dj_gssst

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6 years, 271 days

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These are replies submitted by dj_gssst

I am not able to find the k-th coefficient for C. Please explain me the trick in the following function :

coef:=Sum(Sum(u::nonunit(algebraic)*v::`^`(name,name),eq1::equation),eq2::equation)=
Sum(u,eq2):

 I have found another method by using CoefficientList(), but I was unable to express it by power of k for the k-th coefficient by coefficientList() method. Thank you very much for your help.

I am not able to find the k-th coefficient for C. Please explain me the trick in the following function :

coef:=Sum(Sum(u::nonunit(algebraic)*v::`^`(name,name),eq1::equation),eq2::equation)=
Sum(u,eq2):

 I have found another method by using CoefficientList(), but I was unable to express it by power of k for the k-th coefficient by coefficientList() method. Thank you very much for your help.

Dear all,

 Please provide me a solution.

Dear experts,

 evalf(Sum(..,k=0..infinity)); worked fine. But the same function doesn't work if the summand variable is a solution of two equations. I have given the code

restart; with(LinearAlgebra); with(RootFinding); Digits := 100; lambda := 3; mu := 7; eta := 4; xi := 2

3

 

7

 

4

 

2

(1)

A := proc (z) options operator, arrow; lambda^5/(lambda+mu-mu*z)^5 end proc;

proc (z) options operator, arrow; lambda^5/(lambda+mu-mu*z)^5 end proc

 

proc (z) options operator, arrow; lambda^5/(lambda+xi+eta-eta*z)^5 end proc

 

proc (z) options operator, arrow; xi*(A(z)-B(z))/(xi-(mu-eta)*(1-z)) end proc

(2)

expand(A(z), z);

243/(10-7*z)^5

 

Sum(((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4)*z^k, k = 0 .. infinity)

(3)

convert(B(z), FormalPowerSeries, z);

Sum(((1/243)*(4/9)^k+(25/2916)*(4/9)^k*k+(35/5832)*(4/9)^k*k^2+(5/2916)*(4/9)^k*k^3+(1/5832)*(4/9)^k*k^4)*z^k, k = 0 .. infinity)

(4)

convert(C(z), FormalPowerSeries, z);

Sum((-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4)*z^k, k = 0 .. infinity)

(5)

a := proc (k) options operator, arrow; (243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4 end proc;

proc (k) options operator, arrow; (243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4 end proc

 

proc (k) options operator, arrow; -(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4 end proc

(6)

delta := .1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182;

.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182

 

.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786

 

.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(k-1)*((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4)

 

.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^k*(-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4)

 

proc (k) options operator, arrow; .1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(k-1)*((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4) end proc

 

proc (k) options operator, arrow; .1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^k*(-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4) end proc

(7)

y := evalf(Sum(delta^(k-1)*a(k), k = 1 .. infinity));

0.1075168698791322945131368855542654932195602557245309692494169359920231557707996781154291830071959580e-1

 

0.1075168698791322945131368855542654932195602557245309692494169359920231557707996781154291830071959580e-1

(8)

x := evalf(Sum(theta^k*c(k), k = 1 .. infinity));

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

 

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

(9)

sol := fsolve({k[1] = k[2]*x+k[1]*y, k[1]/(1-delta)+k[2]/(1-theta) = 1}, {k[1], k[2]});

{k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2, k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080}

 

k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2

 

k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080

(10)

p[1] := proc (n) options operator, arrow; s*delta^(n-1) end proc; p[0] := proc (n) options operator, arrow; t*theta^n end proc; evalf(Sum(p[1](n), n = 1 .. 100))+evalf(Sum(p[0](n), n = 0 .. 100)); u := evalf(Sum(delta^(n-1)*sol[1], n = 1 .. infinity)); v := evalf(Sum(theta^n*t, sol[2] = 0 .. infinity)); u+v

proc (n) options operator, arrow; s*delta^(n-1) end proc

 

proc (n) options operator, arrow; t*theta^n end proc

 

Sum(.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1)*k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2*.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1), n = 1 .. infinity)+Sum(.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n*k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080*.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n, n = 0 .. infinity)

 

Sum(.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1)*k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2*.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1), n = 1 .. infinity)

 

Sum(.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n*k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080*.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n, (k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080) = 0 .. infinity)

 

Sum(.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1)*k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2*.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1), n = 1 .. infinity)+Sum(.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n*k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080*.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n, (k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080) = 0 .. infinity)

(11)

