Evaluation Of Certain Functions Research Articles

Conjectures on the evaluation of certain functions with algebraic properties

In this article we consider functions with Moebius-periodic Taylor expansion coefficients. These functions under some conditions take algebraic values and can be evaluated in terms of theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers–Ramanujan continued fraction, Eisenstein series and functions associated with Jacobi symbol coefficients.

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

Affinity labelling of the Ca2+-activated neutral proteinase (calpain) in intact human platelets

Two irreversible calpain inhibitors, benzyloxycarbonyl (Cbz)-Leu-Leu-Tyr-Ch2F and Cbz-Leu-Leu-Tyr-CHN2, were shown earlier [Anagli, Hagmann and Shaw (1991) Biochem. J. 274, 497-502] to penetrate intact platelets and to inactivate calpain. This permitted an evaluation of certain functions attributed to this proteinase. For example, in platelets pretreated with these inhibitors, talin and actin-binding protein were protected from subsequent degradation when the Ca2+ level was raised. On the other hand, additional properties of stimulated platelets attributed to calpain remained unaffected by this treatment, and such hypotheses may be dismissed. Radioiodinated inhibitors permitted confirmation of the labelling of calpain by the procedures used. Although Cbz-Leu-Leu-Tyr-CHN2 is more effective in vitro than the corresponding fluoromethyl ketone, we now show that the latter penetrates more readily. These two inhibitors, and two additional ones, t-butyloxycarbonyl-Val-Lys(Cbz)-Leu-Tyr- CHN2 and Cbz-Leu-Tyr-CH2F, have been radioiodinated to permit a comparison of their intracellular labelling patterns in activated platelets. Calpain is the major target of all four inhibitors. Although they are closely related peptide structures, variations with respect to the labelling of additional proteins were observed. These were minor in the case of the peptidyl diazomethyl ketones, but were major in the case of the fluoromethyl ketones. However, in contrast to calpain, this labelling was neither time-dependent nor Ca(2+)-dependent. Radiolabelling and cellular fractionation studies were used to localize active calpain during platelet activation. Calpain appears to be activated in the cytosol and translocated to the membrane or cytoskeletal sites.

ON IRREGULARITIES OF DISTRIBUTION AND APPROXIMATE EVALUATION OF CERTAIN FUNCTIONS

Journal Article ON IRREGULARITIES OF DISTRIBUTION AND APPROXIMATE EVALUATION OF CERTAIN FUNCTIONS Get access W. W. L. CHEN W. W. L. CHEN Imperial CollegeLondon SW7 Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 36, Issue 2, June 1985, Pages 173–182, https://doi.org/10.1093/qmath/36.2.173 Published: 01 June 1985 Article history Received: 13 February 1984 Published: 01 June 1985

Bounds on the value of information in uncertain decision problems

The cost of obtaining good information regarding the various probability distributions needed for the solution of most stochastic decision problems is considerable. It is important to consider questions such as: (1) what minimal amounts of information are sufficient to determine optimal decision rules; (2) what is the value of obtaining knowledge of the actual realization of the random vectors; and (3) what is the value of obtaining some partial information regarding the actual realization of the random vectors. This paper is primarily concerned with questions two and three when the decision maker has an a priori knowledge of the joint distribution function of the random variables. Some remarks are made regarding results along the lines of question one. Mention is made of assumptions sufficient so that knowledge of means, or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules. The analysis of the second question leads to the development of bounds on the value of perfect information. For multiperiod problems it is important to consider when the perfect information is available. Jensen's inequality is the key tool of the analysis. The calculation of the bounds requires the solution of nonlinear programs and the numerical evaluation of certain functions. Generally speaking, tighter bounds may be obtained only at the expense of additional information and computational complexity. Hence, one may wish to compute some simple bounds to decide upon the advisability of obtaining more information. For the analysis of the value of partial information it is convenient to introduce the notion of a signal. Each signal represents the receipt of certain information, and these signals are drawn from a given probability distribution. When a signal is received, it alters the decision maker's perception of the probability distributions inherent in his decision problem. The choice between different information structures must then take into account these probability distributions as well as the decision maker's preference function. A hierarchy of bounds may be determined for partial information evaluation utilizing the tools of the multiperiod perfect information case. However, the calculation of these bounds is generally considerably more dicult than the calculation of similar boulids in the perfect information case. Most of the analysis is directed towards problems in which the decision maker has a linear utility function over profits, costs or some other numerical variable. However, some of the bounds generalize to the case when the utility function is strictly increasing and concave.

Application of Continued Fractions for Fast Evaluation of Certain Functions on a Digital Computer

The purpose of this paper is to develop a method for evaluation of certain elementary functions on a digital computer by the use of continued fractions. The time required for this evaluation is drastically reduced by using short operations like shift and add, instead of multiplications. Functional consistency is the most important factor that aliows the expansion of a function into a continued fraction. Several cases are discussed; in particular the solution of the quadratic equation is discussed in more detail to demonstrate the convergence of the method.

Approximate Evaluation of Certain Functions Arising in Unsteady Airfoil Theory

appears in the kernel of an integral equation shown by Lawrence and Gerber [1] to govern the aerodynamics of low-aspect-ratio airfoils in unsteady motion. The Cicala function [2], F(p) = M*(oo, 3)/3, plays a similar role in Reissner's unsteady lifting line theory [3]. Although M*(a, () has been partially tabulated [1], a simple approximate expression valid over the entire domain of interest (0 ? a oo. On the other hand, the bracketed portion of the integrand in (1) is a well-behaved function of the single parameter t x/l. We therefore seek an approximation of