Universality quantitative

M. J. Feigenbaum discovered a remarkable regularity in period doublings whereby the ratio of the parameter intervals between successive doublings approaches a universal constant 4.669. See M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19,25-52 (1978). 

Feigenbaum, M. J., 1978, Quantitative university for a class of nonlinear transformations. J. Slat. Phys. 19, 25-52. 

The U-sequence is qualitative it dictates the order, but not the precise parameter values, at which periodic attractors occur. We turn now to Mitchell Feigen-baum s celebrated discovery of quantitative universality in one-dimensional maps. 

Feigenbaum, M., Quantitative Universality for a Class of Nonlinear Transformations, Journal of Statistical Physics, 19 25-33 (1978). 

K. L. E. Kaiser, ed.. Proceedings of the Workshop on Quantitative Structure—Activif Relationships (QSARJ in EnvironmentalToxicology held atMcMaster University, Hamilton, Ontario, Canada, August 16—18, 1983, D. Reidel Pubhshing Co., Dordrecht, Germany, 1983. 

Until the last War, variants of optical emission spectroscopy ( spectrometry when the technique became quantitative) were the principal supplement to wet chemical analysis. In fact, university metallurgy departments routinely employed resident analytical chemists who were primarily experts in wet methods, qualitative and quantitative, and undergraduates received an elementary grounding in these techniques. This has completely vanished now. 

Universality Fei genbaum s remarkable discovery consisted of the following two quantitative observations (see figure 4.6) ... 

James Dewey Watson (1928- ) was born in Chicago, Illinois, and enrolled in the University of Chicago at age 15. He received his Ph.D. in 1950 at the Unwersity of Indiana and then worked at Cambridge University in England from 1951 to 1953, where he and Francis Crick deduced the structure of DNA. After more than 20 years as professor at Harvard University, he moved in 1976 to the Laboratory of Quantitative Biology at Cold Spring Harbor, Long Island, New York. He shared the 1962 Nobel Prize in medicine for his work on nucleic acids. 

This collection of worksheets (about three per chapter) is for students to use in collaborative groups. Designed to stimulate group activity and discussion, the questions provided on each worksheet are equally conceptually and quantitatively oriented. Also included are instructions for use, information on how to guide student discussion, and supporting data on the success of cooperative learning at the University of Connecticut... 

Ion chromatography (IC) is a relatively new technique pioneered by Small et al.25 and which employs in a novel manner some well-established principles of ion exchange and allows electrical conductance to be used for detection and quantitative determination of ions in solution after their separation. Since electrical conductance is a property common to all ionic species in solution, a conductivity detector clearly has the potential of being a universal monitor for all ionic species. 

By mixing suitable indicators together changes in colour may be obtained over a considerable portion of the pH range. Such mixtures are usually called universal indicators . They are not suitable for quantitative titrations, but may be employed for the determination of the approximate pH of a solution by the colorimetric method. One such universal indicator is prepared by dissolving 0.1 g of phenolphthalein, 0.2 g of methyl red, 0.3 g of methyl yellow, 0.4 g of... 

In most cases, the swelling of polyelectrolyte hydrogels depends only on ionic strength of the solution but not on the size and nature of the ions [101]. Therefore, the ionic suppression curves similar to those of Fig. 2 and 3 are to some extent universal and allow to predict quantitatively the swelling of hydrogels for practically any ionic situation. 

It is usually assumed in the derivation of isothermal rate equations based on geometric reaction models, that interface advance proceeds at constant rate (Chap. 3 Sects. 2 and 3). Much of the early experimental support for this important and widely accepted premise derives from measurements for dehydration reactions in which easily recognizable, large and well-defined nuclei permitted accurate measurement. This simple representation of constant rate of interface advance is, however, not universally applicable and may require modifications for use in the formulation of rate equations for quantitative kinetic analyses. Such modifications include due allowance for the following factors, (i) The rate of initial growth of small nuclei is often less than that ultimately achieved, (ii) Rates of interface advance may vary with crystallographic direction and reactant surface, (iii) The impedance to water vapour escape offered by... 

HPLC coupled with an evaporative light scattering (ELS) detector has been used as an universal detection technique also valid for quantitative analysis. Alcohol ether sulfates were analyzed by this method with good results and also at very low concentrations [295]. 

Erosion/Deposition Impact Matrix and Map. Methods applied in this semi-quantitative assessment involved mapping depositional features using high and low altitude imagery, numerical ranking of land use activity impacts and construction of problem matrixes. The information generated in these first steps were applied to the Universal Soil Loss Equation (15-16),... 

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