tomleslie

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These are answers submitted by tomleslie

whilst this approach might work for the trivial problem which you gave, it is doomed to failure whenever you have more than one abs() term in your expression(s).

Why do you think that

because maple may has difficulties when dealing with absolute values

What evidence do you have for this statement?

What you want to plot is a little unconveentional. The attached worksheet produces a couple of plots.

The first is the conventional f(eta) vs eta, and theta(eta) vs eta

The second is theta(eta) vs L, for eta = 0,1,2,3, cos I wasn't sure which value of eta you were interested in

trickPlot.mw

Not sure how the first part of your question relates to the second: converting your integrand to polar coordinates is trivial, but not so for the limits.

Given the quoted limits, your question only makes sense if the integration wrt x is performed first, so the solution to your integral would be given by

int( int(9*x*y^2, x=-sqrt(a^2-y^2)..0), y=0..1);

So why do you want polar coordinates???

Since your polynomials are univariate, I suggest you examine the PolynomialTools[CoefficientList] command. Use of this command ensures that polynomial coefficients appear in defined order (with an option to reverse).

My solution for your problem is shown in the attached

coeffs.mw

@Kitonum 

No real problem with the above but


for k to 9999 do
     q:=k;
     if add
        ( irem(q, 10, 'q')^3,
           k = 1..length(q)
        )
        >= k
     then n:=k
     fi;
od:
n;

runs about 15x faster

restart;
  with(Statistics):
  with(Student[Statistics]):
  with(plots):
#
# generate some data
#
  N1:=NormalRandomVariable[Normal](0,1):
  N2:=NormalRandomVariable[Normal](0.5, 0.5):
  A:=Sample(N1,100):
  B:=Sample(N2,100):
#
# Produce two histograms
#
  P1:=Histogram(A, style=polygon, color="Red"):
  P2:=Histogram(B, style=polygon, color="blue"):
#
# display them
#
  display(P1,P2);

VectorCalculus[CrossProduct](a,b), requires that a, b be 3-D vectors, yours are 2-D. Easiest thing to do is just add a third entry and set it equal to zero as in

restart;
r:=(u,v)->Vector[column]([u*cos(v), u*sin(v),0]);
ru:=diff~(r(u,v),u);
rv:=diff~(r(u,v),v);
VectorCalculus[CrossProduct](ru,rv);

Fairly easy to filter the array with

 

val:=NULL;
Z:=Array(1..5);
Z[1]:=val;
Z[2]:=5;
Z[3]:=val;
Z[4]:=val;
Z[5]:=8;
data:=[seq(`if`(Z[n]<>val,[n,Z[n]],NULL), n=1..5)];
t1:=plot(data, style=point, symbol = solidcircle, color= red):
plots[display]({t1});

Whatever is assigned to val (NULL, a numeric, it doesn't matter) will be omitted from the data plotted

In the code loop

for i from 1 to 2*r+1 do
CCC[i]:=simplify(Tr(X).H[i]-Tr(X0).H[i]-3*(Tr(X).Dtau1.CC[i].Tr(Dtau2).U));
od;

H[i] is a scalar but Tr(X) and Tr(X0) are vectors so the matrix multiplication operation '.' is invalid. Use '*' instead.

Furthermore

Tr(X)*H[i] is a 1*3 vector
Tr(X0)*H[i] is a 1*3 vector, but
-3*(Tr(X).Dtau1.CC[i].Tr(Dtau2).U)) is a scalar -and so cannot be combined with the preceding vectors. If you mean this term to be add to each element of the preceding vectors then use

CCC[i]:=simplify(Tr(X)*H[i]-Tr(X0)*H[i]-~3*(Tr(X).Dtau1.CC[i].Tr(Dtau2).U));

Couldn't check the subsequent code becuase the next execution group contains


J:=3*(Tr(X).X+Tr(U).U);
 MM:=Minimize(J,{seq(CCC[i]=0,i=1..N*M)});
 MINIMUM_IS:=op(1,MM);


Both N and M are undefined, so the seq won't execute - based on the generation of CCC in the preceding loop, 2*r+1 looks like a good bet for the last argument to seq, but I decided that this was piling up too much guesswork about what you wished to achieve

 

I find that if I use

dataplot(L0[()..(), 1], L0[()..(), 2], axes = box, size = [600, 320] ,symbol=point);

in Maple2015, then this gives the same result (at least "visually") as

plot(L0[()..(), 1], L0[()..(), 2], axes = box, size = [600, 320]);

in Maple 18 (apart from the fact that they use different colors by default - trivial to fix).

The dataplot command is new in Maple2015 but AFAIK, it is meant to be a "simpler" interface to the same underlying commands. All of which means that the behaviour of the underlying plot command (ie the ones you used) has been changed - I don't know why, so I think you have a legitimate gripe and if you don't get a better answer here than this one, then I suggest you send it to the Maple techies as a bug.

