Combination Of Elementary Functions Research Articles

Correction for counting losses in X-ray diffractometry

Non-extended and extended dead-time models, a pulse-height analyser (PHA) windowing model, and a model intermediate between the non-extended and extended models for losses in counting methods are compared. The validities of the methods are examined by application to the analysis of powder diffraction peak intensity profiles measured by a foil method. The intermediate model including parameters for the dead-time and degree of extension can reproduce both the non-extended and the extended models and also intermediate dependence between the two models. A convenient approximate formula for the intermediate model, the maximum relative deviation of which is 0.0003, is also proposed. The determination of the parameters and a correction for the measured intensities can easily be achieved by applying the approximate model, because it provides simple formulae for the correction function expressed as a combination of elementary functions. Experimental and analytical methods for precise evaluation of the parameters to specify the counting losses are also presented. Systematic deviations of the observed dependence from the non-extended and extended dead-time models have been detected by the precise analyses of experimental data, while the PHA windowing model, intermediate model and its approximation have reproduced the observed dependence within the experimental errors.

Wavelet-Based Signal Approximation with Genetic Algorithms

In this article, the usability of genetic algorithms for signal approximation is discussed. Due to recent developments in the field of signal approximation by wavelets, this work concentrates on signal approximation by wavelet-like functions. Signals are approximated by a finite linear combination of elementary functions and a genetic algorithm is employed to find the coefficients to such an approximation. The algorithm maintains a population of different approximations, encoded in the form of 'chromosomes'. From this population 'parents' are selected according to their 'fitness', and the 'children' that constitute the next generation are produced from these parents using mutation and crossover operators.Fitness functions employed to evaluate different approximations are the L1, L2, L4, and L∞ norms. Experiments are carried out on several test signals, using Gabor and spline wavelets, both to evaluate the quality of different fitness functions, encoding schemes, and operators, and to assess the usefulness of genetic algorithms in the realm of signal approximation.Although other existing methods are faster while providing comparable approximation quality, the algorithm offers a great deal of flexibility in terms of different elementary functions, fitness criteria, etc.

Stress state of a rock mass in the vicinity of seam containing a working. Three-dimensional problem

Formulas are obtained to determine the stresses at any point of a rock mass. The formulas are derived through double integrals of an auxiliary function represented by a convergent Neumann series. Approximations of the kernels are presented by algebraic combinations of elementary functions.

An improved dynamic solar collector test method for determination of non-linear optical and thermal characteristics with multiple regression

The objective is to characterise the solar collector during a relatively short testing period with no requirement for steady state climatic conditions. This information is then used for predicting annual performance of the collector. A standard collector model that is compatible with the ISO 9806-1 test standard is used with correction terms for beam and diffuse incidence angle modifiers, thermal capacitance, wind speed and sky temperature. This results in a more complete characterisation of the collector. The collector parameters are identified by multiple linear regression, MLR. The method has been tested for characterisation of unglazed collectors, glazed flat plate collectors, evacuated tubular collectors, CPC collectors and concentrating collectors with satisfying results. Typically the correlation coefficient R 2 is better than 0.99 and the standard deviation of the difference between model and measurement is in the range 3–10 W/m 2. In the original method the angular dependence of the optical efficiency and the temperature dependence of the heat losses are supposed to be adjusted to a predetermined function. The most recent development is a routine that makes it possible to accurately identify non-linear optical and thermal performance. This extended MLR method can identify the zero loss efficiency for every angle of incidence interval and the temperature dependent heat losses for every temperature interval. This opens the application of the method to collectors with special incidence angle and heat loss effects that cannot be described easily with a combination of elementary functions. Instead a table of parameter values is determined, which is used directly in standard simulation programmes. This method will further increase the accuracy when comparing different collector designs. It has been used for comparing different glazings and for comparison with spectrophotometric measurements. It has also been used for analysing the heat loss factors for Teflon and honeycomb glazings. Since the total power output of the collector is less dependent on the heat loss coefficient than on the optical efficiency the scattering in this data is larger than for the incidence angle curves. The reflectance of booster mirrors cannot be derived with the MLR-method with acceptable accuracy. The correlation between direct irradiance and irradiance from the reflector exhibit a very strong correlation. Instead the effective reflectance of the mirror can be estimated by comparison of the measured output with calculation by the complete collector and reflector model. This effective reflectance is not compatible with the specular reflectance obtained from spectrophotometric measurements caused by large differences in acceptance angles. Standard multiple linear regression available in most spread sheet and statistical programs can be used for the parameter identification in the extended MLR-procedure. The identification takes only a few seconds. At the Älvkarleby Laboratory the test method is now used as a routine tool for the evaluation of new collector materials and designs. The Swedish National testing institute has evaluated the methods with the conclusion that they have a potential for being used in standardised collector testing.

