Markiyan Hirnyk

Markiyan Hirnyk
10 years, 221 days


These are answers submitted by Markiyan Hirnyk

By inequal

June 21 2016 Markiyan Hirnyk 6956

The inequal command is a tool to this end:

 

with(plots):

plots:-inequal(x^2+y^2 < 1 and (x-1)^2+y^2 < 1, x = -3 .. 3, y = -3 .. 3, gridlines = false);

 

``

 

 

 

 

 

``

 

Download inequal1.mw

 

 

 

Sorry, but the execution of ?Appell  brings nothing. It means that the Appell hypergeometric function is not implemented in Maple as yet (as well as in Mathematica in the general case). Such implementation, AFAIUI, is not a simple task, this is a job for profies.

@Traruh Synred Following Maple help to ?Statistics,Histogram,

>with(Statistics):
>N := RandomVariable(Normal(0, 1)):
>A := Sample(N, 1000): H := Histogram(A, bincount = 30, frequencyscale = absolute);

>TallyInto(A, default, bins = 30);
[HFloat(-2.790335999149854) .. HFloat(-2.578765786130552) = 2,

HFloat(-2.578765786130552) .. HFloat(-2.3671955731112497) = 8,

HFloat(-2.3671955731112497) .. HFloat(-2.155625360091947) = 4,

HFloat(-2.155625360091947) .. HFloat(-1.9440551470726448) = 13,

HFloat(-1.9440551470726448) .. HFloat(-1.7324849340533426) = 21,

HFloat(-1.7324849340533426) .. HFloat(-1.5209147210340401) = 21,

HFloat(-1.5209147210340401) .. HFloat(-1.3093445080147377) = 31,

HFloat(-1.3093445080147377) .. HFloat(-1.0977742949954354) = 43,

HFloat(-1.0977742949954354) .. HFloat(-0.8862040819761332) = 61,

HFloat(-0.8862040819761332) .. HFloat(-0.674633868956831) = 62,

HFloat(-0.674633868956831) .. HFloat(-0.4630636559375283) = 60,

HFloat(-0.4630636559375283) .. HFloat(-0.25149344291822606) = 75, HFloat(-0.25149344291822606) ..

HFloat(-0.03992322989892383) = 85,

HFloat(-0.03992322989892383) .. HFloat(0.17164698312037885) = 84,

HFloat(0.17164698312037885) .. HFloat(0.3832171961396811) = 99,

HFloat(0.3832171961396811) .. HFloat(0.5947874091589833) = 65,

HFloat(0.5947874091589833) .. HFloat(0.8063576221782855) = 58,

HFloat(0.8063576221782855) .. HFloat(1.0179278351975878) = 46,

HFloat(1.0179278351975878) .. HFloat(1.2294980482168905) = 40,

HFloat(1.2294980482168905) .. HFloat(1.4410682612361922) = 43,

HFloat(1.4410682612361922) .. HFloat(1.652638474255495) = 24,

HFloat(1.652638474255495) .. HFloat(1.8642086872747976) = 24,

HFloat(1.8642086872747976) .. HFloat(2.0757789002940994) = 13,

HFloat(2.0757789002940994) .. HFloat(2.287349113313402) = 6,

HFloat(2.287349113313402) .. HFloat(2.4989193263327047) = 6,

HFloat(2.4989193263327047) .. HFloat(2.7104895393520065) = 2,

HFloat(2.7104895393520065) .. HFloat(2.922059752371309) = 1,

HFloat(2.922059752371309) .. HFloat(3.133629965390612) = 2,

HFloat(3.133629965390612) .. HFloat(3.3452001784099137) = 0,

HFloat(3.3452001784099137) .. HFloat(3.5567703914292164) = 1]



Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/example.mw .

Download example.mw

PS. After my comment to the Maple 2016 announce I can't present pictures by copy&paste in MaplePrimes and in many cases I can't present the content of an mw file.