A few (probably irrelevant) comments

  1. Your data file contain roughly 10 identical x-values with different y-values: I say roughly 10 because it seems to vary from about 6 to about 11. when first examining your data I thought this might be an issue and upsetting the plot, but it doesn't *seem* to be. I tried plotting the second column of you matrix against row number: Maple 18 and Maple2015 still gave substantially different results, so I don't think this is an issue
  2. interesting way to select columns of a matrix the L0[()..(),1]. Normally I'd just use L0[..,1]

Couldn't really follow what you code was trying to do. An implemenatation of the logical process you went through in the statement of your question would be

   restart;
#
# Initialise 4 bags with nothing in any of them
# and an index for the "current bag
#
   bags:=table([b1=0, b2=0, b3=0, b4=0]):
   bi:=0:
#
# loop round all sums which have to be achieved
#
   for j from 1 to 15 do
     #
     # check whether j cannot be obtained as a sum
     # of existing bags
     #
     if member( j,
                      map
                      ( add,
                        combinat[choose] ( [entries( bags, 'nolist' ) ] )
                      )
                   )=false
     then
         #
         # j must go in the next available bag
         #
            bi:= bi+1:
            bags[ b||bi ]:= j:
     fi;
   od;
#
# display result
#
    op(bags);

Output from taylor is of datatype series. Generally you need to convert to polnomial form to do further manipulations. Try

restart;
uimj:=taylor(u(x-h,y),h,4);
uipj:=taylor(u(x+h,y),h,4);
convert(uimj,polynom)+convert(uipj,polynom);

int(X^2 + 3*y + 3*z + X^7,X=0..1);

Depends on whether you are allowed to select the same sideLength more than once - or do you specifically wish to exclude all isosceles and equilateral triangles??? Not obvious from your question, although if you are just constructing valid triangles - then why exclude equilaterals and isosceles???

Code provided by Kitonum deliberately excludes isosceles and equilaterals - but do you want to do this????

A minor revision to Kitonum's first code example will allow isosceles and equilaterals to be included

L:=[seq( seq( j,k=1..3), j=1..8)];
P:=combinat[choose](L, 3):
k:=0:
for p in P do
    a:=p[1]; b:=p[2]; c:=p[3];
    if a+b>c and a+c>b and b+c>a
    then k:=k+1:
    fi;
od:
k;

In which case you will get 70 valid triangles (rather than 22) - makes a big difference

The difference actually gets smaller as the number of side lengths goes up. A simple revision to Kitonum's second code example to allow isosceles and equilateral triangles, with a judicious choice of limits for the innermost loop to apeed things up a bit would be

restart;
ts:=time():
k:=0:
#ne:=8: #a check value
ne:= 500:
for a from 1 to ne do
    for b from a to ne do
        for c from b to min(a+b-1, ne) do
            if   a+c>b and b+c>a
            then k:=k+1;
            fi;
        od:
    od:
od:
k;
time()-ts;

results in 10510625 triangles, rather than 10323125, and run in about 6secs on my machine

I accept that the OP was not interested in a built-in command because (s)he wanted to produce a solution as an exercise. However I think it is reasonable to compare the OP's code with the Maple's built-in version, just for verification purposes

Looking at the codes supplied by OP and Kitonum (the two are equivalent), I was struck by thought that they might do interesting things depending on whether the number of points requested was 0 mod 3, 1 mod 3, or 2 mod 3.

The accuracy of the original version of what is now known as "Simpson's 3/8" rule, is completely independent of the number of points (although the name "3/8 rule" probably only makes real sense, when the number of points is a multiple of 3). The Maple help page ?Student[Calculus1][Simpson's 3/8 Rule] is actually quite informative in this respect

Consider the attached worksheet which will allow you to compare/contrast the result of OP's code, Kitonum's code, and the Maple's built-in version of "Simpson's 3/8" rule in two scenarios - where the number of points is small, and where the number of points is large

simpson38.mw

Notice how accurate the Maple built-in function is for small numbers of points, irrespective of whether the number of points is 0 mod 3, 1 mod 3, or 2 mod 3. OPcode (and Kitonum - because they produce identical results) are pretty close when the numbers of points is a multiple of 3 and pretty awful otherwise

Even with a large number of points, it is still noticeable that OPcode/Kitonum agrees much better with the Maple built-in command when the number of points is a multiple of 3.

Actually I expected complete agreement when the number of points is a multiple of 3, but I haven't got the time/energy to figure out why there is a discrepancy in this specific case - let's just say I'm backing the Maple version!

Why do I care??
Well I live quite close to where Thomas Simpson was born and is buried (and have visited the latter). A small plaque in a very old church isn't much reward for torturing generations of students, but hell, it's more than most of us will get, and he deserves some respect (even although many, much earlier, mathematicians had probably produced the same rules)

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