rlopez

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14 years, 320 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

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These are replies submitted by rlopez

@ The worksheet in Maple 7 is constructed with software issued by a company that went out of business sometime after Maple 9 was released. Hence, there is no way to continue developing that form of the product.

From Maple 10 onwards, it is possible to save a file as a "Maple Classic Worksheet (.mws)."

So, which meaning do you attach to "classic"? Do you mean the format used prior to the introduction of the Java-based interface that arrived with Maple 10, or do you mean the "Maple Classic Worksheet (.mws)" that is available from Maple 10 onwards? If you mean the former, does that imply that the latter is not a satisfactory replacement? And if so, why? What are the shortcomings of the extant Maple Classic Worksheet (.mws)?

A final note: Until recently, my home computer was a desktop running Windows XP. The last version of Maple that would install under that OS was Maple 18. It does not surprise me that in a newer operating system, an older version of Maple will not work. However, it is the case that Maple 2017, installed under Windows10, saves a worksheet as a Maple Classic Worksheet (.mws).

 

RJL Maplesoft

I'm not sure that the statement "Maplesoft wants everyone to switch to clickable things" is exactly true. I devised the term "Clickable Calculus" as a banner for the ease-of-use features that were beginning to appear in Maple, from Maple 10 onwards. Maplesoft at first wrinkled it nose at the term, then embraced as when it saw that "clickability" worked with an appropriate audience. (Think of all the users and students who don't need to master Maple commands in order to get value from the product.)

I was never aware of a Maplesoft mindset to get all Maple users to surrender the worksheet with text input, modes that appear to serve coders and programmers best. If there really is such a hidden agenda at Maplesoft, I'm as troubled about that as the many users who consistently report such feelings here on MaplePrimes.

Couldn't stop thinking about this calculation. The line integral on the right in the Divergence form of Green's theorem is actually the line integral of the normal component of <f,g> along a curve. Hence, this is a flux calculation, for which Maple has a built-in tool. So, using Kitonum's structure, it is possible to implement the line integral on the bounding SCROC with the Flux command in the Student VectorCalculus package. Just to prove to myself that this works, I shortened the code to the following, and verified that it gives the same answer(s) as the earlier code. By no means does this detract from the original insight exhibited by Kitonum.

IntOverDomain := proc(f, L)
local n, i, j, m, yk, yb, xk, xb, Q, p, P, R, V;
use Student:-VectorCalculus in
n:=nops(L);
Q:=int(f,x);
V:=VectorField(<Q,0>); 
for i from 1 to n do
if type(L[i], listlist(algebraic)) then
R:=Vector~(L[i])[];
P[i]:=Flux(V,LineSegments(R));
else
P[i]:=Flux(V,Path(Vector(L[i,1]),L[i,2]));fi;
od;
add(P[i], i = 1 .. n);
end use:
end proc:
 

@vv I was under the impression that inverting would change the meaning of the residuals, and hence, the meaning of the least-squares fitting function. Is this a misapprehension on my part?

@rlopez 

Note to self: It's actually in Maple! In the Format menu, select Styles. In the resulting dialog, select "Maple Plot" in the collection of items listed on the left. Then click "Modify". In the dialog that results from this, change "Alignment" to "left", then click OK. Any graph drawn after this will be left-aligned.

 

Kitonum's answer is correct, and finally, I see why. The construction suggested by the OP's question is that of an ellipse. However, the length of the string looped around the given foci is the distance between the foci. That's why the calculations one normally uses to obtain the equation of an ellipse result in the equation of a line. That it becomes a line segment is apparent with some additional inspection. Indeed, I graphed the surface representing the left-hand side of the equation

evalc(abs(a+I*b+3-2*I))+evalc(abs(a+I*b-3-8*I)) = 6*sqrt(2)

and the plane z=6(sqrt(2). These surfaces intersect in a line segment. Why a line segment I kept asking myself. The algebra for obtaining the equation of an ellipse degenerates to (1-b+5)^2=0, so why not a degenerate ellipse which is a line. It's a line segment because the distance between the foci is itself the sum of the distances from the foci. Like a dog with a bone...

