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Kitonum

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17 years, 61 days
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These are answers submitted by Kitonum

CurveFitting:-LeastSquares...

Kitonum 21500
February 17 2019
1 0

Perhaps this is a bug. As a workaround use  CurveFitting:-LeastSquares  instead:

CurveFitting:-LeastSquares(pts1, pts2, x, curve = 3*x+a);
                                     -61/3+3*x

Adjustment...

Kitonum 21500
February 17 2019
1 3 
restart;

Ec := (Ems+I*Eml)*(1+((Ems+I*Eml)/Ef-1)*Zeta*phi/((Ems+I*Eml)/Ef+Zeta))/(1-((Ems+I*Eml)/Ef-1)*phi/((Ems+I*Eml)/Ef+Zeta));

a:=simplify(Re(Ec)) assuming positive;
b:=simplify(Im(Ec)) assuming positive;

a  and  b  are the real and imaginary parts of  Ec .

Adjustment...

Kitonum 21500
February 16 2019
0 0


 

restart

A := binomial(n, k)

`assuming`([limit(A*(lambda/n)^k*(1-lambda/n)^(n-k), n = infinity)], [lambda > 0, lambda < n, k::posint])

lambda^k/(GAMMA(k)*k*exp(lambda))

 (1)

simplify(convert(%, factorial))

lambda^k*exp(-lambda)/factorial(k)

 (2)

`assuming`([limit(eval(A*(lambda/n)^k*(1-lambda/n)^(n-k), k = 0), n = infinity)], [lambda > 0, lambda < n])

exp(-lambda)

 (3)

``


 

Download Poisson.mw

Edit.

Simplier...

Kitonum 21500
February 16 2019
1 1


 

restart

A := binomial(n, k); assume(p >= 0, p <= 1)

binomial(n, k)

 (1)

E := sum(A*p^k*(1-p)^(n-k)*k, k = 0 .. n)

(p/(1-p)+1)^n*p*n*(1-p)^n/((1-p)*(p/(1-p)+1))

 (2)

E := simplify(E)

p*n

 (3)

simplify(sum(A*p^k*(1-p)^(n-k)*(k-E)^2, k = 0 .. n))

-p*n*(-1+p)

 (4)

subs(-1+p = -q, %)

p*n*q

 (5)

NULL


 

Download simpl_new.mw


Edit.

rsolve...

Kitonum 21500
February 14 2019
2 7

You can easily get the explicit formula for this sequence:

rsolve({y(0)=(-sqrt(5)+5)*(1/8), y(n)= 4*y(n-1)*(1-y(n-1))}, y(n));

Adjustment...

Kitonum 21500
February 13 2019
0 0


 

> 

restart;
A[0] := 0;                      
A[1] := sqrt(2*(k[1]^2-w[1]^2))/n;
A[2] := sqrt(2*(k[2]^2-w[2]^2))/n;
c[1] := 1;
c[2] := 1;
c[3] := 1;
c[4] := 1;
c[5] := 1;
c[6] := 1;
k[1] := 10.5;
k[2] := 3.5;
w[1] := 5.05;
w[2] := .5;
m := 1.9;
n := 1.75;
xi[1] := -t*w[1]+x*k[1];
xi[2] := -t*w[2]+x*k[2];
a := m/sqrt(2*(k[1]^2-w[1]^2));
b := m/sqrt(k[2]^2-w[2]^2);
g := a*(c[2]*exp(a*xi[1])+c[3]*exp(-a*xi[1]));
h := c[1]+c[2]*exp(a*xi[1])+c[3]*exp(-a*xi[1]);
G := b*(c[5]*exp(b*xi[2])+c[6]*exp(-b*xi[2]));
H := c[4]+c[5]*exp(b*xi[2])+c[6]*exp(-b*xi[2]);
u := A[0]+A[1]*g/h+A[2]*G/H;

0

  

(2*k[1]^2-2*w[1]^2)^(1/2)/n

  

(2*k[2]^2-2*w[2]^2)^(1/2)/n

  

1

  

1

  

1

  

1

  

1

  

1

  

10.5

  

3.5

  

5.05

  

.5

  

1.9

  

1.75

  

-5.05*t+10.5*x

  

-.5*t+3.5*x

  

.1459402733

  

.5484827558

  

.1459402733*exp(-.7369983802*t+1.532372870*x)+.1459402733*exp(.7369983802*t-1.532372870*x)

  

