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how to solve this integral? ...

Asked by: Aung 65
integration sum
November 16 2024
0  16
> 

restart:
> 

 

> 

inte_eq := int(-(sum(B[i]*beta[i]* exp(-beta[i] * (t-tau)),i=1..n)),tau=0..t);

int(-(sum(B[i]*beta[i]*exp(-beta[i]*(t-tau)), i = 1 .. n)), tau = 0 .. t)

 (1)

how to solve this ?

Download creep_integral_sol.mw

"How do I solve a parametric inequation in Maple...

Asked by: Blanc 20
November 12 2024
2  8

 I don't know how to solve with maple the following inequality with respect to the real variable x, n being an integer
Please tell me the procedure.

(f(x,n)>0 and (f(x,n)<0

f := (x, n) -> 2^n/(1 + n*2^n*x^2) - 2*x^2*(2^n)^2*n/(1 + n*2^n*x^2)^2

Infinity limit of expression with summations?...

Asked by: MaPal93 175
sum assumptions limit
November 10 2024
1  15

I want to compute the limit for a parameter (gamma) to infinity of an expression which includes summations over indexes. Below are two approaches. I believe that both should produce the same output, however:

  1. Approach A generates a signum, but I don't understand where this is coming from
  2. Approach B works smoothly once I fix the final value n of the index to some number but runs very slow if I leave n undefined (I had to interrupt the execution)

Maybe approach B is not wrong, and I just need to wait longer...

Any thoughts? 

Approach A: Compute infinity limit while keeping X and lambda undefined (since they are independent of gamma, see below), then define X and lambda
 

problem: what generates signum?

> 

restart;
> 

local gamma;

gamma

 (1)
> 

assume(0 < gamma, 0 < n, 0 < w, 0 < Delta, 0 < vo[i], -1 < rho and rho < 1, 0 < sigma__v, 0 < sigma__d, 0 < sigma__dr, delta[i]::real, delta[j]::real, delta[k]::real, delta__r::real, X[i]::real, X[j]::real, X__r::real);
interface(showassumed=0);

1

 (2)
> 

W:=sum(X[i]*(-lambda*X[i]-lambda*delta[i]-vo[i]),i=1..n)+X__r*(-lambda__r*X__r-lambda__r*delta__r-w*sum(vo[i],i=1..n))+sum((X[i]+w*X__r)*(vo[i]+(1/(w*n))*(Delta)),i=1..n)-gamma/2*sigma__v^2*(1-(1+rho*(n-1))/(n))*sum((X[i]+w*X__r)^2,i=1..n)-gamma/2*sigma__v^2*(rho-(1+rho*(n-1))/(n))*sum(sum((X[i]+w*X__r)*(X[j]+w*X__r),j=1..i-1)+sum((X[i]+w*X__r)*(X[j]+w*X__r),j=i+1..n),i=1..n);
 

sum(X[i]*(-lambda*X[i]-lambda*delta[i]-vo[i]), i = 1 .. n)+X__r*(-lambda__r*X__r-lambda__r*delta__r-w*(sum(vo[i], i = 1 .. n)))+Delta*X__r+sum(Delta*X[i]/(w*n)+vo[i]*X__r*w+X[i]*vo[i], i = 1 .. n)-(1/2)*gamma*sigma__v^2*(1-(1+rho*(n-1))/n)*(n*X__r^2*w^2+sum(2*X__r*w*X[i]+X[i]^2, i = 1 .. n))-(1/2)*gamma*sigma__v^2*(rho-(1+rho*(n-1))/n)*(n^2*X__r^2*w^2-n*X__r^2*w^2+sum(sum(X__r*w*X[j]+X[i]*X[j], j = i+1 .. n)+sum(X__r*w*X[j]+X[i]*X[j], j = 1 .. i-1)+X__r*w*X[i]*n-X__r*w*X[i], i = 1 .. n))

 (3)
> 

limit(W,gamma=infinity);

-signum(-(sum(sum(X[j]*(X__r*w+X[i]), j = i+1 .. n)+sum(X[j]*(X__r*w+X[i]), j = 1 .. i-1)+X__r*w*(n-1)*X[i], i = 1 .. n))+(n-1)*(sum(X[i]*(2*X__r*w+X[i]), i = 1 .. n)))*infinity

 (4)
> 

#convert(%, piecewise);

Approach B: Define X and lambda first, then compute infinity limit

problem: why last execution so slow?

