Maple Questions and Posts

How to conver a list of strings to a list of vecto...

Asked by: Felipe_123 45

October 21 2020

2 6

Hi friends,

I have the next list of strings and I want to convert it to a list of vectors.

with(StringTools)

Generate(3, "012");
["000", "001", "002", "010", "011", "012", "020", "021", "022",

"100", "101", "102", "110", "111", "112", "120", "121", "122",

"200", "201", "202", "210", "211", "212", "220", "221", "222"]

Or maybe there's an easiest way to generate all the n-ary vectors of a given lenght.

Thank's in advance.

Evaluate an integral...

Asked by: Zeineb 520

October 21 2020

0 5

Dear all

I need your help to evaluate the following integral using Maple

int(x^x,x=0..1)

Thank you for any help

Convert digits of numbers into Arrays...

Asked by: santamartina 40

October 21 2020

1 4

Dear friends,

I have a given number and I need to convert its digits into an array. For example:

a:= 456; and I need an array [4 5 6].

I know how to obtain such a procedure as a list

map(parse, StringTools:-Explode(convert(a, string)));

but not how to get the result as an array. Could you please help me with the right commands?

Many thanks for the help.

searchtext result position...

Asked by: Anthrazit 750

October 21 2020

0 5

I think the help file regarding searchtext should be more specific regarding the result of the search.

Using searchtext(pattern, string, range), the result will be an integer which indicates the position of the first character from the beginning of the range - not from the beginning of the string.

One of the examples actually shows that.

SearchText("ijklm", "abcdefghijklmnopqrstuvWxy", 5..-5);

I have this issue when I try to name my label to D_m1 I get back "VectorCalculus:-D_m1". I know that after I have the figure, I can manually change it but I have many of them and want to save them automatically, therefore it would be necessary to immediately get the right label.

Are there any suggestions? Maybe a different way, how to write subscript?

Thank you for your help!

why DEtools:-particularsol hangs when dsolve works...

Asked by: nm 8552

October 21 2020

1 1

I solved this linear constant coefficients ode by hand, and wanted to use Maple to check my solution.

dsolve works with no problem. I wanted to check the particular solution on its own, so called DEtools:-particularsol(ode,y(x)); but was surprised it hangs.

Why would DEtools:-particularsol(ode,y(x)); hang, when dsolve gives the complete solution (including particular solutin ofcourse) right away? It is 7th order ODE. But nothing special. Linear and constant coefficients. Since dsolve solves it right away, I expected DEtools:-particularsol to have no problem as well.

restart;
ode:=diff(y(x),x$7)-2*diff(y(x),x$6)+9*diff(y(x),x$5)-16*diff(y(x),x$4)+24*diff(y(x),x$3)-32*diff(y(x),x$2)+16*diff(y(x),x)=exp(2*x)+x*sin(x)+x^2;

dsolve(ode);  #no problem.

#this hangs. Why?
DEtools:-particularsol(ode,y(x));

>	interface(version);

>	Physics:-Version()

$`The "Physics Updates" version in the MapleCloud is 851. The version installed in this computer is 847 created 2020, October 17, 17:3 hours Pacific Time, found in the directory C:\Users\me\maple\toolbox\2020\Physics Updates\lib\`$

>	restart; ode:=diff(y(x),x$7)-2diff(y(x),x$6)+9diff(y(x),x$5)-16diff(y(x),x$4)+24diff(y(x),x$3)-32diff(y(x),x$2)+16diff(y(x),x)=exp(2x)+xsin(x)+x^2;

diff(diff(diff(diff(diff(diff(diff(y(x), x), x), x), x), x), x), x)-2*(diff(diff(diff(diff(diff(diff(y(x), x), x), x), x), x), x))+9*(diff(diff(diff(diff(diff(y(x), x), x), x), x), x))-16*(diff(diff(diff(diff(y(x), x), x), x), x))+24*(diff(diff(diff(y(x), x), x), x))-32*(diff(diff(y(x), x), x))+16*(diff(y(x), x)) = exp(2*x)+x*sin(x)+x^2

>	#this works sol:=dsolve(ode)

>	#this hangs. Why? DEtools:-particularsol(ode,y(x));

>

Download particular_sol.mw

Sorting lists with table entries...