1+exp(-.7369983802*t+1.532372870*x)+exp(.7369983802*t-1.532372870*x)

  

.5484827558*exp(-.2742413779*t+1.919689645*x)+.5484827558*exp(.2742413779*t-1.919689645*x)

  

1+exp(-.2742413779*t+1.919689645*x)+exp(.2742413779*t-1.919689645*x)

  

7.439442594*(.1459402733*exp(-.7369983802*t+1.532372870*x)+.1459402733*exp(.7369983802*t-1.532372870*x))/(1+exp(-.7369983802*t+1.532372870*x)+exp(.7369983802*t-1.532372870*x))+2.799416849*(.5484827558*exp(-.2742413779*t+1.919689645*x)+.5484827558*exp(.2742413779*t-1.919689645*x))/(1+exp(-.2742413779*t+1.919689645*x)+exp(.2742413779*t-1.919689645*x))

 (1)
> 

plot3d(u, x = -20 .. .20, t = -20 .. .20);
t := 0;
plot(u, x = -15 .. 15);
 

 

0

  

 
> 

 


 

Download plots.mw

Explore...

Kitonum 21500
February 13 2019
2 1

And what if you just write a single line of code that does the same thing without any packages and plot components :

Explore(plot(sin(a*x)+cos(b*x^2), x=0..10, -3..3), a=0..1., b=0..1.);
                    

 

Adjustment...

Kitonum 21500
February 11 2019
2 0

I removed the square brackets that you use to group expressions. To do this, Maple should use only parentheses. Brackets are used to create lists. Some corrections were also made to improve the quality of the graphs:


 

> 

restart;
A[0] := 0;                           
A[1] := sqrt(2*(k[1]^2-w[1]^2))/sqrt(lambda);                             
A[2] := sqrt(2*(k[2]^2-w[2]^2))/sqrt(lambda);                             
c[1] := 1;
c[2] := 1;
c[3] := 1;
c[4] := 1;
c[5] := 1;
c[6] := 1;
k[1] := 10.5;
k[2] := 3.5;
w[1] := 5.05;
w[2] := .5;
m := 1.9;
lambda := 1.75;
xi[1] := -t*w[1]+x*k[1];
xi[2] := -t*w[2]+x*k[2];
a := m/sqrt(k[1]^2-w[1]^2);
b := m/sqrt(k[2]^2-w[2]^2);
g := a*(c[2]*cos(a*xi[1])-c[3]*sin(a*xi[1]));
h := c[1]+c[2]*sin(a*xi[1])+c[3]*cos(a*xi[1]);
G := b*(c[5]*cos(b*xi[2])-c[6]*sin(b*xi[2]));
H := c[4]+c[5]*sin(b*xi[2])+c[6]*cos(b*xi[2]);
u := A[0]+A[1]*g/h+A[2]*G/H;
  

0

  

(2*k[1]^2-2*w[1]^2)^(1/2)/lambda^(1/2)

  

(2*k[2]^2-2*w[2]^2)^(1/2)/lambda^(1/2)

  

1

  

1

  

1

  

1

  

1

  

1

  

10.5

  

3.5

  

5.05

  

.5

  

1.9

  

1.75

  

-5.05*t+10.5*x

  

-.5*t+3.5*x

  

.2063907138

  

.5484827558

  

.2063907138*cos(1.042273105*t-2.167102495*x)+.2063907138*sin(1.042273105*t-2.167102495*x)

  

1-sin(1.042273105*t-2.167102495*x)+cos(1.042273105*t-2.167102495*x)

  

.5484827558*cos(.2742413779*t-1.919689645*x)+.5484827558*sin(.2742413779*t-1.919689645*x)

  

1-sin(.2742413779*t-1.919689645*x)+cos(.2742413779*t-1.919689645*x)

  

9.841457496*(.2063907138*cos(1.042273105*t-2.167102495*x)+.2063907138*sin(1.042273105*t-2.167102495*x))/(1-sin(1.042273105*t-2.167102495*x)+cos(1.042273105*t-2.167102495*x))+3.703280398*(.5484827558*cos(.2742413779*t-1.919689645*x)+.5484827558*sin(.2742413779*t-1.919689645*x))/(1-sin(.2742413779*t-1.919689645*x)+cos(.2742413779*t-1.919689645*x))

 (1)
> 

plot3d(u, x = -10 ..0.2, t = -10 .. 0.2, view=-50..50, grid=[200,200]);
 

 
> 

plot(eval(u,t=0), x = -15 .. 15, -50..50, numpoints=5000, discont, size=[800,400]);

 
> 

 


 

Download 2plots.mw

expand, factor...