> 

restart;
> 

local gamma;

gamma

 (5)
> 

assume(0 < gamma, 0 < n, 0 < w, 0 < Delta, 0 < vo[i], -1 < rho and rho < 1, 0 < sigma__v, 0 < sigma__d, 0 < sigma__dr, delta[i]::real, delta[j]::real, delta[k]::real, delta__r::real, X[i]::real, X[j]::real, X__r::real);
interface(showassumed=0);

1

 (6)
> 

X[i] := (1/(2*n*w))*sqrt((4*n-3)/(1+rho*(n-1)))*(sigma__d/sigma__v)*Delta-(1/(2*n))*delta[i]-(1/(2*n))*(sum(delta[j],j=1..i-1)+sum(delta[j],j=i+1..n));
X[j] := (1/(2*n*w))*sqrt((4*n-3)/(1+rho*(n-1)))*(sigma__d/sigma__v)*Delta-(1/(2*n))*delta[j]-(1/(2*n))*(sum(delta[k],k=1..j-1)+sum(delta[k],k=j+1..n));
X__r := (1/(2*w*sqrt(n*(1+rho*(n-1)))))*(sigma__dr/sigma__v)*Delta-(1/2)*delta__r;

lambda := sqrt((1+rho*(n-1))/(4*n-3))*(sigma__v/sigma__d);
lambda__r := w*sqrt(n*(1+rho*(n-1)))*(sigma__v/sigma__dr);
 

(1/2)*((4*n-3)/(1+rho*(n-1)))^(1/2)*sigma__d*Delta/(n*w*sigma__v)-(1/2)*delta[i]/n-(1/2)*(sum(delta[j], j = 1 .. i-1)+sum(delta[j], j = i+1 .. n))/n

  

(1/2)*((4*n-3)/(1+rho*(n-1)))^(1/2)*sigma__d*Delta/(n*w*sigma__v)-(1/2)*delta[j]/n-(1/2)*(sum(delta[k], k = 1 .. j-1)+sum(delta[k], k = j+1 .. n))/n

  

(1/2)*sigma__dr*Delta/(w*(n*(1+rho*(n-1)))^(1/2)*sigma__v)-(1/2)*delta__r

  

((1+rho*(n-1))/(4*n-3))^(1/2)*sigma__v/sigma__d

  

w*(n*(1+rho*(n-1)))^(1/2)*sigma__v/sigma__dr

 (7)
> 

W:=sum(X[i]*(-lambda*X[i]-lambda*delta[i]-vo[i]),i=1..n)+X__r*(-lambda__r*X__r-lambda__r*delta__r-w*sum(vo[i],i=1..n))+sum((X[i]+w*X__r)*(vo[i]+(1/(w*n))*(Delta)),i=1..n)-gamma/2*sigma__v^2*(1-(1+rho*(n-1))/(n))*sum((X[i]+w*X__r)^2,i=1..n)-gamma/2*sigma__v^2*(rho-(1+rho*(n-1))/(n))*sum(sum((X[i]+w*X__r)*(X[j]+w*X__r),j=1..i-1)+sum((X[i]+w*X__r)*(X[j]+w*X__r),j=i+1..n),i=1..n):
 

W limit for n=3

> 

Winf := collect(expand(simplify(limit(eval(W,n=3),gamma=infinity))),[Delta,Delta^2]):

quad_term := collect(expand(simplify(coeff(Winf,Delta^2))),[sigma__d,sigma__dr]);
linear_term := expand(simplify(coeff(Winf,Delta)));
coeff_term := collect(expand(simplify(eval(Winf,Delta=0))),delta__r^2);

(-(1/4)/(sigma__v*(1+2*rho)^(1/2)*w^2)+(1/2)*(1/(1+2*rho))^(1/2)/(w^2*sigma__v))*sigma__d+(1/12)*3^(1/2)*sigma__dr/((1+2*rho)^(1/2)*sigma__v*w)

  