Asked by: Anthrazit 750

October 20 2020

1 2

How can I sort lists which have tables as entries, and decide upon which entry the sort should be based.

In the attached example each entry in the list has the values material, name and type.

How would I sort the list based on the material name for example?
Are there any better methods to define and store the data?

Listsort.mw

A question about the Jordan form of a matrix...

Asked by: mmcdara 6149

October 20 2020

1 5

Hi,
I'm surprised by the error Maple returns when asked to find the JordanBlockMatrix or the JordanForm of a matrix whose some entries are floating points.
I don't understand why these operations are valid only with entries of the form a+I*b with a and b are algebraic numbers or with transcendent numbers such as Pi.
Is this a theoretical result or a technical limitation of Maple?
Jordan.mw

Download Jordan.mw

Computing a tetrad in canonical form - automatically...

by: ecterrab 13431 Product: Maple

October 20 2020

5 0

In a recent question in Mapleprimes, a spacetime (metric) solution to Einstein's equations, from chapter 27 of the book of Exact Solutions to Einstein's equations [1] was discussed. One of the issues was about computing a tetrad for that solution [27, 37, 1] such that the corresponding Weyl scalars are in canonical form. This post illustrates how to do that, with precisely that spacetime metric solution, in two different ways: 1) automatically, all in one go, and 2) step-by-step. The step-by-step computation is useful to verify results and also to compute different forms of the tetrads or Weyl scalars. The computation below is performed using the latest version of the Maplesoft Physics Updates.

>

(1)

The starting point is this image of page 421 of the book of Exact Solutions to Einstein's equations, formulas (27.37)

Load the solution [27, 37, 1] from Maple's database of solutions to Einstein's equations

>

$`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (z, zb, r, u)}$

Physics:-g_[mu, nu] = Matrix(%id = 18446744078276690638)

(2)

>

(3)

The assumptions on the metric's parameters are

>

The line element is as shown in the second line of the image above

>

2*r^2*Physics:-d_(z)*Physics:-d_(zb)/P(z, zb, u)^2-2*Physics:-d_(r)*Physics:-d_(u)-2*H(X)*Physics:-d_(u)^2

(4)

Load Tetrads

>

((`Defined as spacetime tensors representing the NP null vectors of the tetrad formalism `*`see <a href='http://www.maplesoft.com/support/help/search.aspx?term=Physics,tetrads`*`,' target='_new'>?Physics,tetrads`*`,</a> `*l[mu]*`, `)*n[mu]*`, `*m[mu]*`, `)*conjugate(m[mu])

(5)

The Petrov type of this spacetime solution is

>

(6)

The null tetrad computed by the Maple system using a general algorithms is

>

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078178770326)

(7)

According to the help page TransformTetrad , the canonical form of the Weyl scalars for each different Petrov type is

So for type II, when the tetrad is in canonical form, we expect only and different from 0. For the tetrad computed automatically, however, the scalars are

>

(8)

The question is, how to bring the tetrad (equation (7)) into canonical form. The plan for that is outlined in Chapter 7, by Chandrasekhar, page 388, of the book "General Relativity, an Einstein centenary survey", edited by S.W. Hawking and W.Israel. In brief, for Petrov type II, use a transformation of to make , then a transformation of making , finally use a transformation of making . For an explanation of these transformations see the help page for TransformTetrad . This plan, however, is applicable if and only if the starting tetrad results in , which we see in (8) it is not the case, so we need, in addition, before applying this plan, to perform a transformation of making

In what follows, the transformations mentioned are first performed automatically, in one go, letting the computer deduce each intermediate transformation, by passing to TransformTetrad the optional argument canonicalform. Then, the same result is obtained by transforming the starting tetrad one step at at time, arriving at the same Weyl scalars. That illustrates well both how to get the result exploiting advanced functionality but also how to verify the result performing each step, and also how to get any desired different form of the Weyl scalars.

