Systems of ordinary differential equations that exhibit chaotic responses have yet to be correctly integrated. So far no ‘convergent’ computational results have been determined for chaotic differential equations. Various computed numbers are not solutions of the continuous differential equations; all chaotic responses are simply numerical noise and have nothing to do with the solutions of differential equations. It would be an exciting contribution if a convergent computed chaotic solution for a Lorenz model could be obtained.
Over the last one-half century due to continuous improvement in computers, numerical simulations have been developed into important tools, if not the only ones, to solve differential equations related to real problems from atomic to astronomic scales. Consequently, the number of engineers and scientists engaged in the use of such tools has increased dramatically. Computation has become, in many ways, the modern version of "mathematical analysis," and the kinds of problems analyzed by numerical methods keeps on growing. Most of the truly challenging problems, such as those related to chaos or turbulence, are routinely approached in this way. Researchers and other practitioners confidently believe that their particular problems are actually being solved by merely running their computers. Unfortunately, the importance of careful checks of convergence, an expected outgrowth from "numerical analysis," is often ignored.The sensitivity of computed results for chaos or turbulence to the size of integration time steps is not completely unknown in the numerical analysis community. However, most people, who recognize the existence of this problem, optimistically believe that the statistical properties of such computed results are relevant to their problem. The apparent reason for this belief is the fact that so many individuals have done similar types of computations and have published vast numbers of papers and reports. How could it be that something is wrong, given all of this effort? Simply stated, it has become common practice to not check computational results since they have been accepted for such a long time. This situation may be partially due to the fact that the problem of error analysis for nonlinear differential equations has not been systematically studied. Under this family of circumstances, raising questions about the validity of numerical simulations of chaos or turbulence is viewed by some as an attack on their integrity.We want to congratulate the members of the meteorology group at the Naval Research Laboratory in Monterey, California, supported by the Office of Naval Research, for publishing this important paper (Teixeira et al. 2007) on their systematic study of time step sensitivities for nonlinear differential equations of chaos and turbulence. Their firm conclusion is that computational chaos results of differential equations, which are sensitive to time steps, are simply errors, as are their ensemble averages. Their work is a landmark contribution and will lead a long train of future studies, which will improve understanding of the amplification of truncation errors in computational chaos and turbulence.Even though it does not alter the conclusion of their paper, an objection about their assumption that the result of the smallest time step is closest to the correct one can be made. The amplification mechanism for truncation errors has been analyzed from a geometric point of view for the Lorenz system (Yao 2005). Two types of error amplifications, exponential and "explosive," were identified.The more...
At the World Summit for Children (New York, 1990), a resolution was passed to eliminate vitamin A and iodine deficiencies and significantly reduce iron-deficiency anemia by the year 2000. In responding to this urgent call, we developed a unique multiple-micronutrient fortification delivery system called "GrowthPlus/CreciPlus." Using this technology, a fortified powder fruit drink has been formulated and extensively evaluated. One serving of the product delivers the following US recommended dietary allowances: 20-30% of iron; 10-35% of vitamin A; 25-35% of iodine; 100-120% of vitamin C; 25-35% of zinc; 15-35% of folate; and 10-50% of vitamins E, B2, B6, and B12. This was accomplished through (a) identifying and selecting the right fortificants, and (b) understanding their chemical and physical properties that contribute to multiple problems (product acceptability, stability, and bioavailability). Data from a home-use test showed fortification with the "Multiple-Fortification Technology" has no effect on the appearance and taste of the eventually consumed powder fruit drink. One-year stability studies demonstrated that iodine and the vitamins have adequate stability. Bioavailability evaluation by using double-isotope labeling technique showed that the iron from the fortified powder drink has excellent bioavailability (23.4% +/- 6.7). In conclusion, a powder fruit drink has been clinically demonstrated to deliver multiple micronutrients, which include adequate levels of bioavailable iron, vitamin A, iodine, zinc, vitamin C, and B vitamins, without compromising taste, appearance, and bioavailability. The critical limiting step in the micronutrient fortification program is the production and distribution of the multiple-micronutrient-fortified product. The fortified powder drink was marketed in Venezuela under the brand name NutriStar.
At the World Summit for Children (New York, 1990), a resolution was passed to eliminate vitamin A and iodine deficiencies and significantly reduce iron-deficiency anemia by the year 2000. In responding to this urgent call, we developed a unique multiple-micronutrient fortification delivery system called "GrowthPlus/ CreciPlus®." Using this technology, a fortified powder fruit drink has been formulated and extensively evaluated. One serving of the product delivers the following US recommended dietary accomplished through (a) identifying and selecting the right fortificants, and (b) understanding their chemical and physical properties that contribute to multiple problems (product acceptability, stability, and bioavailability). Data from a home-use test showed fortification with the "Multiple-Fortification Technology" has no effect on the appearance and taste of the eventually consumed powder fruit drink.One-year stability studies demonstrated that iodine and the vitamins have adequate stability. Bioavailability evaluation by using double-isotope labeling technique showed that the iron from the fortified powder drink has excellent bioavailability (23.4% ± 6.7). In conclusion, a powder fruit drink has been clinically demonstrated to deliver multiple micronutrients, which include adequate levels of bioavailable iron, vitamin A, iodine, zinc, vitamin C, and B vitamins, without compromising taste, appearance, and bioavailability. The critical limiting step in the micronutrient fortification program is the production and distribution of the multiplemicronutrient-fortified product. The fortified powder drink was marketed in Venezuela under the brand name NutriStar®.
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