-(1/2)*delta__r-(1/6)*delta[1]/w-(1/6)*delta[2]/w-(1/6)*delta[3]/w

  

((1/4)*3^(1/2)*sigma__v*w/((1+2*rho)^(1/2)*sigma__dr)+(1/2)*3^(1/2)*sigma__v*w*rho/((1+2*rho)^(1/2)*sigma__dr))*delta__r^2+(1/36)*sigma__v*delta[1]^2/((1+2*rho)^(1/2)*sigma__d)+(1/36)*sigma__v*delta[2]^2/((1+2*rho)^(1/2)*sigma__d)+(1/36)*sigma__v*delta[3]^2/((1+2*rho)^(1/2)*sigma__d)+(1/18)*sigma__v*delta[1]^2*rho/((1+2*rho)^(1/2)*sigma__d)+(1/18)*sigma__v*delta[2]^2*rho/((1+2*rho)^(1/2)*sigma__d)+(1/18)*sigma__v*delta[3]^2*rho/((1+2*rho)^(1/2)*sigma__d)+(1/18)*sigma__v*delta[2]*delta[1]/((1+2*rho)^(1/2)*sigma__d)+(1/18)*sigma__v*delta[1]*delta[3]/((1+2*rho)^(1/2)*sigma__d)+(1/18)*sigma__v*delta[2]*delta[3]/((1+2*rho)^(1/2)*sigma__d)+(1/9)*sigma__v*delta[2]*delta[1]*rho/((1+2*rho)^(1/2)*sigma__d)+(1/9)*sigma__v*delta[1]*delta[3]*rho/((1+2*rho)^(1/2)*sigma__d)+(1/9)*sigma__v*delta[2]*delta[3]*rho/((1+2*rho)^(1/2)*sigma__d)

 (8)

W limit for generic n

> 

Winf := collect(expand(simplify(limit(W,gamma=infinity))),[Delta,Delta^2]):

quad_term := collect(expand(simplify(coeff(Winf,Delta^2))),[sigma__d,sigma__dr]);
linear_term := expand(simplify(coeff(Winf,Delta)));
coeff_term := collect(expand(simplify(eval(Winf,Delta=0))),delta__r^2);
 

NULL

Download Infinity_limit_and_summations.mw

How to diff() on summations and double summations?...

Asked by: MaPal93 175
sum differentiation diff
November 06 2024
2  14
I am trying to compute partial derivatives of some complicated expression which include summations. First, I noticed that sum behaves differently if I use 1D vs. 2D math. Why?

Questions:
  1. Partial derivative of a summation: why is it not just 2*X[i]?
  2. Partial derivative of a double summation: how to define the nested structure of a double summation with j<>i?
  3. System of n+1 equations: how to define and solve for it?

For 3., each i equation is the partial derivative of my complicated expression with summations with respect to X[i], where i ranges from 1 to n. The last equation is the partial derivative with respect to X_r (a fixed variable).

Thanks.

> 

restart

> 

A := sum(X[i]^2, i = 1 .. n); eq[1] := diff(A, X[i]) = 0

sum(X[i]^2, i = 1 .. n)

  

sum(2*X[i], i = 1 .. n) = 0

 (1)

> 

B__wrong := sum(sum((X__r*w+X[i])*(X__r*w+X[j]), j = 1 .. n), i = 1 .. n); B__correct := 'sum(sum((X__r*w+X[i])*(X__r*w+X[j]), j = 1 .. n), i = 1 .. n)'

n^2*X__r^2*w^2+sum(sum(X__r*w*X[j]+X[i]*X[j], j = 1 .. n)+n*X__r*w*X[i], i = 1 .. n)

  

sum(sum((X__r*w+X[i])*(X__r*w+X[j]), j = 1 .. n), i = 1 .. n)

 (2)

> 

eqs := seq(eq[i], i = 1 .. n); vars := seq(X[i], i = 1 .. n)

Error, range bounds in seq must be numeric or character

  

Error, range bounds in seq must be numeric or character

  
 

NULL

Download equations_with_summations.mw

How do i define a tensor with an equation of three...

Asked by: Steve255 15
tensor differentialgeometry physics
October 30 2024
3  4

I am studying linear operators that have vanishing Nijenhuis Torsion, that is a (1-1) tensor L whose corresponding (1-2) tensor N given by a tensor equation of L (see attached) is identically zero.