@Marzio Giaveno 

Try assign~(seq(X[k]=X[0]+k*(X[N]-X[0])/N, k = 1 .. N-1));

This sets up a sequence of equations, and assign~ maps the assignment operator onto each of the equations in the sequence. That does not generate the first error shown in your post.

The list of points in P each contain a function f evaluated at a known X but an unknown Y. But what about the function f? The  sum of distances between adjacent points in SL is then the argument of the Minimize command, which acts numerically. If the unknown function f is in this argument, again errors will be thrown. At this point, either there's the coding error of forgetting to define f, or a logic error in thinking Minimize can do its job without knowing what the function f is.

I did not progress to the third error because the second one is insurmountable.

What are F, V, A, and B?

You define P1, P2, P3, P4 with upper case, but display p1, p2, p3, p4 with lower case. (Maple is case sensitive.)

RJL Maplesoft Kitonum's solution is elegant, but I had to look at (and think about) the output of the discriminant command to understand its role. Upon substitution of y=m*x into the equation of the circle, a quadratic in both m and x results. The discriminant of a quadratic will tell you how many solutions the quadratic has. What is wanted for tangency is that there is just one solution, a condition that results when the discriminant is zero. The solve command is applied to the discriminant, but it is just an expression. Maple assumes that when an expression is given to the solve command the intent was to set the expression to zero. The two values of m that result are the slopes of the two tangent lines that you are looking for.

I was curious to see just when Maple's solve command could handle the inequalities. The command asked about succeeds in every version from Maple 11 through Maple 2016.2. Hence, the OP might have needed to do a restart, etc., to get the command to work - unless a really old version of Maple was being used.

@brian bovril I was trying to clarify the point that there is some regression analysis available for nonlinear fits. Not as many items as for linear fits, but some. I overlooked the OP's specific pointer to "standard error of the coefficients". And no, this is not one of the items available for regression analysis in a nonlinear fit. Sorry. I should have been more careful in reading the OP's question. Thanks for calling the oversight to my attention.

@brian bovril The help page at ?Statistics,Regression,Solution will show that one can obtain regression analysis for both linear and nonlinear fits. For linear fits, there are more items in the analysis, but some of the items in the linear fit analysis appear in the list for nonlinear fits.

OK, during lunch I realized what I needed. I think the Lagrange Expansion theorem applies. My first recourse for a reference is "Perturbation Techniques in Mathematics, Physics, and Engineering" by Bellman. Could probably pull it out of Wiki, but I'm addicted to print. Will have a go at it after some afternoon errands.

Interesting comment. I can obtain that result by applying the asympt command to the equation that I ended up solving numerically. (Bring all terms to one side first.) This gives an asymptotic expansion in terms of r and s. Setting the sum of the first two terms to zero and solving for s gives the result in vv's comment. Solving for s becomes intractable if more terms are taken, so I'm left with the analytic question: How valid is it to expand first, then solve. I'll need to think more about this technique. Perhaps vv's result can be obtained in some other way?

@Markiyan Hirnyk 

OK, you have clarified for me what BSplineCurve returns.

I then went back to example2.mw where you obtain 44 for integral based on a cubic spline built from 200 data points taken from the graph of xydata. Graph that cubic spline and compare its graph to that of "expand(Interpolant(p1))". I would not trust that 44 obtained by integrating this expanded interpolant.

Graph the parametric curve obtained by BSplineCurve obtained in another_way.mw. Although this curve is close to the piecewise-linear curve given by xydata, it does not appear to be defined over the whole interval [0,8]. This might account for the lower value of 34 when integrating.

I think I stand by my 49.54 obtained by integrating under the piecewise-linear spline that approximates the modified data on [0,8.01].

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