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intat...

vv 13065
October 10 2024
3 0

intat(f(x), x=a)  computes an antiderivative F(x) of f(x) and returns F(a).

Two antiderivatives may differ by an additive constant, depending e,g. on the method used to compute it.

 intat is useful when it is desired to subsequently perform a change of variable, or in the case of symbolic manipulation of the solutions of differential equations.

 

So, for example,

> 

intat(x^2, x=1);

1/3

 (1)

has not much sense, unless you know which antiderivative was computed.

 

> 

restart;
> 

f := sin(x)*cos(x) + 1/x;

sin(x)*cos(x)+1/x

 (2)
> 

F1 := intat(f, x=a);

ln(a)+(1/2)*sin(a)^2

 (3)
> 

F2 := intat(simplify(f), x=a);

ln(a)-(1/4)*cos(2*a)

 (4)
> 

simplify(F1-F2);

1/4

 (5)
> 

F3 := Int(f, x=0..a); value(%);

Int(sin(x)*cos(x)+1/x, x = 0 .. a)

  

infinity

 (6)

Note. When Intat(f(x), x=a)  is replaced by Int(f(x). x=0..a), odesolve will verify it, but 0 could be out of dom(f), so, nonsense!

 

Area only but fast...

vv 13065
October 07 2024
2 1 
1/2*evalf(Int(min(-6*cos(t), 2-2*cos(t))^2, t=Pi/2..3*Pi/2));
identify(%);
#                          15.70796327
#                              5 Pi

Using Edge is ok...

vv 13065
October 02 2024
1 0

Using Edge is ok, Firefox too; maybe try some font settings:

 

Pell type reduction...

vv 13065
September 30 2024
4 0

The answer for your previous problem works here too.

restart;
eq:=x^2+y^2-N*(1+x*y):  # [x=0, y=sqrt(N)] and [x=sqrt(N),y=0] are solutions!
XY:=[X=x, Y=2*y-x*N]: xy:=solve(XY,[x,y])[]:
EQ:=simplify(eval(eq,xy)):
for N in [9, 49 , 729] do
  SOL:=isolve(EQ):
  sol[N]:=map(u -> simplify(eval(xy, u)), [SOL]);
  num:={seq}(simplify(sol[N])[],_Z1=0..2);
  print('N'=N, select(type,num, [anything=integer,anything=integer]));
od:

N = 9, {[x = -1497363, y = -13307787], 
  [x = -1497363, y = -168480], [x = -168480, y = -1497363], 
  [x = -168480, y = -18957], [x = -18957, y = -168480], 
  [x = -18957, y = -2133], [x = -2133, y = -18957], 
  [x = -2133, y = -240], [x = -240, y = -2133], 
  [x = -240, y = -27], [x = -27, y = -240], [x = -27, y = -3], 
  [x = -3, y = -27], [x = -3, y = 0], [x = 0, y = -3], 
  [x = 0, y = 3], [x = 3, y = 0], [x = 3, y = 27], 
  [x = 27, y = 3], [x = 27, y = 240], [x = 240, y = 27], 
  [x = 240, y = 2133], [x = 2133, y = 240], 
  [x = 2133, y = 18957], [x = 18957, y = 2133], 
  [x = 18957, y = 168480], [x = 168480, y = 18957], 
  [x = 168480, y = 1497363], [x = 1497363, y = 168480], 
  [x = 1497363, y = 13307787]}

N = 49, {[x = -96687343207, y = -4735705783543], 
  [x = -96687343207, y = -1974033600], 
  [x = -1974033600, y = -96687343207], 
  [x = -1974033600, y = -40303193], 
  [x = -40303193, y = -1974033600], [x = -40303193, y = -822857], 
  [x = -822857, y = -40303193], [x = -822857, y = -16800], 
  [x = -16800, y = -822857], [x = -16800, y = -343], 
  [x = -343, y = -16800], [x = -343, y = -7], [x = -7, y = -343], 
  [x = -7, y = 0], [x = 0, y = -7], [x = 0, y = 7], 
  [x = 7, y = 0], [x = 7, y = 343], [x = 343, y = 7], 
  [x = 343, y = 16800], [x = 16800, y = 343], 
  [x = 16800, y = 822857], [x = 822857, y = 16800], 
  [x = 822857, y = 40303193], [x = 40303193, y = 822857], 
  [x = 40303193, y = 1974033600], [x = 1974033600, y = 40303193], 
  [x = 1974033600, y = 96687343207], 
  [x = 96687343207, y = 1974033600], 
  [x = 96687343207, y = 4735705783543]}

