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).
> 
> 
> 	Emelie Lamothe
> 	lamothee@aecl.ca
> 	----------
> 	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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