Items tagged with academic

The William Lowell Putnam Mathematical Competition, often abbreviated to the Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the world (regardless of the students' nationalities). One can see some problems and answers here. I find it remarkable that a lot of these problems can be done with Maple. Here is a sample (The DirectSearch package should be downloaded from http://www.maplesoft.com/applications/view.aspx?SID=101333 and installed in your Maple.).

 

rsolve({a(k)=a(k-1)^2-2,a(0)=5/2},a)#2014,A-3

NULL

rs := unapply(rsolve({a(0) = 5/2, a(k) = a(k-1)^2-2}, a), k)

proc (k) options operator, arrow; 2*cosh(arccosh(5/4)*2^k) end proc

(1)

(2)

evalf(product(1-1/rs(k), k = 0 .. infinity))

.4285714286

(3)

identify(%)

3/7

(4)

sol := solve({1/x-1/(2*y) = 2*(-x^4+y^4), 1/x+1/(2*y) = (x^2+3*y^2)*(3*x^2+y^2)}, explicit)

sol[1]; evalf(sol)

{x = 1.122865470, y = .1228654698}, {x = -0.39087502e-2+.3661111372*I, y = -1.003908750+.3661111372*I}, {x = .6924760152-.5923802638*I, y = -.3075239848-.5923802638*I}, {x = .6924760152+.5923802638*I, y = -.3075239848+.5923802638*I}, {x = -0.39087502e-2-.3661111372*I, y = -1.003908750-.3661111372*I}, {x = .3469845126+.1168520057*I, y = 0.3796751170e-1+1.067908527*I}, {x = .7773739670+.4755282581*I, y = .4683569607-.4755282736*I}, {x = .2183569726+.2938926261*I, y = 1.027373941-.2938926802*I}, {x = .3469845126+1.067908522*I, y = 0.3796751830e-1+.1168520056*I}, {x = -.2120324818+.8862728900*I, y = .5969845187+.2984876419*I}, {x = -.3494002531+.8416393955*I, y = -.6584172547-.1094171332*I}, {x = -.2120324818+.2984876377*I, y = .5969845144+.8862728969*I}, {x = -.9084172475+.6600037635*I, y = -0.9940025307e-1+0.7221851120e-1*I}, {x = -.3494002531+.1094171208*I, y = -.6584172336-.8416394020*I}, {x = -.9084172475+0.7221851117e-1*I, y = -0.9940025050e-1+.6600037719*I}, {x = -.9084172475-0.7221851117e-1*I, y = -0.9940025050e-1-.6600037719*I}, {x = -.3494002531-.1094171208*I, y = -.6584172336+.8416394020*I}, {x = -.9084172475-.6600037635*I, y = -0.9940025307e-1-0.7221851120e-1*I}, {x = -.2120324818-.2984876377*I, y = .5969845144-.8862728969*I}, {x = -.3494002531-.8416393955*I, y = -.6584172547+.1094171332*I}, {x = -.2120324818-.8862728900*I, y = .5969845187-.2984876419*I}, {x = .3469845126-1.067908522*I, y = 0.3796751830e-1-.1168520056*I}, {x = .2183569726-.2938926261*I, y = 1.027373941+.2938926802*I}, {x = .7773739670-.4755282581*I, y = .4683569607+.4755282736*I}, {x = .3469845126-.1168520057*I, y = 0.3796751170e-1-1.067908527*I}

(5)

plots:-implicitplot([1/x+1/(2*y) = (x^2+3*y^2)*(3*x^2+y^2), 1/x-1/(2*y) = 2*(-x^4+y^4)], x = 0 .. 2, y = 0 .. 1, color = [red, blue], gridrefine = 4)

 

"http://kskedlaya.org/putnam-archive/  and https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition"

Re(convert(int(ln(x+1)/(x^2+1), x = 0 .. 1), polylog))

(1/8)*Pi*ln(2)

(6)

Im(convert(int(ln(x+1)/(x^2+1), x = 0 .. 1), polylog))

0

(7)

NULL

DirectSearch:-GlobalOptima(int(sqrt(x^4+(-y^2+y)^2), x = 0 .. y), {y = 0 .. 1}, maximize)

[.333333333333333, [y = HFloat(0.9999999999999992)], 96]

(8)

rsolve({T(0) = 2, T(1) = 3, T(2) = 6, T(n) = (n+4)*T(n-1)-4*n*T(n-2)+(4*n-8)*T(n-3)}, T)

GAMMA(n+1)+2^n

(9)

floor(10^20000/(10^100+3))