N = 729, {[x = -4052517025117644747, y = -2954279352292037818803], 
  [x = -4052517025117644747, y = -5559018725201760], 
  [x = -5559018725201760, y = -4052517025117644747], 
  [x = -5559018725201760, y = -7625554438293], 
  [x = -7625554438293, y = -5559018725201760], 
  [x = -7625554438293, y = -10460313837], 
  [x = -10460313837, y = -7625554438293], 
  [x = -10460313837, y = -14348880], 
  [x = -14348880, y = -10460313837], [x = -14348880, y = -19683], 
  [x = -19683, y = -14348880], [x = -19683, y = -27], 
  [x = -27, y = -19683], [x = -27, y = 0], [x = 0, y = -27], 
  [x = 0, y = 27], [x = 27, y = 0], [x = 27, y = 19683], 
  [x = 19683, y = 27], [x = 19683, y = 14348880], 
  [x = 14348880, y = 19683], [x = 14348880, y = 10460313837], 
  [x = 10460313837, y = 14348880], 
  [x = 10460313837, y = 7625554438293], 
  [x = 7625554438293, y = 10460313837], 
  [x = 7625554438293, y = 5559018725201760], 
  [x = 5559018725201760, y = 7625554438293], 
  [x = 5559018725201760, y = 4052517025117644747], 
  [x = 4052517025117644747, y = 5559018725201760], 
  [x = 4052517025117644747, y = 2954279352292037818803]}

With sol[N] you may inspect the general solution depending on _Z1.


 

Solution...

vv 13065
September 30 2024
5 1

 

isolve is old and should be updated.

Actually isolve knows to solve generalized Pell equations. It only needs a little help to convert the diophantine quadratic to generalized Pell form.

At the end we must filter the obtained numeric solutions in order to eliminate the non-integer ones due to the form of xy, see below.

 

> 

restart;
> 

eq:=x^2 - 12*x*y + 6*y^2 + 4*x + 12*y - 3:
> 

XY:=[X = 2*x-12*y+4, Y = -5*y+3]; # via complete squares

[X = 2*x-12*y+4, Y = -5*y+3]

 (1)
> 

xy:=solve(XY,[x,y])[];

[x = (1/2)*X+8/5-(6/5)*Y, y = -(1/5)*Y+3/5]

 (2)
> 

EQ:=simplify(eval(eq,xy));

(1/4)*X^2+19/5-(6/5)*Y^2

 (3)
> 

SOL:=isolve( EQ ):
> 

sol:=map(u -> simplify(eval(xy, u)), [SOL]);

[[x = (1/10)*(30^(1/2)+7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-30^(1/2)+7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(3*30^(1/2)-17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-3*30^(1/2)-17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-3*30^(1/2)-17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(3*30^(1/2)-17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-30^(1/2)+7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(30^(1/2)+7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(30^(1/2)-7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-30^(1/2)-7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(3*30^(1/2)+17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-3*30^(1/2)+17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-3*30^(1/2)+17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(3*30^(1/2)+17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-30^(1/2)-7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(30^(1/2)-7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)-12)*(11+2*30^(1/2))^_Z1]]

 (4)
> 

nops(sol)

8

 (5)
> 

num:={seq}(simplify(sol)[],_Z1=0..20): # some numeric solutions
> 

numsols:=select(type,num,  [anything=integer,anything=integer]);

