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there are 2 types of continuous numbers: linear and nonlinear continuous numbers. The continuous numbers is (are) the non-terminating numbers with continuity. we need to understand what the continuous numbers is.
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“Nonlinear change” or “change” in reading proportionality graphs, see Appendix B. Nonlinear face values are the measurement of variables relative to their asymptotes. Proportionality graphs are the graphs with a straight line expressible as a proportionality equation. Leading graphs are the graphs having a parabolic curve with continuous changing of the slope. Primary graphs are the plot of cumulative Y versus cumulative X. primitive elementary graphs are the plot of vertical elementary y versus various horizontal X either as column graph or as line chart. Many researchers are new to the Alpha Beta (αβ) math and need to familiarize with several terminologies and phrases to read this article, thus, we will give the following 6 definitions (A to F) and explain the nonlinear concepts in the followings 5 subsections.ĭefinitions: (A). Their difference is XY =, where Yu, Yb, Xu, and Xb are the upper and bottom asymptotes of Y and X variables. 5 - 10 The Alpha Beta (αβ) math is an extension of the XY math it emphasizes the association between the XY continuous nonlinear numbers and its associated asymptotes while the XY math is helpless in addressing the asymptotes relating to the nonlinear numbers. We introduce a new extended XY math (named Alpha Beta (αβ) math) concept for graphical expression of the experimental data, along with the presentation of simple mathematical equations having meaningful equation parameters. The cumulative primary graphs, but not the primitive elementary graph, should be the foundation for all nonlinear data analyses. That is, we can compare 2 variables graphically with primary graph using the cumulative numbers and mathematically with proportionality equations. 1, 3, 4 This article reveals that when comparing 2 variables mathematically, we need to compare both with the continuous cumulative numbers. When comparing 2 variables, they tend to use statistical manipulation and curve fitting with polynomial equations to relate the 2 variables. Most analyses wrongfully rely on primitive elementary line charts rather than on the reliable xy scatter charts as solid primary graphs. They also made fundamental mistakes in omitting or disregarding the necessary “origin” or “starting zeros”, including the nonlinear zero. Typically, the dose-response and pharmacokinetics analysts over used and abused the first order kinetic equation (as exponential equation) and ignored the need to unlocking the nonlinear nature of the experimental data. 1, 2 They do not even know what the linear numbers is and what the nonlinear numbers is. In the past, researchers in life and biomedical sciences do not have reliable nonlinear mathematical concepts for comprehensive understanding and in-depth analysis of the experimental data, resulting in inconsistent data presentations and miss-interpretations.
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(Note: q = log, Yu = upper asymptote of Y, Xb = bottom asymptote of X) It demonstrates a straightforward methodology for solving the key upper asymptotes (Yu) for the proportionality equation using the Microsoft Excel via determining the “coefficient of determination”. The article emphasizes the building of a straight-line proportionality relationship for the dose-response data in a log-linear and/or log-log graphs. We initially presented the data as various primitive elementary graphs then extended them to the primary graphs, leading graphs, and the proportionality graphs. The author obtained a set of data from late Professor Cohen on the lung-cancer mortality rate versus indoor radon level collected from 1,597 counties and territory of the USA. We conclude that their relationship is governed by the proportionality law where the cumulative lung cancer mortality Y is negatively proportional to the cumulative radon intensity X or specifically, the nonlinear change of nonlinear face value (qYu – qY) is negatively proportional to the nonlinear change of nonlinear face value (X – Xb). We analyze the relationship between the lung cancer mortality and the indoor radon intensity from the viewpoint of nonlinear mathematics.