Kitonum 21500
February 10 2019
1 0

is(expand(B*exp(I*Pi/3))=expand(A));
                                                                     true

# Or

factor(expand(A/B));
polar(%);


Edit.
 

Closed formula...

Kitonum 21500
February 10 2019
0 1

sum(2^k, k=0..n);
                                               2^(n+1)-1

# Check for n=5
sum(`2`^k, k=0..5)=sum(2^k, k=0..5);
                              2^5+2^4+2^3+2^2+2+1 = 63

We see that  63=2^6-1

This result  2^(n+1)-1  is the same as your answer, because  2*2^n-1 = 2^(n+1)-1

Imaginary unit...

Kitonum 21500
February 10 2019
0 3

The imaginary unit in Maple should be coded as  I  (not as i). i is just a symbol in Maple.

Re((1+I)^2); 
Im((1+I)^2);

                                           0
                                           2

plots:-inequal...

Kitonum 21500
February 07 2019
2 0

You can use  plots:-inequal  command:

restart;
A:=plot(sin(x), x=-3*Pi/4-0.3..9*Pi/4+0.3, color=red, thickness=3):
B:=plot(cos(x), x=-3*Pi/4-0.3..9*Pi/4+0.3, color=blue, thickness=3):
C:=plots:-inequal((y-cos(x))*(y-sin(x))<0, x=-3*Pi/4..9*Pi/4, y=-1.2..1.2, color=yellow):
plots:-display(A,B,C, scaling=constrained, size=[1000,300]);
         


Addition. The  plots:-inequal  command especially useful when a plane region is given by inequalities of the form  F(x,y)<0  or  F(x,y)>0 . Here is an example of painting over the regions obtained by different intersections of three circles:

with(plots):
C1:= (x+sqrt(7))^2+y^2-16: 
C2 := x^2+(y-3)^2-9: 
C3 := (x-4)^2+y^2-25:
Range:=x=-7..9.5, y=-5.5..6.5:
P:=implicitplot([C1,C2,C3], Range, color=black, thickness=2):
R1:=inequal({C1<0,C2<0,C3<0},  Range, color=blue, nolines):
R2:=inequal({{C1<0,C2<0},{C1<0,C3<0},{C2<0,C3<0}},  Range, color=yellow, nolines):
R3:=inequal({{C1<0,C2>0,C3>0},{C1>0,C2<0,C3>0},{C1>0,C2>0,C3<0}}, Range, color=green, nolines):
display(P,R1,R2,R3, scaling=constrained, size=[600,500]);
                 

Adjustment...

Kitonum 21500
February 07 2019
0 0


 

> 

restart
> 

"P(t):=(r*Q[inf])/(2+2 cosh(b-r*t));"

proc (t) options operator, arrow, function_assign; r*Q[inf]/(2+2*cosh(b-r*t)) end proc

 (1)
> 

"Q(t):=(&int;)[0]^(t)P(t) &DifferentialD;t assuming real;"

proc (t) options operator, arrow, function_assign; `assuming`([int(P(t), t = 0 .. t)], [real]) end proc

 (2)
> 

value(Q(t))

(1/2)*Q[inf]*tanh((1/2)*b)-(1/2)*Q[inf]*tanh(-(1/2)*t*r+(1/2)*b)

 (3)
> 

 


 

Download Hubbert-new.mw

Correction...

Kitonum 21500
February 07 2019
0 0

Maybe the following is what you want:


 

> 

restart;
v := 145000;
thetavn := (1/6)*Pi;
omegac := .1;
x:=omegac*t;
s := cos(2*thetabn)*x+2*sin(thetabn)^2*sin(x);
plots:-implicitplot(s = 0, t = -200 .. 200, thetabn = 43*Pi*(1/180) .. 88*Pi*(1/180), tickmarks = ["decimalticks","piticks"], gridrefine = 4, size = [800, 600]);

145000

  

(1/6)*Pi

  

.1

  

.1*t

  

.1*cos(2*thetabn)*t+2*sin(thetabn)^2*sin(.1*t)

  

 
> 

 


 

Download plot.mw

Edit.

Adjustment...

Kitonum 21500
February 07 2019
0 6

I fixed only 1 line:

GAUSSIAN_new.mw

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