>SOL[j] := DirectSearch:-SolveEquations([seq(EQ[16117][n], n = 1 .. 3)], evaluationlimit = 60000, AllSolutions, solutions = 1, method = quadratic, number = 250);

[[6.02975324949749 10^(-15),[[[` 1 .. 26 `Vector[column]],[`Data Type: `anything],[`Storage: `rectangular],[`Order: `Fortran_order]]],[HLa[1,16117]=0.909386711735506,HLa[2,16117]=0.406251423909525,HLa[3,16117]=-17024.1382192605,HRa[1,16117]=0.909386711740191,HRa[2,16117]=0.862064656271464,HRa[3,16117]=-4917.27772529676,NLa[1,16117]=1.06616948224172,NLa[2,16117]=0.946371669593081,NLa[3,16117]=-3437.41803435019,NRa[1,16117]=1.06616948222820,NRa[2,16117]=1.14739851463955,NRa[3,16117]=1902.05830347325,SL[2,16118]=0.00000173372622663711,SL[3,16118]=0.00000343122730279778,SL[4,16118]=0.00000530436428099936,SR[1,16118]=0.00000174897850262596,W[1,16117]=82.4755760350782,W[2,16117]=67.0920422567193,W[3,16117]=0.00902226892959830,fra[1,16117]=0.503135287892994,fra[2,16117]=17025.0002839167,fra[3,16117]=-4917.27772529676,ga[1,16117]=-4.81853475084196 100^(-12),ga[2,16117]=-0.468749318829457,ga[3,16117]=-12450.4560367357,v[1,16118]=8.50355911487117 10^(-7)],60000]]

DS.mw

By Explore

June 10 2016 Markiyan Hirnyk 6956

Here is an example.

Explore(plots:-implicitplot(ln(x)-(C+x)*(1+sin(y)), x = -10 .. 10, y = -10 .. 10,gridrefine=2));

 

Explore1.mw

Edit. The gridrefine=2 option was added.

The series of Fibonacci numbers is divergent. Maple finds its generalized sum (see ?sum,details for info) in such a way.

_EnvFormal := true:

(10-4*5^(1/2))/(5*5^(1/2)+5)+(-10-4*5^(1/2))/(5*5^(1/2)-5)

(1)

``

``

 

Download generalized_sum.mw

This is  a very narrow topic of algebraic geometry which does not belong to the mainstream of modern math (see http://mathoverflow.net/questions/60255/information-about-publishing-and-citations concerning statistics). The one is not described in Wiki. Magma has tools to this end. 

No problem

June 03 2016 Markiyan Hirnyk 6956

>A := [seq(i, i = 0.1e-2 .. 2, 0.1e-2)];

>map(c-> evalf(eval(diff(KelvinBei(0, x), x), x = c)), A);

[0.0004999999999,0.0009999999995,0.001500000000,0.001999999999,...,.9167201808, .9170136132]

eval.mw

are needed. For example,


int(I*sqrt((R*exp(I*theta)+1)/(R*exp(I*theta)-a)), theta = 0 .. Pi)assuming 0 < R, 0 < a, R < 1, a < R;``

