Markiyan Hirnyk

Markiyan Hirnyk

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11 years, 21 days

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These are answers submitted by Markiyan Hirnyk

For example, the output of

restart; printlevel := 10: Student[MultivariateCalculus]:-LagrangeMultipliers(x*y, [x^2+y^2-1], [x, y]);

contains, in particular,

{x^2+y^2-1, -2*y*lambda[1]+x = 0, -2*x*lambda[1]+y = 0}

LM1.mw

because s and N may be  complex numbers so branch cuts should be taken into consideration. If we simplify matters, then Maple does the job

simplify((1/s^2)^N, symbolic);
                             (-2 N)
                            s      

 

if I correctly understand you. The matter is the same plot is demonstrated from different viewing angles. Here is a simple example 


 

plot3d(x^2+2*y^2, orientation = [20, 40, 70]);plot3d(x^2+2*y^2, orientation = [20, 40, 70])

 

plot3d(x^2+2*y^2, orientation = [20, 70, 40]);plot3d(x^2+2*y^2, orientation = [20, 70, 40])

 

``


 

Download orientation.mw

This can be done in Maple 2016 as follows:


 

with(NumberTheory):``

q := NumberTheory:-RepeatingDecimal(-37/31)

_m400140804

(1)

``


 

Download recurring_decimal.mw

-


 

Try


NULL

NULL

NULL

NULL

NULL

Digits := 15; -1; Fct := (1/8)*sqrt(3)*(-4*exp((2*I)*beta)*beta^2*Ei(1, (2*I)*beta)+beta^2*(-(7/4*I)*beta+(5/8)*beta^2-1)*exp(I*beta)*Ei(1, I*beta)+(5/8*I)*beta^3-(5/2*I)*beta+(9/8)*beta^2+9/4); -1; with(plots); -1; semilogplot(Re(Fct), beta = 10^(-3) .. 10^4, 0 .. .6)

 

NULL

NULL

NULL

 

 

NULL


Download digits.mw

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.

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.

 

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.

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