Although it is possible to perform both computations, automatically and step-by-step, departing from the tetrad (7), that tetrad and the corresponding Weyl scalars (8) have radicals, making the readability of the formulas at each step less clear. Both computations, can be presented in more readable form without radicals departing from the tetrad shown in the book, that is

>	$e_[a, mu] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (1, 4) = -1, (2, 1) = 0, (2, 2) = r/P(z, zb, u), (2, 3) = 0, (2, 4) = 0, (3, 1) = r/P(z, zb, u), (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = -H(X)}))$

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078621688766)

(9)

>

(10)

The corresponding Weyl scalars free of radicals are

>

(11)

So set this tetrad as the starting point

>

(12)

All the transformations performed automatically, in one go

To arrive in one go, automatically, to a tetrad whose Weyl scalars are in canonical form as in (31), use the optional argument canonicalform:

>

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

(13)

Note the length of

>

(14)

That length corresponds to several pages long. That happens frequently, you get Weyl scalars with a minimum of residual invariance, at the cost of a more complicated tetrad.

The transformations step-by-step leading to the same canonical form of the Weyl scalars

Step 0

As mentioned above, to apply the plan outlined by Chandrasekhar, the starting point needs to be a tetrad with , not the case of (9), so in this step 0 we use a transformation of making . This transformation introduces a complex parameter E and to get any value of E suffices. We use :

>

Matrix(%id = 18446744078634914990)

(15)

>

Matrix(%id = 18446744078634940646)

(16)

Indeed, for this tetrad, :

>

(17)

Step 1

Next is a transformation of to make , that in the case of Petrov type II also implies on .According to the the help page TransformTetrad , this transformation introduces a parameter B that, according to the plan outlined by Chandrasekhar in Chapter 7 page 388, is one of the two identical roots (out of the four roots) of the principalpolynomial. To see the principal polynomial, or, directly, its roots you can use the PetrovType command:

>

(18)

The first two are the same and equal to -1

>

(19)

So the transformed tetrad is

>

Matrix(%id = 18446744078641721462)

(20)

Check this result and the corresponding Weyl scalars to verify that we now have and

>

(21)

>

(22)

Step 2

Next is a transformation of that makes . This transformation introduces a parameter E, that according to Chandrasekhar's plan can be taken equal to one of the roots of Weyl scalar that corresponds to the transformed tetrad. So we need to proceed in three steps:

a.	transform the tetrad introducing a parameter E in the tetrad's components

b.	compute the Weyl scalars for that transformed tetrad

c.	take and solve for E

d.	apply the resulting value of E to the transformed tetrad obtained in step a.

a.Transform the tetrad and for simplicity take E real

>

Matrix(%id = 18446744078624751238)

(23)

>

(24)

b. Compute for this tetrad

>

(25)

c. Solve discarding the case which implies on no transformation

>

$simplify(solve({rhs(psi__4 = (r^2*P(z, zb, u)*(E-1)^2*(diff(diff(H(X), r), r))-2*r*P(z, zb, u)^2*(E-1)*(diff(diff(H(X), r), z))+P(z, zb, u)^3*(diff(diff(H(X), z), z))-2*P(z, zb, u)^2*(E-1)^2*(diff(diff(P(z, zb, u), z), zb))+2*r*P(z, zb, u)*(E-1)*(diff(diff(P(z, zb, u), u), z))-2*r*P(z, zb, u)*(E-1)^2*(diff(H(X), r))+4*P(z, zb, u)^2*(E+(1/2)*(diff(P(z, zb, u), z))-1)*(diff(H(X), z))+2*((P(z, zb, u)*(E-1)*(diff(P(z, zb, u), zb))-(diff(P(z, zb, u), u))*r)*(diff(P(z, zb, u), z))+H(X)*P(z, zb, u)*(E-1))*(E-1))/(r^2*P(z, zb, u))) = 0, E <> 0}, {E}, explicit)[1])$