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999970000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999997300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000809999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999757000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000072899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999978130000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000006560999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998031700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000590489999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999822853000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000053144099999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999984056770000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004782968999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998565109300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000430467209999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999870859837000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000038742048899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999988377385330000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003486784400999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998953964679700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000313810596089999999999999999999999999999999999999999999999999999999999999999999999999999999999999999905856821173000000000000000000000000000000000000000000000000000000000000000000000000000000000000000028242953648099999999999999999999999999999999999999999999999999999999999999999999999999999999999999991527113905570000000000000000000000000000000000000000000000000000000000000000000000000000000000000002541865828328999999999999999999999999999999999999999999999999999999999999999999999999999999999999999237440251501300000000000000000000000000000000000000000000000000000000000000000000000000000000000000228767924549609999999999999999999999999999999999999999999999999999999999999999999999999999999999999931369622635117000000000000000000000000000000000000000000000000000000000000000000000000000000000000020589113209464899999999999999999999999999999999999999999999999999999999999999999999999999999999999993823266037160530000000000000000000000000000000000000000000000000000000000000000000000000000000000001853020188851840999999999999999999999999999999999999999999999999999999999999999999999999999999999999444093943344447700000000000000000000000000000000000000000000000000000000000000000000000000000000000166771816996665689999999999999999999999999999999999999999999999999999999999999999999999999999999999949968454901000293000000000000000000000000000000000000000000000000000000000000000000000000000000000015009463529699912099999999999999999999999999999999999999999999999999999999999999999999999999999999995497160941090026370000000000000000000000000000000000000000000000000000000000000000000000000000000001350851717672992088999999999999999999999999999999999999999999999999999999999999999999999999999999999594744484698102373300000000000000000000000000000000000000000000000000000000000000000000000000000000121576654590569288009999999999999999999999999999999999999999999999999999999999999999999999999999999963527003622829213597000000000000000000000000000000000000000000000000000000000000000000000000000000010941898913151235920899999999999999999999999999999999999999999999999999999999999999999999999999999996717430326054629223730000000000000000000000000000000000000000000000000000000000000000000000000000000984770902183611232880999999999999999999999999999999999999999999999999999999999999999999999999999999704568729344916630135700000000000000000000000000000000000000000000000000000000000000000000000000000088629381196525010959289999999999999999999999999999999999999999999999999999999999999999999999999999973411185641042496712213000000000000000000000000000000000000000000000000000000000000000000000000000007976644307687250986336099999999999999999999999999999999999999999999999999999999999999999999999999997607006707693824704099170000000000000000000000000000000000000000000000000000000000000000000000000000717897987691852588770248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(10)

int(exp(-1985*(t+1/t))/sqrt(t), t = 0 .. infinity)

(1/1985)*Pi^(1/2)*exp(-3970)*1985^(1/2)

(11)

l := [seq(LinearAlgebra:-Determinant(Matrix(n, proc (i, j) options operator, arrow; 1/min(i, j) end proc)), n = 1 .. 10)]

[1, -1/2, 1/12, -1/144, 1/2880, -1/86400, 1/3628800, -1/203212800, 1/14631321600, -1/1316818944000]

(12)

with(gfun):

rec := listtorec(l, u(n))

[{u(n+1)+(n^2+5*n+6)*u(n+2), u(0) = 1, u(1) = -1/2}, ogf]

(13)

rsolve(rec[1], u)

(-1)^n*(n+1)/GAMMA(n+2)^2

(14)

``

Hope the reader will try to continue the above.

Download Putnam_done_with_Maple.mw

Hi all!

 

I do a small calculation and get a system of 6
nonlinear equations.
And "n" is the degree of the equation is float.

Here are the calculations that lead to the system.

 

restart;
 with(DirectSearch):
 B:=1: 
 q:=1: 
 l:=1: 
 n:=4.7:
 V:=0.05:
 N:=1200:
 
 
 kappa:=Vector(N+1,[]):
 theta:=Vector(N+1,[]):
 u:=Vector(N,[]):
 M:=Vector(N,[]):
 Z:=Vector(N,[]):
 
 M_F:=q*(6*l*(z-l)-z^2/2):
 M_1:=piecewise((z<l), l-z, 0):
 M_2:=piecewise((z<2*l), 2*l-z, 0):
 M_3:=piecewise((z<3*l), 3*l-z, 0):
 M_4:=piecewise((z<4*l), 4*l-z, 0):
 M_5:=piecewise((z<5*l), 5*l-z, 0):
 M_6:=6*l-z:
 M_finish:=(X_1,X_2,X_3,X_4,X_5,X_6,z)->M_1*X_1+M_2*X_2+M_3*X_3+M_4*X_4+M_5*X_5+M_6*X_6+M_F:
 
 
 kappa_old:=0:
 theta_old:=0:
 u_old:=0:
 M_old:=0:
 
 
 step:=6*l/N:
 u[1]:=0:
 kappa[1]:=0:
 theta[1]:=0:
 
