transformation has been neglected by previous authors and this article provides a complete set of formulas that are needed to implement transformations of observations. The transformations are, however, in general less useful than parameter transformations but may have applications in particular situations.

The propensity for metastatic testicular teratocarcinoma to undergo differentiation, particularly when associated with successful cancer chemotherapy, means that the histology of residual disease following chemotherapy cannot be assumed without biopsy. We present 4 cases in which biopsy of a metastasis was important in determining the management of the patient. We conclude that because of the variable nature of the transformation of metastases in testicular germ cell tumors, particularly after chemotherapy, residual disease will often need further biopsy to establish whether it represents residual malignant disease or benign transformation.

An intensive study on the relationships among the intra-individual, the inter-individual and the population distributions has been performed in order to give further understanding to the ‘normality’ of test results. The test results of more than 15 regular visits by 737 patients to an automated multiphasic testing system are used to estimate the inter-individual distribution which would have been impossible to attempt otherwise. This paper concentrates on glucose results to clarify the situation.

In this paper, we present a sufficient condition for uniform convergence of means to their expectations over the classes of real functions. Our proof is very simple via connecting this convergence to uniform convergence over the class of indicator functions. Furthermore, we obtain a convergent result concerning a non=VC class with an application to variable transformation for fitting regression model.