${E = ((diff(diff(H(X), r), r))*r^2*P(z, zb, u)+(diff(diff(H(X), r), z))*P(z, zb, u)^2*r-2*(diff(H(X), r))*P(z, zb, u)*r-(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u)+(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r-2*P(z, zb, u)^2*(diff(H(X), z))-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)+2*H(X)*P(z, zb, u)+(-P(z, zb, u)^4*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X))*(diff(diff(H(X), z), z))+P(z, zb, u)^4*(diff(diff(H(X), r), z))^2*r^2+(-2*r^2*(diff(diff(P(z, zb, u), u), z))*P(z, zb, u)^3+2*r^2*P(z, zb, u)^2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))-4*r*(diff(H(X), z))*P(z, zb, u)^4)*(diff(diff(H(X), r), z))-2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(H(X), r), r))*r^2+P(z, zb, u)^2*(diff(diff(P(z, zb, u), u), z))^2*r^2+(-2*r^2*P(z, zb, u)*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))+4*r*(diff(H(X), z))*P(z, zb, u)^3)*(diff(diff(P(z, zb, u), u), z))+4*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(P(z, zb, u), z), zb))+4*(diff(H(X), z))^2*P(z, zb, u)^4+4*P(z, zb, u)^2*(diff(P(z, zb, u), z))*((diff(H(X), r))*P(z, zb, u)*r-(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)-(diff(P(z, zb, u), u))*r-H(X)*P(z, zb, u))*(diff(H(X), z))+(diff(P(z, zb, u), u))^2*(diff(P(z, zb, u), z))^2*r^2)^(1/2))/(P(z, zb, u)*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X)))}$

(26)

d. Apply this result to the tetrad (23). In doing so, do not display the result, just measure its length (corresponds to two+ pages)

>

$T__3 := simplify(eval(T__2, {E = ((diff(diff(H(X), r), r))*r^2*P(z, zb, u)+(diff(diff(H(X), r), z))*P(z, zb, u)^2*r-2*(diff(H(X), r))*P(z, zb, u)*r-(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u)+(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r-2*P(z, zb, u)^2*(diff(H(X), z))-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)+2*H(X)*P(z, zb, u)+(-P(z, zb, u)^4*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X))*(diff(diff(H(X), z), z))+P(z, zb, u)^4*(diff(diff(H(X), r), z))^2*r^2+(-2*r^2*(diff(diff(P(z, zb, u), u), z))*P(z, zb, u)^3+2*r^2*P(z, zb, u)^2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))-4*r*(diff(H(X), z))*P(z, zb, u)^4)*(diff(diff(H(X), r), z))-2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(H(X), r), r))*r^2+P(z, zb, u)^2*(diff(diff(P(z, zb, u), u), z))^2*r^2+(-2*r^2*P(z, zb, u)*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))+4*r*(diff(H(X), z))*P(z, zb, u)^3)*(diff(diff(P(z, zb, u), u), z))+4*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(P(z, zb, u), z), zb))+4*(diff(H(X), z))^2*P(z, zb, u)^4+4*P(z, zb, u)^2*(diff(P(z, zb, u), z))*((diff(H(X), r))*P(z, zb, u)*r-(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)-(diff(P(z, zb, u), u))*r-H(X)*P(z, zb, u))*(diff(H(X), z))+(diff(P(z, zb, u), u))^2*(diff(P(z, zb, u), z))^2*r^2)^(1/2))/(P(z, zb, u)*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X)))}[1]))$

>

(27)

Check the scalars, we expect

>

(28)

Step 3

Use a transformation of making . Such a transformation changes , where we need to take A*exp(-I*Omega) = 1/`Ψ__3` , and without loss of generality we can take

Check first the value of in the last tetrad computed

>

(29)

So, the transformed tetrad to which corresponds Weyl scalars in canonical form, with and , is

>

(30)

>

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

(31)

These are the same scalars computed in one go in (13)

>

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

(32)

>

Download The_metric_[27_37_1]_in_canonical_form.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

How to create a function f(x) where f(x)=0 if x<0,...

Asked by: chenaiier 25

October 19 2020

1 6

As the title states, I want to have an equation f(x), and f(x) = 0 if x < 0, f(x) = x if x >= 0. How could I accomplish this?

I'm actually trying to generating a differential equation something like y'(x) + k*h(y) = sin(x) where h(y) is what I described above. Is there any convenient way to do this?

How to list all (finitely many) matrices with cert...

Asked by: tayyild 20

October 19 2020

1 1

I want to construct a list/set of square symmetric matrices (or Toeplitz etc.) matrices with entries from a certain finite field, GF(2) or GF(5).

Thank you in adavnce.

A small bug in pdsolve...