(1/2)*(a^(5/2)*(-R^2-R*a+R+a)^(1/2)*(R+1)^(1/2)*ln(8*R*(R^2-R*a+R-a)^(1/2)-4*(R^2-R*a+R-a)^(1/2)*a+8*R^2-8*R*a+a^2+4*(R^2-R*a+R-a)^(1/2)+8*R-6*a+1)*(R+a)^(1/2)-2*a^(5/2)*(-R^2-R*a+R+a)^(1/2)*(R+1)^(1/2)*ln(2)*(R+a)^(1/2)-a^(3/2)*(-R^2-R*a+R+a)^(1/2)*(R+1)^(1/2)*ln(8*R*(R^2-R*a+R-a)^(1/2)-4*(R^2-R*a+R-a)^(1/2)*a+8*R^2-8*R*a+a^2+4*(R^2-R*a+R-a)^(1/2)+8*R-6*a+1)*R*(R+a)^(1/2)+2*a^(3/2)*(-R^2-R*a+R+a)^(1/2)*(R+1)^(1/2)*ln(2)*R*(R+a)^(1/2)+2*(-R^2-R*a+R+a)^(1/2)*(R+1)^(1/2)*arctan((1/2)*(R*a-R+2*a)/(a^(1/2)*(R^2-R*a+R-a)^(1/2)))*R*a*(R+a)^(1/2)-2*(-R^2-R*a+R+a)^(1/2)*(R+1)^(1/2)*arctan((1/2)*(R*a-R+2*a)/(a^(1/2)*(R^2-R*a+R-a)^(1/2)))*a^2*(R+a)^(1/2)-2*(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*ln(-((2*I)*signum(0, 2*R+a-1, -1)*(-R^2-R*a+R+a)^(1/2)+2*R+a-1)/(4*R^2+4*R*a+a^2-4*R-2*a+1-4*signum(0, 2*R+a-1, -1)^2*R^2-4*signum(0, 2*R+a-1, -1)^2*R*a+4*signum(0, 2*R+a-1, -1)^2*R+4*signum(0, 2*R+a-1, -1)^2*a)^(1/2))*a^(5/2)*(R-a)^(1/2)-(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*a^(5/2)*ln(4*R^2+4*R*a+a^2-4*R-2*a+1-4*signum(0, 2*R+a-1, -1)^2*R^2-4*signum(0, 2*R+a-1, -1)^2*R*a+4*signum(0, 2*R+a-1, -1)^2*R+4*signum(0, 2*R+a-1, -1)^2*a)*(R-a)^(1/2)+2*(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*a^(5/2)*ln(2)*(R-a)^(1/2)-2*(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*ln(-((2*I)*signum(0, 2*R+a-1, -1)*(-R^2-R*a+R+a)^(1/2)+2*R+a-1)/(4*R^2+4*R*a+a^2-4*R-2*a+1-4*signum(0, 2*R+a-1, -1)^2*R^2-4*signum(0, 2*R+a-1, -1)^2*R*a+4*signum(0, 2*R+a-1, -1)^2*R+4*signum(0, 2*R+a-1, -1)^2*a)^(1/2))*R*a^(3/2)*(R-a)^(1/2)-(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*R*a^(3/2)*ln(4*R^2+4*R*a+a^2-4*R-2*a+1-4*signum(0, 2*R+a-1, -1)^2*R^2-4*signum(0, 2*R+a-1, -1)^2*R*a+4*signum(0, 2*R+a-1, -1)^2*R+4*signum(0, 2*R+a-1, -1)^2*a)*(R-a)^(1/2)+2*(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*R*a^(3/2)*ln(2)*(R-a)^(1/2)-(2*I)*(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*arctanh((1/2)*(R*a-R-2*a)*signum(0, 2*R^2+R*a-R, -1)/((-R^2-R*a+R+a)^(1/2)*a^(1/2)))*R*a*(R-a)^(1/2)-(2*I)*(1-R)^(1/2)*signum(0, 2*R^2+R*a-R, -1)*(R^2-R*a+R-a)^(1/2)*arctanh((1/2)*(R*a-R-2*a)*signum(0, 2*R^2+R*a-R, -1)/((-R^2-R*a+R+a)^(1/2)*a^(1/2)))*a^2*(R-a)^(1/2))/((R-a)^(1/2)*(R+a)^(1/2)*a^(3/2)*(R^2-R*a+R-a)^(1/2)*(-R^2-R*a+R+a)^(1/2))

(1)

``


Download assumptions.mw

 

By transform

June 01 2016 Markiyan Hirnyk 6956

This can be done as follows (The plane x=15 is omitted for a better visualization.).

 

restart

with(plots):

f := transform(proc (x, y, z) options operator, arrow; if x <= 15 then [x, y, z] else [30-x, y, z] end if end proc):``

display(f(animate(implicitplot3d, [[x^2+y^2+z^2 = r^2], x = -30 .. 30, y = -30 .. 30, z = -30 .. 30], r = 0.1e-1 .. 30, scaling = constrained, style = surface, grid = [60, 60, 60])), insequence = true);

 

 

display(f(animate(plots:-implicitplot3d, [[x^2+y^2+z^2 = r^2], x = -30 .. 30, y = -30 .. 30, z = -30 .. 30], r = 0.1e-1 .. 30, scaling = constrained, style = surface, grid = [60, 60, 60])), insequence = true)

Download reflection.mw

can be obtained by

>sol:=LinearAlgebra:-LeastSquares({A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9], A[10], A[11], A[12], A[13], A[14], A[15], A[16], A[17], A[18], A[19], A[20]},{C[1], C[2], C[3], C[4], C[5], C[6], C[7], C[8], C[9], C[10], C[11], C[12], C[13], C[14], C[15], C[16], C[17], C[18], C[19], C[20]});

