FW: Proper use of "linear"

["Lamothe, Emelie" <lamothee@aecl.ca>]

Well, being a French person...

I scratched my head and went to my dictionary of mathematics because I
couldn't remember seeing the term "affine" before (getting old, I guess).
So, yes, the semantics are correct but as Jim points out in the last
paragraph, usage does get corrupted, even by mathematicians. "Affine" is
most often used in the context of transformations.  Even the spell checker
is ignorant of this term, even though it's been around since 1918.

> Affine Equation:
> 
> A nonhomogeneous Linear Equation or system of nonhomogeneous Linear
> Equations is said to be affine.
> 
> 
> Affine Transformation 
> 
> Any Transformation preserving Collinearity (i.e., all points lying on a
> Line initially still lie on a Line after Transformation).
> 
> 
> 	----------
> 	From: 	Dukelow, James S Jr[SMTP:jim.dukelow@pnl.gov]
> 	Reply To: 	radsafe@romulus.ehs.uiuc.edu
> 	Sent: 	Tuesday, May 11, 1999 7:53 PM
> 	To: 	Multiple recipients of list
> 	Subject: 	Proper use of "linear"
> 
> 
> 
		[snip]
> 	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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