I am new to maple, i have used it to plot vector fields and solve systems of equations in the past but i am unfamiliar with the DifferentialGeometry and Physics packages.

Attached is my best effort at solving this problem directly for a simple 2d case without the use of any packages and i am wondering if it is possible to do it all in one line without having to define tensor components one by one.

nijenhuis_torsion.mw

Singular vectors of a 2x2 matrix ...

Asked by: rzarouf 45
linearalgebra eigenvectors singularvalues
October 24 2024
3  9

Hi, I have a very concrete task, which is to find the singular vector associated with the largest singular matrix of the matrix M below. Here, a is an arbitrary complex number, l is of magnitude stritcly less than 1, and k>0. The command 'SingularValues' returns the 2 singular values of M. But when I search for the corresponding singular vectors, there is a bug. Those vectors are the eigenvectors of M*M. When I use the command Eigenvectors, the second argument returned is the 2 by 2 null matrix. And when I use the command eigenvects I have an error message (see below). Anyone could help me to fix this issue please?

Many thanks in advance. 

> 

with(LinearAlgebra); assume(k > 0); assume(abs(l) < 1); M := Matrix(2, 2, {(1, 1) = l-(1-l*conjugate(l))*k, (1, 2) = -a*(1-l*conjugate(l))*k, (2, 1) = 0, (2, 2) = l+(1-l*conjugate(l))*k})

Matrix(%id = 36893488151898158132)

 (1)
> 

simplify(SingularValues(M)[1])

(1/2)*(2*(-abs(l)^2+1)*k*(k^2*(abs(l)-1)^2*(abs(l)+1)^2*abs(a)^4+(4*k^2*abs(l)^4+(-8*k^2+4)*abs(l)^2+4*k^2)*abs(a)^2+4*l^2+8*abs(l)^2+4*conjugate(l)^2)^(1/2)+2*k^2*(abs(a)^2+2)*abs(l)^4+4*(-abs(a)^2*k^2-2*k^2+1)*abs(l)^2+2*k^2*(abs(a)^2+2))^(1/2)

 (2)
> 

MatrixMatrixMultiply(HermitianTranspose(M), M)

Matrix(%id = 36893488151996380692)

 (3)
> 

Eigenvectors(MatrixMatrixMultiply(HermitianTranspose(M), M))

Vector[column](%id = 36893488151975984532), Matrix(%id = 36893488151975984652)

 (4)
> 

with(linalg)
> 

eigenvectors(Matrix(2, 2, {(1, 1) = (l-(1-l*conjugate(l))*k)*conjugate(l-(1-l*conjugate(l))*k), (1, 2) = -conjugate(l-(1-l*conjugate(l))*k)*a*(1-l*conjugate(l))*k, (2, 1) = -k*conjugate(a*(1-l*conjugate(l)))*(l-(1-l*conjugate(l))*k), (2, 2) = a*(1-l*conjugate(l))*k^2*conjugate(a*(1-l*conjugate(l)))+(l+(1-l*conjugate(l))*k)*conjugate(l+(1-l*conjugate(l))*k)}))

Error, (in linalg:-eigenvectors) eigenvects only works for a matrix of rationals, rational functions, algebraic numbers, or algebraic functions at present

  
 

``

Download singular_vect_2_by_2_matrix.mwsingular_vect_2_by_2_matrix.mw

How can I make maple compute faster?...

Asked by: GFY 65
performance incomplete-question
October 21 2024
0  2

Recently a new configuration of a computer, with a better cpu - AMD Ryzen 7 8700G w/ Radeon 780M Graphics, but maple computation is particularly slow, cpu call special less computation is particularly slow, is there any way to improve?

what does this result from Maple mean?...

Asked by: jrive 210
rootof summation
October 08 2024
0  2

I do not understandwhat this result from Maple means.