 
 
 
 for i from 2 to N do
 
 z:=i*step:
 kappa_new:=kappa_old+B/V*(M_finish(X_1,X_2,X_3,X_4,X_5,X_6,z))^n*step:
 
 theta_new:=theta_old+1/2*(kappa_old+kappa_new)*step:
 
 u_new:=u_old+1/2*(theta_old+theta_new)*step:
 
 Z[i]:=z:
 kappa[i]:=kappa_new:
 theta[i]:=theta_new:
 u[i]:=u_new:
 kappa_old:=kappa_new:
 theta_old:=theta_new:
 u_old:=u_new:
 
 end do:
 
 So,my system:


 u[N/6]=0;
 u[N/3]=0;
 u[N/2]=0;
 u[2*N/3]=0;
 u[5*N/6]=0;
 u[N]=0;

 

I want to ask advice on how to solve the system.
I wanted to use Newton's method, but I don't know the initial values X_1..X_6.

Tried to set the values X_1..X_6 and to minimize the functional
Fl:=(X_1,X_2,X_3,X_4,X_5,X_6)->(u[N/6])^2+(u[N/3])^2+(u[N/2])^2+(u[2*N/3])^2+(u[5*N/6])^2+(u[N])^2:

with the help with(DirectSearch):
GlobalOptima(Fl);
But I don't know what to do next

Please, advise me how to solve the system! I would be grateful for examples!

 

I use maple worksheet to organize a derivation, etc.

But most of the paragraphs and subparagraphs should normally be closed.

 

But after executing the file (using !!!) every paragraph is opened.   

We need better ability to control the opening/closing of paragraphs.

 

How do I close ALL paragraphs.

 

Also, how do I close all TOP level paragraphs (but not subparagraphs).

 

Thanks, Chee

 

Where u and v are the displacement components in x and y directions respectively.

Today science professionals in engineering software used to only work on the desktop and even just looking to download and use mobile apps math; but they are not able to design their own applications.Maplesoft to set the solution to it through its Maple package; software supports desktop and mobile; solves problems of analysis and calculation with Embedded Components. To show this we have taken the area of different mathematical topics; fixed horizontally to a certain range of parameters and not just a constant as it is customary to develop. This paper shows how the Embedded Components allow us to develop mathematics in all areas. Achieving build applications that are interactive in mobile devices such as tablets; which are used at any time. Maple gives us design according to our university or research need, based on contemporary and modern mathematics.With this method we encourage students, teachers and researchers to use graphics algorithms.

 

CSMP_PUCP_2014.pdf

Coloquio_PUCP.mw

 

Lenin Araujo Castillo

Physics Pure

Computer Science

Hi,

I used to use windows 32 bit, but I have 64 bit windows now. I installed the MapleSim 6.4 and Maple 18 and try to run the simulations that I created with the 32 bit windows. I have this error 'Unable to compile (rc=1), please try again, and if that fails verify your Windows compiller installation'. Could you please let me know what should I do to eliminate this problem?

Best

Onder

Can maple solve maximization problem like

q := proc (a, b, c) options operator, arrow; .2*b+.1*c end proc;
print(`output redirected...`); # input placeholder
(a, b, c) -> 0.2 b + 0.1 c
w := proc (a, b, c) options operator, arrow; .7*a+.1*c end proc;
print(`output redirected...`); # input placeholder
(a, b, c) -> 0.7 ab + 0.1 c
e := proc (a, b, c) options operator, arrow; .7*a+.2*b*c end proc;
print(`output redirected...`); # input placeholder
(a, b, c) -> 0.7 a + 0.2 b c

with(Optimization)

Maxmize(int(min(100+(.7*a+q)*(1/2), a), q)+int(min(100+(.2*b+w)*(1/2), b), w)+int(min(100+(.1*c+e)*(1/2), c), e)-a-b-c-ab-ac-bc)

Error, (in Optimization:-NLPSolve) cannot convert procedures to piecewise

 

 

Thanks alot if you can help me.Urgent! Really appreciate.

 

For different reasons I need to ocasionally export a number of Maple worksheet in a folder to pdf files. Is there a way to automate this? I would want that the worksheet is opened, output removed, then executed and eventually exported to pdf. It can take quite a while to do this manually for about 50 worksheets.

I have two Reissner Nordstrom black holes that are near extreme. How do I show they move? 

y(t) = _C1*exp(-1.*t)*sin(.57736*t)+_C2*exp(-1.*t)*cos(.57736*t)

The answer i got for a DE raised in mapleprime is given above.

What command do i write now to get a plot of the same?

Ramakrishnan V

Presentations of the first national congress of civil engineering developed at the University Cesar Vallejo. From 10 to 12 November 2014.

