What is happening at f(0)?

Look at PlotProblem.mw

Writing an explanatory paragrah i want to use an approximately equal symbol. How can I do this? The palette has lots of symbols but not what in latex would be done with \approx. 

My following code:

restart;
PDE := diff(u(x, t), t) + u(x, t)*diff(u(x, t), x) - 0.1*diff(diff(u(x, t), x), x) = 0;
IC := u(x, 0) = sin(x);
BC := [u(0, t) = u(2*Pi, t)];
pds := {BC, IC, PDE};
ds := {pds, t = 0 .. 10, x = 0 .. 2*Pi};
p := pdsolve(ds, numeric);
u := rhs(p[1]);
u(x, t);

producing following 2 errors, don't know how to handle?

Error, (in anonymous procedure called from pdsolve/numeric) invalid terms in sum: 0 .. 10
Error, invalid input: rhs received p[1], which is not valid for its 1st argument, expr

Hello guys
I'm having trouble converting the RootOf function. Attached is a simple notebook.

Sincerely,

Oliveira

Example.mw

Dear Maple Users:

Could you help in the following question?

How do I use a symbol as a subscript, e.g.  A_*, to label an axis?

When taking notes I sometimes use the palette to insert a matrix into a worksheet. When I do this, the main aspect that is useful to me is being able to visualize matrix expressions as I would write them.

I would like to do the same but for determinants of matrices. Is there a way to get almost the same thing as with the matrix palette, but with vertical bars denoting a determinant rather than the brackets used for matrices?

How do we compute integrals of functions or expressions that have units attached?
For example

The results of the integrals have the wrong units.

Download integration_with_units.mw

Let's say we have four equations and four unknowns and we use solve to find a solution.

The return value of solve is a set.

Here is an example

If we want to make the four expressions above procedures, the manual way is to basically copy the right-hand side of each expression and then write

T__1 := (R,m__1, m__2, g, I__s) -> ...

T__2 := (R,m__1, m__2, g, I__s) -> ...

a := (R,m__1, m__2, g, I__s) -> ...

alpha := (R,m__1, m__2, g, I__s) -> ...

Is there a way to do this automatically from the return value of solve?

If the return value were a list, I would use something like

T__1 := unapply(result[1], R, m__1, m__2, g, I__s)

T__2 := unapply(result[2], R, m__1, m__2, g, I__s)

a := unapply(result[3], R, m__1, m__2, g, I__s)

alpha := unapply(result[4], R, m__1, m__2, g, I__s)

But the return value is not a list.

So, in summary my quesions are

1) what, in general, is the best way to obtain the desired procedures?

2) is there a way to use the strategy I suggested if the result were a list, but for sets?

Download solveEqs.mw

Dear Maple users

I have encountered a very strange error at one of my students. When using functions from the Expression Palette, she receives an error. It happens every time she is using this Palette. A reinstallation of Maple didn't help solve this issue. When using Command Completion for the very same function (for example cosine or square root), it works properly. 

She did send a Maple file to me in order for me to investigate it. I opened it and reexecuted the entire document. The error messages remained. I tried applying the functions from the Expression Palette myself resulting in the same errors. I even opened a new document and tried it again. Still an error. Then I closed Maple completely and opened it again. Within a new document, I applied a function from the Expression Palette. Now it worked!! I can't seem to understand  the logic behind this behavior. The file sent to me by the student obviously contain some kind of error, but why will this error spread to a new document, which I have opned on my own computer? My own computer does not have any issues with Maple. 

I have attached the Maple file with the errors and a screen shot. I hope someone can explain this strange behavior and tell me how I can help this student make Maple work again. 

Kind regards,

Erik V.

Maple file:

Error.mw

After saving and closing my maple file, it was corrupted when I tried to open it again later. 

I tried searching for a solution without any luck...can someone help me repair it?

Regnede_Eksamenssæt.mw

(When it asks to save it as Regnede_Eksamenssæt_1 only a small part of the document is included)

The documentation on plots:-densityplot says

Any additional arguments are interpreted as options which are specified as equations of the form option = value. For example, the option grid = [m, n] where m and n are positive integers specifies that the plot is to be constructed on an m by n grid at equally spaced points in the ranges a..b and c..d respectively. By default a 49 by 49 grid is used; thus 2401 points are generated, and 2401 colored cells are displayed.

Consider the following simple example

If we look at the output from the print statement inside of g we see

Download DensityPlotGridQuestions.mw

That is, there are 16 points in the grid, not 5x5=25.

Is the documentation incorrect or did I miss something?

This question is about an aspect of the code below. In fact, the code could be made simpler to get at the question, but I chose to show you how the problem originated. 

We start with a 4x3 matrix A. There is a procedure, convertArrayToTable, that takes this matrix and creates a table. The table is such that the keys are the numbers in the first column of A. For each such key k1, the associated value is another table. The latter table has as keys the values in the second column that have k1 in the first column.

