Markiyan Hirnyk

Markiyan Hirnyk
10 years, 311 days


These are answers submitted by Markiyan Hirnyk

The apply rule command was likely introduced in Maple V R5 (see screen05.09.16.docx).

As far as I understand your question, you are interested only in real solutions. The answer is complicated. The parametric plane (m,u) can be splitted into 24 domains in each of those the number of the real solutions of

F(z):=m*z^4-4*m*z^3+(3*m+3)*z^2-6*u in z is constant. This can be done as follows.



with(RootFinding[Parametric]):

sys := [m*z^4-4*m*z^3+(3*m+3)*z^2-6*u = 0]:

s := CellDecomposition(sys, [z]);

[[[Equations,=, [m z^4-4 m z^3+(3 m+3) z^2-6 u]],[Inequalities,=, []],[Filter,=, 0<>1],[Variables,=, [z]],[Parameters,=, [m,u]],[DiscriminantVariety,=, [[m],[u],[3 m^4-48 m^3 u-64 m^2 u^2+192 m^2 u-18 m^2-48 m u-24 m-9]]],[ProjectionPolynomials,=, [[u,2 u-9,16 u-9,2 u-1],[m,3 m^4-48 m^3 u-64 m^2 u^2+192 m^2 u-18 m^2-48 m u-24 m-9]]],[SamplePoints,=, [[m=-21,u=-1],[m=-10,u=-1],[m=2,u=-1],[m=5,u=-1],[m=-1,u=1/4],[m=-117438644421/1099511627776,u=1/4],[m=1,u=1/4],[m=3,u=1/4],[m=-1,u=17/32],[m=-41246484769/549755813888,u=17/32],[m=1,u=17/32],[m=147009845109/68719476736,u=17/32],[m=3,u=17/32],[m=5,u=17/32],[m=-1,u=3],[m=-14774057851/549755813888,u=3],[m=24,u=3],[m=49,u=3],[m=-3,u=5],[m=-1,u=5],[m=-1054275706337/4398046511104,u=5],[m=-5162024001/274877906944,u=5],[m=41,u=5],[m=83,u=5]]]]

(1)

RootFinding:-Parametric:-CellPlot(s, 'samplepoints')

 

NumberOfSolutions(s)

[[1, 4], [2, 2], [3, 0], [4, 2], [5, 2], [6, 4], [7, 2], [8, 4], [9, 2], [10, 4], [11, 2], [12, 4], [13, 2], [14, 4], [15, 2], [16, 4], [17, 2], [18, 4], [19, 2], [20, 0], [21, 2], [22, 4], [23, 2], [24, 4]]

(2)

``


See RootFinding,Parametric for more info.

Download number_of_real_solutions.mw

Similar questions were asked and answered a mountain.

You wrote "Only difference is that I wrote  (a+b)/sqrt(b^2-a^2) in one, and  sqrt(a+b)/sqrt(b-a) in the other. But these two expressions are the same".

This statement is not true. The parameters a and b are treated as complex by default. Let us consider

evalc(eval((a+b)/sqrt(-a^2+b^2) = sqrt(a+b)/sqrt(b-a), [a = -I, b = -3*I]))

-(1/2)*sqrt(8) = sqrt(2)

evalf(%);
-1.414213562 = 1.414213562

substitution.mw

See Maple help to this command for info.

Another way

July 20 2016 Markiyan Hirnyk 7001

Just an idea. One can create an empty plot with a background e.g. plot(10,x=0..16, view=[0..16,0..9],background=...).

After that one can apply the plottools:-transform command to it. I am busy today. Because of this reason, an example will be presented later.

 

By inequal

June 21 2016 Markiyan Hirnyk 7001

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 7001

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 7001

>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 7001

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

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