Hi,
How can I simplify this relation(See uploaded .mw file)?
For example, the second term is simplified as: 

deltae*(1-phi0/(kappa-3/2))^(-kappa+1/2)+(1/2)*deltab*(1-sqrt(2)*sqrt(1/(m*ub^2))*sqrt(-phi0));

di1.mw

In Latest Maple 2024.2, I found that when doing z:=%  where % is result on integration, causes internal error 

          Error, unexpected result from Typesetting

But when the interface is set to standard, no such error.

This not only happen in worksheet, but also when code is run in command line!

Worksheet below.

Download extended_interface_causes_internal_bug_dec_13_2024.mw

ps. also reported to Maple support

I was rejected because the editor said my equation is too long. My question is: Is there a way to rewrite the equation in a more concise form? Additionally, is there a package in Maple that allows for automatic simplification or collection of terms without using specific commands? Any suggestions for addressing this issue would be appreciated.

Download coment.mw

Major deficiency in Physics[Vectors]; Distinct sets of basis vectors are not recognized!

You can't define vectors in alternative bases like: {\hat{i}',\hat{j}',\hat{k}'} or {\hat{i}_{1},\hat{j}_{2},\hat{k}_{3}}.

This deficiency has been around for a while. I have found other posts regarding this problem.

The deficiency greatly reduces the allowable calculations with Physics[Vector].

Are there any plans to fix this?

Here is my example which shows this deficiency in more detail.

physics_vectors_and_multiple_unit_vectors.mw
 

Download physics_vectors_and_multiple_unit_vectors.mw

Where can I found details about Statistics:-Sample(..., method=envelope).

It would be nice to have a link to a description of the envelope method Sample uses.
For instance does it share some features of the Cuba library for numeric integration? Does it use the same envelope method evalf/Int(..., method=_CubaSuave)) uses?

Thanks in advance.

I'm having trouble solving this system of differential equations. I haven't solved systems of differential equations before but i tried defining the system and then using dsolve, but it couldn't solve all the equations.

Hope you can help.

Download System_Of_Differential_Equations.mw

I'm trying to transform a partial differential equation (PDE) into an ordinary differential equation (ODE) as demonstrated in the paper. However, I find some steps confusing and difficult to follow. The process often feels chaotic, and managing the complexity of the equations is overwhelming. Could you suggest an effective and systematic method to handle such transformations more easily?

Download transform-pde-to-ode-hard_example.mw

I have expected the opposite. Is exp already optimised that hardware floats do not make sense or does the conversion of the argument to hardware floast eats up all the benefit of using hardware floats?

restart;
CodeTools:-Usage( for i from 1 to 100 by 0.1 do exp(i) end do):
CodeTools:-Usage( for i from 1 to 100 by 0.1 do (evalhf@exp)(i) end do):
memory used=1.54MiB, alloc change=0 bytes, cpu time=16.00ms, real time=15.00ms, gc time=0ns
memory used=5.88MiB, alloc change=32.00MiB, cpu time=31.00ms, real time=33.00ms, gc time=0ns

The Advent of Code 2024 Day 11 problem featured labeled rocks which duplicate according to simple rules:

• Rocks labeled 0 turn into 1's
• Rocks with a even number of digits in the label split into two rocks with the first and last half of the digits
• Other rocks have their labels multiplied by 2024

Your puzzle task is to count the number of rocks after 25, then 75 iterations of these rules.

For 25 iterations, a very simple recursive count suffices

count := proc(n, s)
local f, l;
    if n = 0 then return 1;
    elif s=0 then return count(n-1,1);
    end if;
    l := length(s);
    if l mod 2 = 0 then
         f := floor( s/10^(l/2) );
         return count(n-1,f) + count(n-1,s-f*10^(l/2));
    else return count(n-1, 2024*s);
    end if;
end proc:

For the puzzle input, a collection of 8 numbers (the stone labels) the largest with 7 digits, you can call count(25,s) on each s in the input and add them together in about 1 second, but trying that for 75 "hangs" and clearly will not work.

Generally for something like this, the first trick in my puzzle solving toolbox is memoization, so that the results of recursive calls are stored so they don't have to be recomputed.  In Maple, there are two main ways to add memoization to recursive procedures: option cache and option remember.  The remember option is older and uses a simple table that gets populated with (input,return) pairs whenever the procedure returns. The cache option is newer and uses size limited LRU tables that are more memory friendly.  Typically in the Maple math library we use cache tables since they won't fill up your working memory with cached results, and will remove old cached elements in a smart way (the double option option remember, system also limits table growth by allowing entries to be garbage collected, but it is not as flexible as a cache).

