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Letters...

vv 14112
February 24 2018
1 3

Drawing letters is a trivial task in Maple.

It's a matter of defining the polygon(s). It would be nice if someone with artistic abilities would design them.

Note that for all the letters except i and j a single polygon is enough.

For example:

> 

with(plottools):with(plots):
> 

A := [[3,0],[4,0],[.5,10.5],[-.5,10.5],[-4,0],[-3,0],[-2,3],[2, 3],[1.65,4.05],[-1.65,4.05],[0, 9]]:
> 

display(polygon(A), axes=none);

 

After that, in Maple 2017 we can do this:

> 

AA:=extrude(display(polygon([A]), scaling=constrained), 0..1.2,style=surface, axes=none):
> 

animate(display@rotate,[AA,t, [[0,0,0],[0,1,0]] ], t=0..2*Pi,paraminfo=false,orientation=[-90,-15,25]);
> 

 

 

42 indeed...

vv 14112
February 23 2018
0 0
> 

# Next number
> 

f:=proc(n::{posint,name})
local t, k, f1, i;
f1 := (t,k) -> mul(t-i,i=1..k-1)*mul(t-i,i=k+1..nops(f(0)));
convert(add(f(0)[k]*f1(n,k)/f1(k,k),k=1..nops(f(0))),horner)
end:
> 

f(0):=[3,10,2,7,7,42]:
seq(f(k),k=1..7);
f(x);

3, 10, 2, 7, 7, 42, 314

  

-197+(6212/15+(-3571/12+(195/2+(-179/12+(13/15)*x)*x)*x)*x)*x

 (1)
> 

f(0):=[3,10,2,7,7,13]:
seq(f(k),k=1..7);
f(x);

3, 10, 2, 7, 7, 13, 140

  

-168+(4175/12+(-5837/24+(1847/24+(-271/24+(5/8)*x)*x)*x)*x)*x

 (2)
> 

f(0):=[1,2,4,8,16];
seq(f(k),k=1..6);
f(x);

[1, 2, 4, 8, 16]

  

1, 2, 4, 8, 16, 31

  

1+(-3/4+(23/24+(-1/4+(1/24)*x)*x)*x)*x

 (3)
> 

f(0):=[1,2,4,8,16,32];
seq(f(k),k=1..9);

[1, 2, 4, 8, 16, 32]

  

1, 2, 4, 8, 16, 32, 63, 120, 219

 (4)
> 

 

Factorization...

vv 14112
February 22 2018
1 0

Your  algebraic fractions al already simplified (but there are of course other possibilities, e.g. partial fractions).
For a factorization, the field K is needed.
For K=Q, the polynomials are already factorized.
If you want K=C (complete factorization) the simplest way is to find the roots with solve. E.g.

p:=(k+2)*(k^2+5)*(k^3-5*k^2+13*k-8)/(6*k*(k^2-3*k+8)):
p1,p2:=(numer,denom)(p):
R1,R2:=[solve(p1,k,explicit)],[solve(p2,k,explicit)]:
mul(k-r,r=R1)/mul(k-r,r=R2);

Solution...

vv 14112
February 22 2018
5 2

Actually, the solution of your ODE is

A(t) = a * exp( I * ( a^2*t + b ) ),  where  a, b are real constants.

This can be obtained with Maple writing A(t) in polar form:

restart;
ode := diff(A(t),t) - I * conjugate(A(t))*A(t)^2:
simplify(eval(ode, A(t)=R(t)*exp(I*phi(t)) )  /exp(I*phi(t)) ) assuming real:
evalc([Re,Im](%)):
dsolve(%);

Workaround...

vv 14112
February 21 2018
1 1

restart;
a := Matrix(3, 3, [[x, y, z], [y, z, x], [z, x, y]]):
ex:= Matrix(3, 3, [[x, y, z], [y, z, x], [z, x, y]]) + B:
#eval(ex, a=A); #does not work
evalindets(ex, 'Matrix', u -> `if`(EqualEntries(u,a),A,u));
                             B + A
evalindets(ex, 'Matrix', u -> `if`(LinearAlgebra[Equal](u,a),A,u));
                             B + A

 

Open Newton–Cotes...

vv 14112
February 19 2018
0 1 
# Open Newton–Cotes (N=n-1)
ONC:=proc(n::posint)
local nodes:=[seq(k/n,k=1..n-1)], B,w,i,k,x;
B:=i -> [seq(`if`(k=i,1,0),k=1..n-1)]:
w:=[seq(int(CurveFitting:-PolynomialInterpolation(nodes,B(i),x),x=0..1),i=1..n-1)] ;
`1/(b-a)`.'c'=w,'p'=2*floor(n/2)+1
end:

