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\leftline{Gabby and Eliza,} 
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The presentation on Lagrange Multipliers as an application of Gr\"obner basis
methods, with economics applications was OK, but it felt as though you hadn't
practiced giving the talk enough (or hadn't worked much on it between the 
run-through on Thursday and when you presented on Monday) 
to feel really comfortable explaining the technical material.
There were a number of small slips and somewhat unclear statements.  The basic
ideas were clear enough and I think everyone understood where you were coming
from and how the mathematics we learned in the seminar could be applied.

The paper is better than the talk, although I have a lot of comments about 
various points.  The first and general comment is that there are a lot of 
typos and small language issues.  You need to proofread a paper for a math
course just the way you do for other courses.  Your capitalization is
really inconsistent, for one small thing.  All instances of Lagrange 
and Gr\"obner should be capitalized because they are proper names.  The
second general comment is that you need to be more careful about giving
credit for graphs and other images you take from sources.  They should be
attributed with footnotes just the way direct quotations and paraphrases
of information from your sources are footnoted.  I'm not taking off any 
credit for this, but you should be aware that using figures without 
attribution is often seen as a form of plagiarism.  

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\noindent
{\it Specific Comments}
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\item{1.}  I don't think it's correct to say that Cobb-Douglas production
functions are ``specific to econometrics.''  I saw them in the Introduction
to Economics course I took as an undergraduate.  It's probably better to 
say they are a widely-used class of model functions because they are 
relatively simple.  You address some of the ways they might be unrealistic
at the end of the paper, and that is good. 
\item{2.}  It's not exactly clear how this diagram relates to the 
situation where you have two distinct inputs and the graph would be
a surface in 3-dimensional space (if you included a coordinate axis
for the output).  How do increasing/constant/decreasing returns
connect with the exponents in a Cobb-Douglas function?
\item{3.}  Economists probably do always make the assumption that increasing
$X$ and $Y$ simultaneously takes you to an indifference curve with a 
higher value of utility.  However, from a more philosophical viewpoint, 
it's probably worthwhile to ask whether having more of everything increases
utility or well-being for people in real life.  Are really rich people
always happier than those with less money and possessions?
\item{4.}  There's no theoretical reason why a utility function 
has to have the form $U(x,y) = xy$.  This is just a convenient 
example.  
\item{5.}  What is a ``tautochrone particle?'' (I think you are referring
to something called the tautochrone problem which was a famous question
that stimulated the mathematical field called the {\it calculus of variations}
(and that was the context for Lagrange's invention of the multiplier method). 
A {\it tautochrone} is a descending curve in a vertical plane with the property that 
if you have a bead or particle sliding along the curve, acted on 
by gravity, then it takes the same amount of time for the particle to slide
down to the end no matter what point it is released from.  (The name {\it tautochrone}
comes from Greek words meaning ``same time'' in fact.)  The problem was to find
the equation of such a curve in terms of known functions.
\item{6.}  This is a big overstatement.  People were using and applying
Cobb-Douglas functions in constrained optimization long before Gr\"obner 
bases were even invented(!)
\item{7.}  What does it mean to say this was ``only a theory?''  Isn't everthing
in mathematics a theory?  (I mean that every statement in mathematics has
the form "if some hypotheses are true, then some conclusions follow."  Mathematicians
never claim that they have absolute truth or a description of the 
way the universe really works!)
\item{8.}  I don't know what you mean by ``finessed'' here.  Macaulay 
wasn't the only one to work on Elimination Theory.  This was a central area
of mathematics in the 19th century and lots of people did lots of different 
things related to eliminating variables from systems of equations.  One of 
the most important ideas we didn't discuss in the course uses determinants
called {\it resultants} to do elimination.  This is discussed in Chapter 3 
of IVA.
\item{9.} Saying ``optimal number of attacks'' without more explanation
 sounds really weird and 
cold-hearted.  The use of the words ``utility'' or ``well-being'' strikes
me the same way.  Wouldn't the optimal number of attacks be {\it zero} from
the point of view of the {\it victims'} well-being?  It would be good to say that this 
whole discussion is from the point of view of the terrorists, and the 
issue is how they can use the resources at their disposal to do the most
damage.  The whole point is to understand how that damage would change
if the cost of hitting a political target goes up.  The number of civilian
attacks would go up, while the total damage done would decrease (that's
the shift to the lower ''indifference curve'' for the terrorists).  
\item{10.}  This is not correct (it wasn't correct in the talk slides either). 
The idea is that if the ideal is $I = \{0\}$, then to say you have
a Gr\"obner basis too in that case, you would take the Gr\"obner
basis to be $G = \emptyset$.
\item{11.}  You didn't really finish this example!  What are the 
corresponding values of $y,x$?  Which point $(x,y,z)$ maximizes $f(x,y,z) = x^3yz$ 
on the constraint set?
\item{12.}  This is not clear at all.  I think what you are trying
to say is that some industries (like agriculture) involve many more
inputs than just two as in the basic Cobb-Douglas function, so 
that is not a realistic model.  Is that what you meant?
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\noindent
Final Project Presentation:  82 (B-)
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\noindent
Final Project Paper:  85 (B)  
\bye