Application of the Fokker-Planck equation to particle-beam injection into e- storage rings.

Nonlinear forces in the longitudinal accelerating field---or in the transverse magnetic fields---lead to filamentation of the injected emittance and to the decoherence of the center-of-mass motion. The dynamics of the particle distribution function in the presence of synchrotron radiation is governed by the Fokker-Planck equation. We derive the time evolution of the distribution function after injection as an approximate solution to the Fokker-Planck equation. The approximation assumes the injected emittance to be considerably larger than the equilibrium emittance, which is fulfilled for a certain class of storage rings, e.g., damping rings. In the limit of no quantum excitation, this distribution function will then be an exact solution. Higher moments of the distribution can be expressed in combinations of elementary functions and agree very well with multiparticle simulations.

A 50-MHz CMOS geometrical mapping processor

A recently developed microprogrammable, high-resolution, real-time geometrical mapping processor VLSI is presented. The processor computes pixel addresses within the frame-buffer memory according to user-specified geometrical mapping functions. Its architecture permits high-speed operations and library extensions through a combination of elementary functions. It includes a CORDIC function generator, consisting of a one-dimensional pipeline array with high-speed parallel arithmetic circuits, and a pipeline control method. This results in a 50-MHz throughput rate with an accuracy of 20 bits using 1.2- mu m CMOS technology. The processor will be useful in high-definition television (HDTV) systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the relaxation effects with a gaussian time spectrum

The tabulation of the normalized coefficient (Q−1)G which characterizes the dissipation of elastic energy due to a relaxation effect with a Gaussian time spectrum has been extended to a range of values of the independent variablex and of the distribution parameter β considerably wider than that considered before. For all values of (Q−1)G presently available, their dependence uponx has been represented by a simple combination of elementary functions with an accuracy sufficient for the analysis of experimental data. Moreover, for β≥1.25, the dependence of (Q−1)G upon bothx and β has been represented completely in closed form with the same accuracy.

Some Expansions in Number Theory

This expansion which represents P involves a finite combination of elementary functions but in the form given is not useful for the purpose of calculation by hand because of the labor involved. It is the authors' opinion that simplification is possible. Many interesting problems are posed here for the analysis expert. We will also construct finite expansions in terms of n for the function r(n) which represents the number of primes less than or equal to n, for 77'(n) which represents the number of pairs of twin primes (i. e. primes differing by two) whose larger member is less than or equal to n, and for L(n), the least prime greater than or equal to n. Let there be given a function y = f(n) whose domain is the set of positive integers 1n} with 1 < n < k, k finite or infinite, and whose range is a subset A of the set of positive integers:

The use of series of bessel functions in problems connected with cylindrical wind-tunnels

1. There are various series of the types associated with the names of Fourier-Bessel and Dini which arise in the discussion of the problem of a body (the Rankine "Ovoid") placed in a cylindrical wind-tunnel. For Such series are (1) S 1 = Ʃ ∞ m = 1 J 0 ( k m r )/J 1 2 ( k m a ) e - k m x , (2) S 2 = Ʃ ∞ m = 1 J 1 ( k m r )/ k m a J 1 2 ( k m a ) e - k m x , (3) S 3 = Ʃ ∞ m = 1 J 0 ( k m r )/J 0 2 ( k m a ) e - k m x , (4) S 4 = Ʃ ∞ m = 1 J 0' ( k m r )/ k m a J 0 2 ( k m a ) e - k m x , where J 0 and J 1 denote Bessel functions of orders 0 and 1; k 1 , k 2 , k 3 , ....., are the positive roots of the equation J 0 ( ka ) = 0; k 1 , k 2 , k 3 , ....., are the positive roots of the equation J 1 ( ka ) = 0; and, so far as we are concerned, r , a and x are positive with 0 &lt; r &lt; a . It may be mentioned that a is the radius of the tunnel while r and x are respectively radial and axial coordinates. Is is not my object to discuss the origin of these series, which will be found elsewhere. It is obvious that the series are rapidly convergent and are well adapted for computation when x is large; but convergence is slow when x is small and (in the case of the first and third) is non-existent when x = 0. Mr. C. N. H. Lock, of the National Physical Laboratory, has asked me whether it is possible to express the series in forms which are suitable for computation when x is small-a problem which is evidently of some physical importance-and the investigation presents various features of mathematical interest. Accordingly in this paper I shall show how to express the series as combinations of elementary functions and convergent series of ascending powers of x and r with coefficients in forms which are fairly well adapted for computation.