 

CONIC_UCV.pdf

(in spanish)

Lenin Araujo Castillo

Physics Pure

Computer Science

 

 

 

 


#DeHoog f4(t)

F4 := proc (F::procedure, Tol, M, t) local T, gamma1, Fc, e, q, d, A, B, i, r, m, n, h2M, R2M, A2M, B2M, a, z; T := evalf(2*t); gamma1 := evalf((-1)*.5*ln(Tol)/T); Fc := Array(0 .. 2*M); e := Array(0 .. 2*M, 0 .. 2*M); q := Array(0 .. 2*M, 0 .. 2*M); d := Array(0 .. 2*M); A := Array(-1 .. 2*M); B := Array(-1 .. 2*M); Fc[0] := evalf(.5*F(gamma1)); for i to 2*M do a := gamma1+I*i*Pi/T; Fc[i] := evalf(F(a), 25) end do; for i from 0 to 2*M do e[0, i] := 0 end do; for i from 0 to 2*M-1 do q[1, i] := evalf(Fc[i+1]/Fc[i]) end do; for r to M do for i from 2*M-2*r+1 by -1 to 0 do if 1 < r then q[r, i] := evalf(q[r-1, i+1]*e[r-1, i+1]/e[r-1, i]) end if; if i < 2*M-2*r+1 then e[r, i] := evalf(q[r, i+1]-q[r, i]+e[r-1, i+1]) end if end do end do; d[0] := Fc[0]; for m to M do d[2*m-1] := -q[m, 0]; d[2*m] := -e[m, 0] end do; z := evalf(exp(I*Pi*t/T)); A[-1] := 0; B[-1] := 1; A[0] := d[0]; B[0] := 1; for n to 2*M do A[n] := A[n-1]+d[n]*z*A[n-2]; B[n] := B[n-1]+d[n]*z*B[n-2] end do; h2M := evalf(.5+((1/2)*d[2*M-1]-(1/2)*d[2*M])*z); R2M := evalf(-h2M-h2M*sqrt(1+d[2*M]*z/h2M^2)); A2M := A[2*M-1]+R2M*A[2*M-2]; B2M := B[2*M-1]+R2M*B[2*M-2]; evalf(exp(gamma1*t)*Re(A2M/B2M)/T) end proc;

proc (F::procedure, Tol, M, t) local T, gamma1, Fc, e, q, d, A, B, i, r, m, n, h2M, R2M, A2M, B2M, a, z; T := evalf(2*t); gamma1 := evalf((-1)*.5*ln(Tol)/T); Fc := Array(0 .. 2*M); e := Array(0 .. 2*M, 0 .. 2*M); q := Array(0 .. 2*M, 0 .. 2*M); d := Array(0 .. 2*M); A := Array(-1 .. 2*M); B := Array(-1 .. 2*M); Fc[0] := evalf(.5*F(gamma1)); for i to 2*M do a := gamma1+I*i*Pi/T; Fc[i] := evalf(F(a), 25) end do; for i from 0 to 2*M do e[0, i] := 0 end do; for i from 0 to 2*M-1 do q[1, i] := evalf(Fc[i+1]/Fc[i]) end do; for r to M do for i from 2*M-2*r+1 by -1 to 0 do if 1 < r then q[r, i] := evalf(q[r-1, i+1]*e[r-1, i+1]/e[r-1, i]) end if; if i < 2*M-2*r+1 then e[r, i] := evalf(q[r, i+1]-q[r, i]+e[r-1, i+1]) end if end do end do; d[0] := Fc[0]; for m to M do d[2*m-1] := -q[m, 0]; d[2*m] := -e[m, 0] end do; z := evalf(exp(I*Pi*t/T)); A[-1] := 0; B[-1] := 1; A[0] := d[0]; B[0] := 1; for n to 2*M do A[n] := A[n-1]+d[n]*z*A[n-2]; B[n] := B[n-1]+d[n]*z*B[n-2] end do; h2M := evalf(.5+((1/2)*d[2*M-1]-(1/2)*d[2*M])*z); R2M := evalf(-h2M-h2M*sqrt(1+d[2*M]*z/h2M^2)); A2M := A[2*M-1]+R2M*A[2*M-2]; B2M := B[2*M-1]+R2M*B[2*M-2]; evalf(exp(gamma1*t)*Re(A2M/B2M)/T) end proc

(1)

F4(proc (p) options operator, arrow; int(12*Dirac(a)*BesselJ(0, 2*a)/(a*p), a = 0 .. .100) end proc, 0.1e-4, 4, 10);

Float(undefined)

(2)

``

``


Download dehoog.mw

How is it possible to numerically integrate the function containing Diract delta function.

The following code for example is failed to answer and makes computer memory full.

 

One of my post graduate students chose the theme "Discrete Wavelet analysis of medical signals with Maple" for her thesis. I test, is it possible to support such reearch be Maplesoft &  what may be the form of such support, if it is possible?

Take a look at below. I was expecting maple to give me "g'(1)"! :)



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