Some remarks on the calculation of Born partial wave integrals

1987 ◽  
Vol 101 (2) ◽  
pp. 375-381 ◽  
Author(s):  
B. Piraux ◽  
Colm T. Whelan

The Born partial wave integrals of Seaton [3] are considered. It is shown that the treatment of Whelan [5] may be further simplified and that these integrals may be expressed as a finite sum of associated Legendre polynomials of the second kind. This provides a convenient method for their numerical evaluation and allows the relationship between the Born and Bethe reactance matrices to be given in explicit analytic form.

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.


1956 ◽  
Vol 52 (3) ◽  
pp. 442-448 ◽  
Author(s):  
S. C. Das

Consider the integralwhere x1, x2, …, xN are jointly distributed in a multivariate normal distribution f(x1, x2, …, xN) with (pij) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.


1961 ◽  
Vol 57 (2) ◽  
pp. 288-303
Author(s):  
E. L. Albasiny

ABSTRACTSeveral writers have dealt with the use of Chebyshev polynomials as aids to interpolation. The present paper considers the theory with associated Legendre polynomials replacing those of Chebyshev. Interpolation formulae of both Bessel and Everett type are derived, and formulae and expansions are given for evaluating the coefficients required in them either directly or in terms of derivatives or central differences. The introduction summarizes a number of formulae for polynomial interpolation and indicates the relationship between the new formulae and the old.


2013 ◽  
Vol 706-708 ◽  
pp. 1837-1840 ◽  
Author(s):  
Xiao Hong Fan ◽  
Bin Xu ◽  
Jing Li ◽  
Yong Xu ◽  
Shi Lei ◽  
...  

Several methods were introduced for practicing and studying the method of TTT curve digitization by using Plot Digitizer (version 1.9) and OriginPro (version 8.0724). After digitizing, original TTT images, which can not be inquired in computer procedures, were transformed into the data series and digital patterns, providing a convenient way for inquiring the data points from TTT curves. It offers a handy tool for studying the relationship among the time, temperature and transformation. Furthermore, it provides a convenient method for thermal treatment teaching through courseware. It is believe that, after a systemic data processing, we can obtain a digitizing TTT data base, which can be inquired automatically by PC procedures, promoting the automatic level of the thermal treatment for industrial applications.


2015 ◽  
Vol 794 ◽  
pp. 166-173 ◽  
Author(s):  
Markus Bergmann ◽  
Andreas Sterzing ◽  
Dirk Landgrebe

Designing material characteristics by grain refinement using Severe Plastic Deformation (SPD) is an attractive way to create outstanding material properties. This paper presents a unique method which combines SPD and impact extrusion. The extrusion die is designed to create additional material deformation to a defined depth, resulting in a gradient from ultra-fine grained to coarse grained microstructure. Due to the large gradient the method is called gradation extrusion. The paper presents a new analytical calculation method and a numerical evaluation of the strain, showing the relationship between tool design and achievable effects and provides initial experimental results.


1974 ◽  
Vol 52 (8) ◽  
pp. 731-742 ◽  
Author(s):  
Robert C. Brunet

We present detailed numerical evaluations of the partial wave projections of Feynman diagrams of second- and fourth-order in perturbation for the πN–πN scattering in the [Formula: see text] theory. Perturbative contributions to the S, P, and D waves of isospin 1/2 and 3/2 are given in tables of numerical values. Figures regrouping these results show surprising behavior for the ratios Re(4)/Re(2). These tables and figures allow easy calculations with models using low order perturbation terms such as Padé approximants.


1941 ◽  
Vol 14 (2) ◽  
pp. 458-469
Author(s):  
John V. Weaver

Abstract There are in use today several methods for determining the scorching point of rubber and synthetic compounds. Some compounders use the Williams plastometer method, while others press-cure a thin slab of compound for increasing periods of time at 220° F or thereabouts until an examination of the slab shows that vulcanization has taken place. Another method consists of heating several pieces of the compound in an oven and removing at constant intervals until the compound scorches. The test-pieces in the latter method are quite often sheeted out on a mill to determine if any vulcanization has taken place. These methods all give good indications of the scorching point; however, each one is a rather long and tedious test. The Mooney plastometer scorch test cuts down the time considerably and gives a picture of the scorching rate, when the plasticity is plotted against the time. It is necessary, as a rule, to run this test only 10 or 12 minutes at 250° F to determine just how scorchy a compound really is, while the other methods cited sometimes require an hour or more to determine the scorching characteristics. It is desirable also to express the results of any test in the form of a graph or curve, so that one may make comparisons more easily without reviewing a lot of data. The Mooney scorch test gives results which can be plotted in the form of a curve, which also shows some very interesting facts aside from the scorching characteristics. The plasticity of the compound, the point at which vulcanization begins, and the rate of vulcanization after it begins, are shown. Summarizing the advantages of this test in a few words, one can say that it affords a quick, convenient method for determining scorching characteristics, and that it gives a clearer and more informative picture of scorching than other methods used today. Just a few illustrations demonstrating the use of the Mooney plastometer as an instrument for determining scorch have been given in this report. They do, however, point out the value of this test in the following ways: (1) The Mooney scorch test gives a quick method for studying the effect of various accelerators and accelerator combinations on the scorching characteristics of rubber compounds. It can also be used to show the effect of a retarder with various accelerators, and the effect of any other compounding change which might have a bearing on the scorching characteristics. (2) The Mooney scorch test may be used to show the relationship between scorching and rate of curing. As has been pointed out, there appears to be a definite relationship; however, before any general statements can be made, a considerable amount of work would be necessary. In the particular example cited here (Curve Sheet No. 2), one could visualize, from the shape of the Mooney scorch curves, the rate of curing of these compounds from the first signs of vulcanization to the peak cure.


1983 ◽  
Vol 15 (2) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.


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