{[x = 3, y = 1], [x = 3, y = 3], [x = 5, y = 1], [x = 5, y = 7], [x = 29, y = 3], [x = 29, y = 53], [x = 75, y = 7], [x = 75, y = 141], [x = 603, y = 53], [x = 603, y = 1151], [x = 1613, y = 141], [x = 1613, y = 3083], [x = 13205, y = 1151], [x = 13205, y = 25257], [x = 35379, y = 3083], [x = 35379, y = 67673], [x = 289875, y = 25257], [x = 289875, y = 554491], [x = 776693, y = 67673], [x = 776693, y = 1485711], [x = 6364013, y = 554491], [x = 6364013, y = 12173533], [x = 17051835, y = 1485711], [x = 17051835, y = 32617957], [x = 139718379, y = 12173533], [x = 139718379, y = 267263223], [x = 374363645, y = 32617957], [x = 374363645, y = 716109331], [x = 3067440293, y = 267263223], [x = 3067440293, y = 5867617361], [x = 8218948323, y = 716109331], [x = 8218948323, y = 15721787313], [x = 67343968035, y = 5867617361], [x = 67343968035, y = 128820318707], [x = 180442499429, y = 15721787313], [x = 180442499429, y = 345163211543], [x = 1478499856445, y = 128820318707], [x = 1478499856445, y = 2828179394181], [x = 3961516039083, y = 345163211543], [x = 3961516039083, y = 7577868866621], [x = 32459652873723, y = 2828179394181], [x = 32459652873723, y = 62091126353263], [x = 86972910360365, y = 7577868866621], [x = 86972910360365, y = 166367951854107], [x = 712633863365429, y = 62091126353263], [x = 712633863365429, y = 1363176600377593], [x = 1909442511888915, y = 166367951854107], [x = 1909442511888915, y = 3652517071923721], [x = 15645485341165683, y = 1363176600377593], [x = 15645485341165683, y = 29927794081953771], [x = 41920762351195733, y = 3652517071923721], [x = 41920762351195733, y = 80189007630467743], [x = 343488043642279565, y = 29927794081953771], [x = 343488043642279565, y = 657048293202605357], [x = 920347329214417179, y = 80189007630467743], [x = 920347329214417179, y = 1760505650798366613], [x = 7541091474788984715, y = 657048293202605357], [x = 7541091474788984715, y = 14425134656375364071], [x = 20205720480365982173, y = 1760505650798366613], [x = 20205720480365982173, y = 38650935309933597731], [x = 165560524401715384133, y = 14425134656375364071], [x = 165560524401715384133, y = 316695914147055404193], [x = 443605503238837190595, y = 38650935309933597731], [x = 443605503238837190595, y = 848560071167740783457], [x = 3634790445362949466179, y = 316695914147055404193], [x = 3634790445362949466179, y = 6952884976578843528163], [x = 9739115350774052210885, y = 848560071167740783457], [x = 9739115350774052210885, y = 18629670630380363638311], [x = 79799829273583172871773, y = 6952884976578843528163], [x = 79799829273583172871773, y = 152646773570587502215381], [x = 213816932213790311448843, y = 18629670630380363638311], [x = 213816932213790311448843, y = 409004193797200259259373], [x = 1751961453573466853712795, y = 152646773570587502215381], [x = 1751961453573466853712795, y = 3351276133576346205210207], [x = 4694233393352612799663629, y = 409004193797200259259373], [x = 4694233393352612799663629, y = 8979462592908025340067883], [x = 38463352149342687608809685, y = 3351276133576346205210207], [x = 38463352149342687608809685, y = 73575428165109029012409161], [x = 103059317721543691281150963, y = 8979462592908025340067883], [x = 103059317721543691281150963, y = 197139172850179357222234041], [x = 844441785831965660540100243, y = 73575428165109029012409161], [x = 2262610756480608595385657525, y = 197139172850179357222234041]}

 (6)
> 

nops(numsols);

82

 (7)
> 

{seq}( eval(eq, s), s=numsols); #check

{0}

 (8)
> 

 

 

Download dioph-sols-vv.mw

andmap...

vv 13065
September 26 2024
3 0

 

select(u -> nops(u)=m and andmap(isprime,u), L);

 

DirectSearch...

vv 13065
September 24 2024
3 1

 

> 

restart;
> 

dis:= (A,B) -> sqrt( (A[1]-B[1])^2 + (A[2]-B[2])^2 ):
P:= unapply(dis(A,B)+dis(B,C)+dis(C,A), [A,B,C]):
> 

A0:=[0,0]:
A1:=[u1,0]:
A2:=[u2,u3]:
A3:=v1*~A1:
A4:=A1+v2*~(A2-A1):
A5:=A1+v3*~(A2-A1):
A6:=A0+v4*~(A2-A0):
A7:=A0+v5*~(A2-A0):
A8:=A6+v6*~(A4-A6):
> 

eps:=0.05:
> 

perim:=
P(A0,A3,A6)=10, P(A6,A3,A4)=15, P(A3,A1,A4)=11,
P(A6,A8,A7)=9,  P(A8,A5,A7)=13, P(A8,A4,A5)=12,
P(A7,A5,A2)=20:
> 

vars := u1=1..50, u2=1..50, u3=1..50,
v1=eps ..1-eps, v2=eps ..1-eps, v3=eps ..1-eps, v4=eps ..1-eps, v5=eps ..1-eps, v6=eps ..1-eps:
> 

inisol:=[u1 = 8, u2 = 3, u3 = 12, v1 = 0.5, v2 = 0.3, v3 = 0.5, v4 = 0.09, v5 = 0.3, v6 = 0.5]:
> 

sol:=DirectSearch:-SolveEquations([perim], [vars], initialpoint=inisol );