Asked by: Rouben Rostamian 7711

October 19 2020

2 2

Here is a division-by-zero bug in a solution produced by pdsolve. Admittedly, this sort of problem can be difficult to avoid in a CAS, but here it is, in case there is a chance to get it fixed somehow.

>

restart;

>	kernelopts(version);

>	pde := diff(u(x,y,t),t,t) = diff(u(x,y,t),x,x) + diff(u(x,y,t),y,y);

>	bc := u(x,0,t)=0, u(x,1,t)=0, u(0,y,t)=0, u(1,y,t)=0;

>	ic := u(x,y,0) = xysin(Pix)sin(Pi*y), D[3](u)(x,y,0)=0;

>	pdsol := pdsolve({pde, bc, ic});

$"pdsol:=u(x,y,t)=(∑) (∑)-(2 sin(n Pi x) sin(n1 Pi y) cos(Pi sqrt(n^2+n1^2) t) ({[[Pi^2,n=1],[-(8 n ((-1)^n+1))/((n-1)^2 (n+1)^2),1<n]]) n1 ((-1)^n1+1))/(Pi^4 (-1+n1)^2 (n1+1)^2)"$

>	eval(pdsol, infinity=4); value(%);

$"u(x,y,t)=(∑) (∑)-(2 sin(n Pi x) sin(n1 Pi y) cos(Pi sqrt(n^2+n1^2) t) ({[[Pi^2,n=1],[-(8 n ((-1)^n+1))/((n-1)^2 (n+1)^2),1<n]]) n1 ((-1)^n1+1))/(Pi^4 (-1+n1)^2 (n1+1)^2)"$

Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

Download pdsolve-bug.mw

How Does Maple Actually Solve an Identity, Say, fo...

Asked by: mapleatha 160

October 19 2020

0 0

Since I am a mathematician, I am wondering how Maple goes about solving an identity for 3 functions.
Let's say we have af1(t)+bf_2(t)+cf_3(t) = 0 for all t. How does maple actually find a triplet a,b,c that works for all real t?
It does with solve(identity( ),[a,b,c]). But what is the theory behind it?
We know, of course, a priori, that such a triplet exists.

Thank you!

mapleatha

why ArrayTools:-AllNonZero do not seem to check th...

Asked by: nm 8552

October 18 2020

1 2

I was saying to myself today, that one nice thing in Maple, is that it has good type system so one can check type of arguments aginst wrong types being used in the call.

So I was surprised when I called ArrayTools:-AllNonZero using list as argument, where this function is supposed to accept only Matrix,Vector or Array. And it worked. But it gave wrong answer at the same time.

Am I doing something wrong here? Should not have this call failed to go through?

restart;
interface(warnlevel=4);
kernelopts('assertlevel'=2):

A:=[0,0,0];
ArrayTools:-AllNonZero(A)

Maple replied

true

But

A:=Array([0,0,0]);
ArrayTools:-AllNonZero(A)

Now Maple gives the expected result false

The question is why Maple did not detect the wrong type? Should it have detected wrong type?

And why it even gave wrong answer (but this is not as important, since the call should not be allowed in first place).

btw, this applied to all the other API's listed

why the way solution is written affects odetest re...

Asked by: nm 8552

October 18 2020

2 1

I found a new strange thing with odetest that I have not seen before.

Would you say that these two solutions are the same or not?

They look the same to me. One just have all the terms on one side, that is all.

Why would Maple odetest verify the first one but not the second? I did not know that all terms have to be on one side before. Is this documented somewhere?

Please see worksheet below.

>

restart;

>	interface(version);

>	Physics:-Version();

>	ode:=diff(y(x), x) - 2y(x) = 2sqrt(y(x)); ic:=[y(0)=1]; maple_sol:=dsolve([ode,op(ic)],implicit); odetest(maple_sol,[ode,op(ic)]) assuming x>0

>	#now move some terms to one side, and odetest no longer verifies it my_sol:=y(x)^(1/2) = -1+2*exp(x); odetest(my_sol,[ode,op(ic)]) assuming x>0

>	#the solution is to put all terms on one side my_sol:=y(x)^(1/2) = -1+2*exp(x); odetest(rhs(my_sol)-lhs(my_sol)=0,[ode,op(ic)]) assuming x>0

>

Download odetest_issue_oct_18_2020.mw

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