sol := {C[1] = -47924741305/102784672458-_t2[9], C[2] = 1720500130/3953256633, C[3] = 11884850/3953256633, C[4] = -50258975515/68523114972, C[5] = 541534543997/411138689832+2*_t2[9], C[6] = 16133935319/15813026532+_t2[9], C[7] = -2978296060/3953256633-_t2[9], C[8] = -678014633201/411138689832-_t2[9], C[9] = _t2[9], C[10] = 1205383690663/411138689832-_t2[9], C[11] = -414364093825/102784672458-_t2[9], C[12] = -183552176881/31626053064-5*_t2[9], C[13] = 317846441245/45682076648+2*_t2[9], C[14] = 1174900140593/205569344916+4*_t2[9], C[15] = -134222887433/15813026532-_t2[9], C[16] = 3368371335205/411138689832+_t2[9], C[17] = -4023197448145/205569344916-5*_t2[9], C[18] = 4880488105321/205569344916+_t2[9], C[19] = -3600468868531/102784672458+3*_t2[9], C[20] = 5044889409173/102784672458+2*_t2[9]}

>evalf(eval([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9], A[10], A[11], A[12], A[13], A[14], A[15], A[16], A[17], A[18], A[19], A[20]], sol));#verification
[20.05368301 = 20., 9.902120141 = 10., 14.09313678 = 14.,

36.10750974 = 36., 74.70219227 = 75., 58.95343161 = 59.,

179.9968379 = 180., 123.0106360 = 123., 209. = 209.,

251.9896515 = 252., 295. = 295., 177.8791953 = 178.,

128.2699242 = 128., 194.9197414 = 195., 137. = 137.,

475.1387566 = 475., 277.3828957 = 277., 79.99353217 = 80.,

395.0077614 = 395., 375.5989944 = 376.]

and

>DirectSearch:-SolveEquations([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9], A[10], A[11], A[12], A[13], A[14], A[15], A[16], A[17], A[18], A[19], A[20]], AllSolutions, solutions = 1);

[[0.544447729933713,[[[` 1 .. 20 `Vector[column]],[`Data Type: `anything],[`Storage: `rectangular],[`Order: `Fortran_order]]],[C[1]=781.067303954156,C[2]=0.435210834444738,C[3]=0.00300634094034436,C[4]=-0.733460109096348,C[5]=-1561.74997708889,C[6]=-780.513273529185,C[7]=780.780189574613,C[8]=779.884453305800,C[9]=-781.533567447190,C[10]=784.465385170961,C[11]=777.502186818234,C[12]=3901.86400922012,C[13]=-1556.10934178345,C[14]=-3120.41892306702,C[15]=773.045446348276,C[16]=-773.340781401454,C[17]=3888.09684003972,C[18]=-757.792244705146,C[19]=-2379.62994154859,C[20]=-1513.98501970966],3842]]

AS.mw

@Carl Love Or the Explore command:

 Explore(plot3d( [(5+w*cos(v))*cos(u), (5+w*cos(v))*sin(u), w*sin(v)], u= 0..2*Pi, v= 0..2*Pi))

@Carl Love 

>P := varepsilon^3+varepsilon^2+varepsilon+1;
>convert(map(c ->piecewise(2 <= degree(c), 0, c), [op(P)]), `+`);
1 + varepsilon
>selectremove(c -> 2 <= degree(c) , P)[2];
1 + varepsilon

PS. I think that was asked and answered a lot.

In order to obtain a concrete solution, one should evaluate parameters. For example,


restart; x := -(1/2*(-lambda^2*sigma^2-2*lambda*mu+2*RootOf(-exp(_Z)*erf((1/4)*sqrt(2)*(lambda^2*sigma^2+2*_Z)/(lambda*sigma))+exp(_Z)+erf((1/4)*sqrt(2)*(-lambda^2*sigma^2+2*_Z)/(lambda*sigma))+2*y-1)))/lambda
``

evalf(allvalues(eval(x, [lambda = 2, sigma = 1, mu = 2, y = 3/10])))

1.903267698

(1)

allvalues(eval(x, [lambda = 2, sigma = 1, mu = 2, y = .3]))

1.903267698

(2)

``

If values of some parameters are given as floats, then evalf is not necessary.

Download evaluating.mw

This can be done as follows, making use of the shadebetween command.

plots:-shadebetween(-y^2, x^2, y = 0 .. x, x = 0 .. 1, axes = frame, filled, style = surface);

 

plots:-shadebetween(-y^2, x^2, y = 0 .. x, x = 0 .. 1, axes = frame, filled, style = surface)

 

NULL

Edit. Ranges.

Download shadebetween1.mw

 

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