Proper use of "linear"

["Dukelow, James S Jr" <jim.dukelow@pnl.gov>]

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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