I don't know what it means when a result has "Root of", and in this case, there is also a big summation sign for which I can't tell if the variables underneath it are part of the summartion, or what.  I'd appreciate any guidance to make sense of what this means..

restart

with(inttrans)

assump := R1::real, R2::real, L1::real, C1::real, C2::real, L2::real, omega::real, M::real, R1 > 0, R2 > 0, L1 > 0, C1 > 0, L2 > 0, C2 > 0, omega > 0, M > 0

R1::real, R2::real, L1::real, C1::real, C2::real, L2::real, omega::real, M::real, 0 < R1, 0 < R2, 0 < L1, 0 < C1, 0 < L2, 0 < C2, 0 < omega, 0 < M

 (1)

expr := Vin/(s*(R1+s*L1-1/(s*C1)+(omega*M)^2/(R2+s*L2-1/(s*C2))))

Vin/(s*(R1+s*L1-1/(s*C1)+omega^2*M^2/(R2+s*L2-1/(s*C2))))

 (2)

invlaplace(expr, s, t)

Vin*C1*(sum(exp(_alpha*t)*(-1+C2*_alpha*(L2*_alpha+R2))/(4*C1*C2*L1*L2*_alpha^3+2*C1*C2*M^2*_alpha*omega^2+3*C1*C2*L1*R2*_alpha^2+3*C1*C2*L2*R1*_alpha^2+2*C1*C2*R1*R2*_alpha-2*C1*L1*_alpha-2*C2*L2*_alpha-C1*R1-C2*R2), _alpha = RootOf(1+C1*C2*L1*L2*_Z^4+(C1*C2*L1*R2+C1*C2*L2*R1)*_Z^3+(C1*C2*M^2*omega^2+C1*C2*R1*R2-C1*L1-C2*L2)*_Z^2+(-C1*R1-C2*R2)*_Z)))

 (3)

NULL

Download for_help2.mw

How do you solve an equation for two unknowns coup...

Asked by: GFY 65
ode syntax dsolve
September 27 2024
0  5

How to use maple to compute the solution of two coupling equations that both have higher order derivatives? I used dsolve and couldn't solve it.

question928.mw

How do you draw an image with proc?...

Asked by: GFY 65
fsolve plotting
September 24 2024
2  6

The result of the function was computed successfully using the proc function, but how to plot the computed image? Using the seq command but the step size can not get the desired, using the for loop is too cumbersome, what should be done?fuxian.mwfuxian.mw

Typo in ?InvertedPendulum...

Asked by: Thomas Dean 327
2dmath documentation
September 21 2024
0  1

Executing the code in ?InvertedPendulum produces an error when

sysLin := Linearize(convert(sysEqs, list), [u(t)], [x(t), theta(t)], lin_point)

is executed. The problem is the line where EQ4 is assigned a value.

Something is wrong with the line where EQ4 is assigned a value. It is not executed.
It is like the mode changes to text for that line????

Why is it that the two results of the higher order...

Asked by: GFY 65
ode homework numeric
September 20 2024
0  2

I reduced the order of two nonlinear equations and solved them numerically in this way. Why are the two results obtained the same?
directdsolve919.mw

Is there a Maple command that can match powers acr...

Asked by: Hullzie16 343
solve coefficients
September 17 2024
1  5

I have a differential equation which I am looking for a series solution in inverse powers of r. I am doing this by matching inverse powers between the RHS and LHS of the differential equation and then finding the coefficient in the series solution that would solve this. I have written a little procedure which returns the coefficients, I am just curious if there is a command that will do this for me or if i can improve on this technique? I tried the Solve command in PDETools but it just kept evaluating and did not return an answer. 

Thanks in advance. 

 

EDIT: More detail provided in the first reply. 

coefficient_question.mw

Centering plots in an array plot...

Asked by: Scot Gould 1039
pdf export arrayplot
September 15 2024
1  1

I'm sure this has been asked and answered, but I can't find the correct MaplePrimes page on it. How does one generate an array plot where the plots are centered within each box and not left justified? I need to make a PDF out of the document with an array plot of 1 column and 2 rows. (The following is just an example.) 

Why can't a variable in a procedure be called?...

Asked by: GFY 65
substitution global local
September 15 2024
1  6

Why can't the variables in my procedure be called? Omega1, Omega2, Q1, Q2 are all expressed in the form of local variables in the loop, why can't we solve the equation because there are too many variables? At the same time, I would like to ask how to output the results of Q1 and Q2?

equation915.mw

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