``


Download Quest_FormalPowerSe.mw

Dear experts,

 evalf(Sum(..,k=0..infinity)); worked fine. But the same function doesn't work if the summand variable is a solution of two equations. I have given the code

restart; with(LinearAlgebra); with(RootFinding); Digits := 100; lambda := 3; mu := 7; eta := 4; xi := 2

3

 

7

 

4

 

2

(1)

A := proc (z) options operator, arrow; lambda^5/(lambda+mu-mu*z)^5 end proc;

proc (z) options operator, arrow; lambda^5/(lambda+mu-mu*z)^5 end proc

 

proc (z) options operator, arrow; lambda^5/(lambda+xi+eta-eta*z)^5 end proc

 

proc (z) options operator, arrow; xi*(A(z)-B(z))/(xi-(mu-eta)*(1-z)) end proc

(2)

expand(A(z), z);

243/(10-7*z)^5

 

Sum(((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4)*z^k, k = 0 .. infinity)

(3)

convert(B(z), FormalPowerSeries, z);

Sum(((1/243)*(4/9)^k+(25/2916)*(4/9)^k*k+(35/5832)*(4/9)^k*k^2+(5/2916)*(4/9)^k*k^3+(1/5832)*(4/9)^k*k^4)*z^k, k = 0 .. infinity)

(4)

convert(C(z), FormalPowerSeries, z);

Sum((-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4)*z^k, k = 0 .. infinity)

(5)

a := proc (k) options operator, arrow; (243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4 end proc;

proc (k) options operator, arrow; (243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4 end proc

 

proc (k) options operator, arrow; -(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4 end proc

(6)

delta := .1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182;

.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182

 

.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786

 

.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(k-1)*((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4)

 

.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^k*(-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4)

 

proc (k) options operator, arrow; .1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(k-1)*((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4) end proc

 

proc (k) options operator, arrow; .1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^k*(-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4) end proc

(7)

y := evalf(Sum(delta^(k-1)*a(k), k = 1 .. infinity));

0.1075168698791322945131368855542654932195602557245309692494169359920231557707996781154291830071959580e-1

 

0.1075168698791322945131368855542654932195602557245309692494169359920231557707996781154291830071959580e-1

(8)

x := evalf(Sum(theta^k*c(k), k = 1 .. infinity));

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

 

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

(9)

sol := fsolve({k[1] = k[2]*x+k[1]*y, k[1]/(1-delta)+k[2]/(1-theta) = 1}, {k[1], k[2]});

{k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2, k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080}

 

k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2

 

k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080

(10)

p[1] := proc (n) options operator, arrow; s*delta^(n-1) end proc; p[0] := proc (n) options operator, arrow; t*theta^n end proc; evalf(Sum(p[1](n), n = 1 .. 100))+evalf(Sum(p[0](n), n = 0 .. 100)); u := evalf(Sum(delta^(n-1)*sol[1], n = 1 .. infinity)); v := evalf(Sum(theta^n*t, sol[2] = 0 .. infinity)); u+v

proc (n) options operator, arrow; s*delta^(n-1) end proc

 

proc (n) options operator, arrow; t*theta^n end proc

 

Sum(.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1)*k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2*.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1), n = 1 .. infinity)+Sum(.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n*k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080*.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n, n = 0 .. infinity)

 

Sum(.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1)*k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2*.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1), n = 1 .. infinity)

 

Sum(.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n*k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080*.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n, (k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080) = 0 .. infinity)

 

Sum(.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1)*k[1] = 0.1637935477698018732276851530308612938004131156280124580147547622858728899732152932156471141157968415e-2*.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(n-1), n = 1 .. infinity)+Sum(.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n*k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080*.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^n, (k[2] = .8738880449255668655493761487129800277295707224229803653538821365150932040118518225052273427259289080) = 0 .. infinity)

(11)

``


Download Quest_FormalPowerSe.mw

Dear experts,

  Thank you for your prompt reply and help. I am able to compute the sum upto 100. But unable to compute it for infinity althought the expression is converging.