Now, further below, if you are curious I write about why I am doing this (what the exact real problem is). For now, I am generating a matrix B which in is formed by the concatenation of A and 2A, along the column (ie, A on top of 2A).

So for this matrix B, the table T that is generated has, for example a key 1 and the associated value is a table with keys 0.2 and 0.400000000000000.

I wish to be able to access T[1][0.400000000000000] but this is not working and I don't know why.
 

Download tableDecimalIndex.mw

So now just a brief note on why I want to create such a table.

I want to create a density plot based on the values in the rows of B. In the case of the simple example shown above, the grid of values would be based on values of x from 1 to 4 and of y from 0.2 to 0.400000000000000. Then I would specify, say "grid=[5,3]" for an eight point grid. 

In this example I am assuming the points used by the densityplot procedure would be exactly the ones in the rows of B, ie (1,0.2), (2,0.2), (3,0.2), (4,0.2),(1,0.400000000000000),(2,0.400000000000000),(3,0.400000000000000),(4,0.400000000000000).

The function I would pass in the signature

densityplot(f, a..b, c..d)

would be a custom procedure that I am assuming would be called as f(x,y) and would simply look up T[x][y] in the table I created to get the value.

Of course in my real problem the grid has way more points.

I am making a lot of assumptions about this densityplot procedure, but the documentation isn't very clear at all.

Here is a proof of concept of what I am trying to achieve (note that below, I am avoiding the issue of the decimal keys in tables by using integers instead):
 

In the above output, the output from the print statement in the f procedure is not being shown. Locally on my computer it is shown and f is called at the expected values (the ones in the rows of B). So that is all good. The issue seems to always go back to keying in to tables with certain (but not all) decimal numbers

Download DensityPlot.mw

Let us begin with few simulations: 
 

Download iDistributionVector.mws

Well, I'd like to prove (through the use of Maple): 

transform((x1, x2, x3, x4, x5) -> [x1 - x3, x2 - x4, x5])(inequal(And((x || (1 .. 5)) >=~ 0, norm([x || (1 .. 5)], 1) = 1))) # not Maple syntax

is equivalent to a filled pyramid

ImplicitRegion((X, Y, Z), 0 <= Z <= 1 - abs(X) - abs(Y)) # not SymPy syntax

, 

transform((x1, x2, x3, x4, x5) -> [x1 - x3, x2 - x4, x5])(inequal(And((x || (1 .. 5)) >=~ 0, norm([x || (1 .. 5)], 2) = 1))) # not Maple syntax

is equivalent to a hemi-ball

ImplicitRegion((X, Y, Z), 0 <= Z <= sqrt(1 - X**2 - Y**2)) # not SymPy syntax

, and 

transform((x1, x2, x3, x4, x5) -> [x1 - x3, x2 - x4, x5])(inequal(And((x || (1 .. 5)) >=~ 0, norm([x || (1 .. 5)], 'infinity') = 1))) # not Maple syntax

is equivalent to a solid cuboid

ImplicitRegion((X, Y, Z), -1 <= X <= 1 & -1 <= Y <= 1 & 0 <= Z <= 1) # not SymPy syntax

. (Here, for the convenience of the descriptions, I utilize some non-standard notation from .) 
Note that ”two regions are equal" is a two-way property, which means the following proof 

is(Z >= 0) and is(Z <= 1 - abs(X) - abs(Y)) assuming (X, Y, Z) =~ (x1 - x3, x2 - x4, x5), x || (1 .. 5) >=~ 0, add(x || (1 .. 5)) = 1;
                              true
(*Accordingly, the latter region is a subset of the former one.*) 

is incomplete (because it's hard to determine whether the is routine always performs equivalent transformations in internal evaluation). 

So, can I execute such eliminations in Maple?

As the code:

poly := x^4 + 8*x + 12:
galois(poly, x)

"4T4", {"A(4)"}, "+", 12, {"(1 2 4)", "(2 3 4)"}

 Then I know it's Galois group has to be (isomorphic to) A4. And I can draw its Subgroup Lattice:

DrawSubgroupLattice(GaloisGroup(poly, x), 'indices')

But according to Galois's theory, each subgroup represents an intermediate field. As far as I know, ⑤⑥⑦⑧ are Q(r₁),Q(r₂),Q(r₃) and Q(r₄), respectively, where r_i is the root of equation x^4+8x+12. But I have no idea what fields ②③④⑨ means. How do you calculate out those intermediate fields with maple?

I am a bit confused that why the color option in the following plot3d does not work.

plot3d( [ b, ( b^2 ) / 4, -b / 2 ], b = -4 .. 4, c = -4 .. 4, thickness = 5, color = red  ); 

The output is the following.