[7.928429359693041*10^(-17), Vector(7, {(1) = 0.1798504456e-8, (2) = -0.1393335225e-8, (3) = -0.7304805649e-9, (4) = 0.7992083439e-8, (5) = -0.1359651947e-8, (6) = 0.1542570516e-8, (7) = 0.2339469063e-8}), [u1 = 8.211758237046912, u2 = 1.8442345202064532, u3 = 12.033442054587029, v1 = .5526589984562774, v2 = .26313930832995064, v3 = .5414796127553622, v4 = 0.7943775626427665e-1, v5 = .2941739734925204, v6 = .485843681048224], 5405]

 (1)
> 

plots:-display(plot(eval([A0,A1,A2,A0],sol[3])), plot(eval([A3,A4,A6,A3],sol[3])), plot(eval([A5,A7,A8,A5],sol[3])), scaling=constrained);

 
> 

eval([perim], sol[3]);

[HFloat(10.000000001798504) = 10, HFloat(14.999999998606665) = 15, HFloat(10.99999999926952) = 11, HFloat(9.000000007992083) = 9, HFloat(12.999999998640348) = 13, HFloat(12.00000000154257) = 12, HFloat(20.00000000233947) = 20]

 (2)
> 

eval(seq("A"||i=A||i,i=0..8), sol[3])

"A0" = [0, 0], "A1" = [HFloat(8.211758237046912), 0], "A2" = [HFloat(1.8442345202064532), HFloat(12.033442054587029)], "A3" = [HFloat(4.538302082851432), 0], "A4" = [HFloat(6.536212450422957), HFloat(3.166471619072571)], "A5" = [HFloat(4.763873960641556), HFloat(6.515863543831874)], "A6" = [HFloat(0.14650185231032542), HFloat(0.9559096369525809)], "A7" = [HFloat(0.5425257968612042), HFloat(3.539925463989865)], "A8" = [HFloat(3.2509023701302153), HFloat(2.029897207531015)]

 (3)
> 

 


(Edited - a typo).

Download perims-vv.mw

infinite...

vv 13065
September 20 2024
3 1

The curve length is infinite. The integrand has a singularity at t=ln(Pi/2):

restart;
f:= t -> sqrt((sin(t/cos(exp(t))) + t*(1/cos(exp(t)) + t*exp(t)*sin(exp(t))/cos(exp(t))^2)*cos(t/cos(exp(t))))^2 + (cos(t) - t*sin(t))^2 + 2*t):
a:=ln(Pi/2): evalf(a);
#                          0.4515827054

Digits:=15:
:-int(f, a..a+1/10, numeric, epsilon=1e-5);
#                     3.33001574946940* 10^13

D...

vv 13065
September 19 2024
0 3 
Optimization:-Minimize(AD, {CD=AB-2, BC=AB+2, AD=AB+BC+CD, AB>=1, BC>=1, CD>=1}, assume=integer);
#        [9, [AB = 3, AD = 9, BC = 5, CD = 1]]

 

D...

vv 13065
September 18 2024
2 0 
eval(E-A, solve({C-A + D-B + E-C = a, D-B = b}, {A,B,C,D,E}));
#                             -b + a

 

A more precise version and a simple proo...

vv 13065
September 17 2024
4 3

 

A more precise version and a simple proof

 

The cited result is true only when the curbe is not "too flat" and in those cases the result of the limit is different.

 

To simplify things we shall consider the Ox axes as one of the tangents to the curve.

So, we shall take  f(0) = 0, f'(0) = 0.