So this preable is to say, that the first memoization option I tend to reach for is option cache but best practice for Maple math library development is not always the thing that solves a code puzzle.  In this case, adding option cache to our procedure count also "hangs".  If you are smart, you will immediately try again with option remember, and you'll find that works easily:

count := proc(n, s)
option remember;
    if n = 0 then return 1;
    elif s=0 then return count(n-1,1);
    end if;
    local l := length(s);
    if l mod 2 = 0 then
         local f := floor( s/10^(l/2) );
         return count(n-1,f) + count(n-1,s-f*10^(l/2));
    else return count(n-1, 2024*s);
    end if;
end proc:
ans1 := CodeTools:-Usage( add( map2(count, 25, input) ) ); # 17ms
ans2 := CodeTools:-Usage( add( map2(count, 75, input) ) ); # 825ms

So why does remember work, when cache fails so spectacularly? Well, you can create a function with a cache, call it once create it and the examine the Cache object with op:

> cacheFunc := proc() option cache; args; end proc:
> cacheFunc(1): op( 4, eval(cacheFunc) );

                       Cache(512, 'temporary' = [1 = 1])

You can see here that the default cache is 512 entries which is a pretty good default for general use, but way way too small for this ridiculous puzzle problem.  Fortunately, the cache option allows you to specify a size to create a much bigger cache for a highly recursive procedure like this one.  I tested caches of various sizes (cache always rounds up the next power of 2) and compared to option remember and remember, system.

1. option remember  - 707ms
2. option remember, system - 4.22s
3. option cache  - too slow, killed after hours
4. option cache(1024) - 30.21s
5. option cache(16384) - 5.60s
6. option cache(131072)- 726ms
7. option cache(1048576)- 806ms

So, for this problem, it looks like a 2^17=~100,000 entry cache (#6) gives about the same performance as a remember table, so it's not surpring a larger cache does no better. 

In fact, since results are not removed from an option remember table, you can actually check exactly how many results were cached and see it is just less than 2^17.

> add( map2(count, 75, input) ): numelems( op(4, eval(count) ) );
                                    120076

And the take away is, for this sort of puzzle solving you should try option remember even if option cache is usually a better choice when implementing a more serious mathematical algorithmn that wants to occasionally needs to reuse results.

Is there a way to determine how we can construct a system of equations from this complex PDE? Also, moderator, you mentioned I could create a new question using the branch option, but you deleted my previous question, which led me to delete my earlier post. don't delete this one.

Download PDE-Hard.mw

The output  means that there is no real solution, but clearly, both and  satisfy the original equation: 

So, why does `solve` lose the real solutions without any warning messages? 
Code: 

restart;
eq := 9*(x^2 + 2*cos(x)) = Pi^2 + 9;
RealDomain:-solve(eq, [x]);
                               []

:-solve({eq, x >= 0}, [x]); # as (lhs - rhs)(eq) is an even function 
                               []

this equation will be solve by changing variable but when  i found the function and substitute the ODE is not zero where  is mistake?

Download problem.mw

I'm trying to solve system of ODE (Temperature changing with time) which are going to use the heat capacity obtained from thermophysical package (heat capacity is changing with temperature).

In the support page there is an example in which they were able to integrate the heat capacity from the package. So I wondering if it is possible to include it in an ODE system.

I used their same approach, I tried defining the call to the package as a function but I'm getting an error:

"Error, (in dsolve/numeric/process_input) input system must be an ODE system, found {ThermophysicalData:-CoolProp:-PropsSI(C,P,101325,T,T1(t),"hydrogen"), T1(t), T2(t), T3(t)}"

Attached question.mw

Dear Maple user i want to extract the data from the given graph and store in excel file. where the first column contain the value of lambda in that substitude the values of delta2 ranging from 0.002 to 0.1 (atleast 20 values) and second column  contain Nb =0.1, third column Nb= 0.2 and third column Nb=0.3. Thanks in advance

restart:
h:=z->1-(delta2/2)*(1 + cos(2*(Pi/L1)*(z - d1 - L1))):
K1:=((4/h(z)^4)-(sin(alpha)/F)-h(z)^2+Nb*h(z)^4):
lambda:=Int(K1,z=0..1):
L1:=0.2: F:=10:
d1:=0.2:
alpha:=Pi/6:
plot( [seq(eval(lambda, Nb=j), j in [0.1,0.2,0.3])], delta2=0.02..0.1);

Hello sir, how are you?
Sorry to bother you, I needed your help. I have the script from your textbook "3D Graph Equation of Motorcycle run on Maple Software". It's not working. I'd appreciate it if you could take a look.

with(plots);
implicitplot3d(((49.80*x + 19.44*y + 133.2300 - 19.08*sqrt(x^2 + 8.30*x + 19.8469 + y^2 + 3.24*y) - 66.6150*abs(-3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42) + 0.5625*abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2) = ((((((((((((2 + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42) + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(x^2 + 8.30) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(y^2 + 3.24)) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(x^2 + 8.30) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42) + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(x^2 + 3.24) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(y^2 - 3.18)) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(x^2 + 8.30) - 3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42) + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(x^2 + 8.30) - 3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(x^2 + 3.24)) - 3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(y^2 + 3.24) + 0.42*abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2) and ((((((((((((2 + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42) + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(x^2 + 8.30) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(y^2 + 3.24)) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(x^2 + 8.30) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42) + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(x^2 + 3.24) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(y^2 - 3.18)) + abs(3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2))) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(x^2 + 8.30) - 3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42) + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(x^2 + 8.30) - 3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y - 2)*(x^2 + 3.24)) - 3.9*sqrt((x - 1.7)^2 + (y - 1.35)^2)) + 0.42 + abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2)*(y^2 + 3.24) + 0.42*abs(0.00390625*(x^8 + y^8) - 0.5*x + 2*y + 2) = 0, x = -7 .. 7, y = -4 .. 3, z = -3 .. 3, numpoints = 350000, style = surface, color = "Niagara Azure");