Dear experts,

  Thank you for your prompt reply and help. I am able to compute the sum upto 100. But unable to compute it for infinity althought the expression is converging.


restart; with(LinearAlgebra); with(RootFinding); Digits := 100; lambda := 3; mu := 7; eta := 4; xi := 2

3

 

7

 

4

 

2

(1)

erlang := proc (m) options operator, arrow; lambda^m*t^(m-1)*exp(-lambda*t)/factorial(m-1) end proc; erlang(2); f := proc (k) options operator, arrow; sum(xi*exp(-xi*x)*(eta*x)^j*exp(-eta*x)*(mu*(t-x))^(k-j)*exp(-mu*(t-x))/(factorial(j)*factorial(k-j)), j = 0 .. k) end proc; f(4); g := int(f(4), x = 0 .. t); h := unapply(g, t); l := int(erlang(2)*h(t), t = 0 .. infinity); f := proc (k) options operator, arrow; sum(xi*exp(-xi*x)*(eta*x)^j*exp(-eta*x)*(mu*(t-x))^(k-j)*exp(-mu*(t-x))/(factorial(j)*factorial(k-j)), j = 0 .. k) end proc; f(k); g := int(f(k), x = 0 .. t); h := unapply(g, t, k); h(t, k); c := proc (k) options operator, arrow; int(h(t, k), t = 0 .. infinity) end proc; x := 0; c(5)

proc (m) options operator, arrow; lambda^m*t^(m-1)*exp(-lambda*t)/factorial(m-1) end proc

 

9*t*exp(-3*t)

 

proc (k) options operator, arrow; sum(xi*exp(-xi*x)*(eta*x)^j*exp(-eta*x)*(mu*(t-x))^(k-j)*exp(-mu*(t-x))/(factorial(j)*factorial(k-j)), j = 0 .. k) end proc

 

(1/12)*exp(-2*x)*exp(-4*x)*(7*t-7*x)^4*exp(-7*t+7*x)+(4/3)*exp(-2*x)*x*exp(-4*x)*(7*t-7*x)^3*exp(-7*t+7*x)+8*exp(-2*x)*x^2*exp(-4*x)*(7*t-7*x)^2*exp(-7*t+7*x)+(64/3)*exp(-2*x)*x^3*exp(-4*x)*(7*t-7*x)*exp(-7*t+7*x)+(64/3)*exp(-2*x)*x^4*exp(-4*x)*exp(-7*t+7*x)

 

-(2401/12)*exp(-7*t)*t^4-162*exp(-7*t)-378*exp(-7*t)*t-441*exp(-7*t)*t^2-343*exp(-7*t)*t^3+162*exp(-6*t)+216*exp(-6*t)*t+144*exp(-6*t)*t^2+64*exp(-6*t)*t^3+(64/3)*exp(-6*t)*t^4

 

proc (t) options operator, arrow; -(2401/12)*exp(-7*t)*t^4-162*exp(-7*t)-378*exp(-7*t)*t-441*exp(-7*t)*t^2-343*exp(-7*t)*t^3+162*exp(-6*t)+216*exp(-6*t)*t+144*exp(-6*t)*t^2+64*exp(-6*t)*t^3+(64/3)*exp(-6*t)*t^4 end proc

 

434677387/5904900000

 

proc (k) options operator, arrow; sum(xi*exp(-xi*x)*(eta*x)^j*exp(-eta*x)*(mu*(t-x))^(k-j)*exp(-mu*(t-x))/(factorial(j)*factorial(k-j)), j = 0 .. k) end proc

 

2*exp(-2*x)*exp(-4*x)*exp(-7*t+7*x)*(-3*x+7*t)^k/GAMMA(k+1)

 

int(2*exp(-2*x)*exp(-4*x)*exp(-7*t+7*x)*(-3*x+7*t)^k/GAMMA(k+1), x = 0 .. t)

 

proc (t, k) options operator, arrow; int(2*exp(-2*x)*exp(-4*x)*exp(-7*t+7*x)*(-3*x+7*t)^k/GAMMA(k+1), x = 0 .. t) end proc

 

int(2*exp(-2*x)*exp(-4*x)*exp(-7*t+7*x)*(-3*x+7*t)^k/GAMMA(k+1), x = 0 .. t)

 

proc (k) options operator, arrow; int(h(t, k), t = 0 .. infinity) end proc

 

Error, (in int) integration range or variable must be specified in the second argument, got 0 = 0 .. t

 

NULL


Download MaplePrimes_Questio.mwMaple_Prime_Questio.pdfMaple_Prime_Questio.pdf

Hi,

 I have given permission to view the files in googledocs. I have uploaded the files many times in this website, but it never shows up. Please let me know how to make the uploaded files visible.

@Kitonum 

 Dear Experts,

  I have attached the PDF and Maple code in googledocs. Here is the link

https://docs.google.com/open?id=0B1_ZWGeTPJnoQk1NSk02NzlTZHM for PDF of the formula

https://docs.google.com/open?id=0B1_ZWGeTPJnob0xVZXhrN1lnUm8 for Maple code

 

Please let me know if you didn;t get it.

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