 

 

The tangent at the point P(a, f(a)) intersects the Ox axis for the abscissa  c = a - f(a)/f'(a);

 

So, the ratio r of the two areas is

 

> 

restart;
> 

r := (a*f(a)/2 - int(f(t),t=0..a))/ (1/2 * (a-f(a)/D(f)(a))* f(a));

2*((1/2)*a*f(a)-(int(f(t), t = 0 .. a)))/((a-f(a)/(D(f))(a))*f(a))

 (1)
> 

# f(0) = 0, f'(0)=0

0

 (2)
> 

eval( r,  f = (u -> u^2*g(u)) ):
series(%, a);

series(2/3+(2*(-(1/30)*((D@@2)(g))(0)+(1/12)*(D(g))(0)^2/g(0))/g(0))*a^2+O(a^3),a,3)

 (3)

Hence  limit(r, a=0)  = 2/3   indeed.

 

What happens if f is more "flat".  E.g. f(0) = 0, f'(0)=0, f''(0)=0

 

> 

eval( r,  f = (u -> u^3*g(u)) ):
series(%, a);

series(3/4+((1/40)*(D(g))(0)/g(0))*a+O(a^2),a,2)

 (4)

And so on; if the first nonzero derivative (at 0) is the 10th we obtain:

 

> 

eval( r,  f = (u -> u^10*g(u)) ):
series(%, a, 14);

series(10/11+((2/297)*(D(g))(0)/g(0))*a+(2*((7/2574)*((D@@2)(g))(0)-(31/10692)*(D(g))(0)^2/g(0))/g(0))*a^2+O(a^3),a,3)

 (5)

So, the limit will be  k/(k+1)  it  the first nonzero derivative of f at 0 is the k-th.


 

Download voller-vv.mw

NLPSolve...

vv 13065
September 07 2024
2 9

Optimization:-NLPSolve works very well and fast without the need of any simplification.

restart;
Digits:=15:
e:=(x,y) -> x^2 + 2*y^2 - 1:
h:=(x,y) -> (x-sin(x))^2 + (y-sin(y))^2 - 1:
constr:=
e(x1,y1)=0, h(x2,y2)=0, h(x3,y3)=0, r>=0,
D[1](e)(x1,y1)*(y1-y0) - D[2](e)(x1,y1)*(x1-x0) = 0,
D[1](h)(x2,y2)*(y2-y0) - D[2](h)(x2,y2)*(x2-x0) = 0,
D[1](h)(x3,y3)*(y3-y0) - D[2](h)(x3,y3)*(x3-x0) = 0,
(x1-x0)^2 + (y1-y0)^2 = r^2,
(x2-x0)^2 + (y2-y0)^2 = r^2,
(x3-x0)^2 + (y3-y0)^2 = r^2,
x3 >= x2 + 1/10:

sol:=Optimization:-Maximize(r, {constr},   x0=1..1.4, y0=1..1.4, x1=0..1, y1=0..1, x2=1..2, y2=1..2, x3=1..2, y3=1..2);

sol := [0.769863723979874459, [r = 0.769863723979874, x0 = 1.13805882552303, x1 = 0.699729170630851, x2 = 1.27446720511737, x3 = 1.89574141571275, y0 = 1.13805882552303, y1 = 0.505162888466810, y2 = 1.89574141571275, y3 = 1.27446720511737]]

p1:=plots:-implicitplot(e, -2..2, -2..2): p2:=plots:-implicitplot(h, -2..2, -2..2):
p3:=plottools:-circle(eval([x0,y0],sol[2]), eval(r,sol[2]), color=red):
p4:=plots:-pointplot(eval([[x0,y0],[x1,y1],[x2,y2],[x3,y3]],sol[2]), symbolsize=8, color=blue):
p5:=plots:-textplot(eval({seq}([x||i,y||i,P__||i], i=0..3), sol[2]), align={below, left}, color=blue):
plots:-display(p1,p2,p3,p4,p5);

row...

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September 04 2024
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data_new := DataFrame(data_2, rows=[1,2]);

 

Bezout...

vv 13065
August 28 2024
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For a system of m polynomial equations of (total) degrees d_1, ..., d_m in n unknowns over C (or other algebraically closed field), the standard result is given by Bezout' theorem:

For m=n, the number of solutions is either infinite or <= d_1* d_2 * ...* d_m.

For details see: Bézout's theorem - Wikipedia

 

COLOR...

vv 13065
August 28 2024
2 0 
restart;
# f := floor;
f := x -> x^2:
X := [seq(x, x=-3..3,0.02)]: n:=nops(X):
Y := map(f, X):
C := Array(1..n, i -> X[i], datatype=float[8]):
plot(X,Y, color=COLOR(HUE, C), thickness=6);

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