Re: Proper use of "linear"

[Bob Hearn <rah@america.net>]

So linear and quadratic equations are constrained to intercept the origin?
I think your definitions are not as generally applicable as you imply here.

Bob Hearn
rah@america.net

At 06:54 PM 5/11/99 -0500, Dukelow, James S Jr wrote:
>
>
>While catching up on some old RADSAFE Digests, I came across the 
>discussion of the proper usage of "linear" and "linear no-threshold" 
>in Digest 2438. 
>
>The following is offered in the spirit of Goethe, who wrote: "The 
>mathematicians are a sort of Frenchman: when you talk to them, they 
>immediately translate it into their own language, and right away it 
>becomes something entirely different." 
>
>When a mathematician talks about a linear function f, he/she means one 
>which satisfies the linearity properties: 
>
>        f(x + y) = f(x) + f(y)
>
>and     f(a*x) = a*f(x) .
>
>They use the same definition whatever the x's and y's and the scalar 
>a's are.  f might be a function from the real numbers to the real 
>numbers (the case of interest for dose-response curves), or f might be 
>a function from a space of vectors of some dimension (Euclidean n-
>space) to the real numbers, or a function between two Euclidean n-
>spaces, or a function between a couple of infinite dimensional 
>functions spaces. 
>
>These linearity conditions are the reason that the sum of two solutions 
>(i.e., the "superposition" of the two solutions) of a homogeneous 
>linear differential equation or a homogeneous linear integral equation 
>is again a solution of the same equation. 
>
>For dose-response curves, the linear functions are precisely those of 
>the form  R = a * D, whose graphs are straight lines passing through 
>the origin, so linear no-threshold dose-response functions are linear 
>functions.  
>
>Mathematicians call functions of the form  R = a * D + b, affine 
>functions.  They have graphs that may or may not pass through the 
>origin, depending on whether or not b = 0.
>
>By these definitions, the equation  x + y = 2  defines an affine 
>function, not a linear function. 
>
>What we would understand to be a linear-threshold dose-reponse curve, 
>the mathematician would call a continuous, piecewise-linear function.  
>Its graph would have two straight line pieces, consisting of a linear 
>function  R = 0 * D, for D running from zero up to the threshold T, and 
>an affine function  R = a * D + b, for D greater than the threshold 
>(with the additional requirement that a and b are related by the 
>equation  0 = a * T + b, that is, that the affine function passes 
>through the point D = T and R = 0). 
>
>It could be argued that these definitions are far too picky for use by 
>ordinary people, and, in fact, mathematicians themselves are not 
>bothered by the hobgoblin "consistency".  They talk about solving a 
>system of linear equations, when they almost always mean solving a 
>system of affine equations (i.e., equations with non-zero right hand 
>sides). 
>
>Best regards.
>
>Jim Dukelow
>Pacific Northwest National Laboratory
>Richland, WA
>jim.dukelow@pnl.gov
>
>These comments are mine and have not been reviewed and/or approved by 
>my management or by the U.S. Department of Energy.
>
>
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