ONC(5);
       `1/(b-a)` . c = [11/24, 1/24, 1/24, 11/24], p = 5

Solution...

vv 14112
February 17 2018
1 1 
Optimization:-Minimize(
0,
{
cos(q[1])*cos(q[3])-sin(q[1])*sin(q[3])+sin(q[1])*q[2]+cos(q[1]) - 1 = 0,
sin(q[1])*cos(q[3])+cos(q[1])*sin(q[3])-cos(q[1])*q[2]+sin(q[1]) -1 = 0
},


q[1] = 2 ..  Pi, q[2] = 1 .. 1.5, q[3] = -3 .. -2

#   q[1] = 0 ..  Pi, q[2] = 0.1 .. 1.5, q[3] = -Pi/2 .. 4*Pi/3
# ,  initialpoint = [q[1]=2.0944, q[2]=1.4, q[3]=-2.7925]

);

       [0., [q[1] = 2.30431169727399, q[2] = 1.03640893001776, q[3] = -2.75622305752699]]

Expansion...

vv 14112
February 17 2018
1 2
> 

int_part:=proc(f,h,t,n::integer)
local k,u,v,s;
u:=f;
v:=h;
s:=0;
for k from 1 to n do;
  u:=int(u,t);
  s:=s-(-1)^(k)*u*v;
  v:=diff(v,t);
od;
s;
end:

> 

int_part(exp(-x*t), 1/sqrt(t*(t+1)), t, 6):
A:=unapply(simplify(-eval(%,t=1)),x);

proc (x) options operator, arrow; (1/2048)*exp(-x)*2^(1/2)*(1024*x^5-768*x^4+1216*x^3-3024*x^2+10404*x-46035)/x^6 end proc

 (1)
> 

J:= x -> Int((exp(-x*t))/sqrt(t*(t+1)),t=1..infinity):
> 

evalf[20](eval( [A(x), A(x)-J(x)], x=20));  # check

[0.70333100226385465863e-10, -0.211984348249630e-15]

 (2)
> 

 

workaround...

vv 14112
February 17 2018
0 1

It seems that Maple needs help here.

solve({combine(7*cos(2*t)=7*cos(t)^2-5), t>=0, t<=2*Pi}, t, allsolutions, explicit);

by transform...

vv 14112
February 17 2018
1 5
> 

restart;
> 

with(plots):with(plottools):
p:=display(textplot([0,0,"A"]), 'font'=["times","roman",200],size=[210,200],axes=none ):
q:=display(textplot([0,0,"B"]), 'font'=["times","roman",200],size=[210,200],axes=none ):
FA:="A.png": FB:="B.png":  plottools:-exportplot(FA,p):  plottools:-exportplot(FB,q):
A:=plot3d(1, x=-1..1, y=-1..1,  image =FA): B:=plot3d(1, x=-1..1, y=-1..1,  image =FB):
> 

f := (u,a,b) -> transform( (x,y,z) -> [a+x*cos(u)-y*sin(u),b+x*sin(u)+y*cos(u)]):
display(f(0,0,0)(A),f(Pi/6,4,0)(A),f(-Pi/3,2,-2)(B), axes = none,scaling=constrained);

 
> 

 

semilog...

vv 14112
February 16 2018
2 3
> 

u:=Re(exp(1/4-(1/4)*signum(x))*cos((1/2)*ln(abs(x))/Pi)-I*exp(1/4-(1/4)*signum(x))*sin((1/2)*ln(abs(x))/Pi)):
> 

u1:=simplify(u) assuming x>0;
u2:=simplify(u) assuming x<0;

cos((1/2)*ln(x)/Pi)

  

exp(1/2)*cos((1/2)*ln(-x)/Pi)

 (1)

limsup(u, x=infinity) = limsup(u1, x=infinity) =1  should be clear now. Also for liminf.

 

To see the oscillations, a semilog plot is needed.

> 

plots:-semilogplot(u,x=10..10^100);

 
> 

 

3d letters...

vv 14112
February 11 2018
0 0

With Maple the approach should be completely different, see:
https://www.mapleprimes.com/questions/204335-Can-Rotate-3d-Text-Like-This-Be-Done-In-Maple

Cholesky...

vv 14112
February 10 2018
2 4

 
> 

A := Matrix(3, 3, [[2, -3, 1], [-3, 5, 0], [1, 0, 5]]):
> 

L:=LinearAlgebra:-LUDecomposition(A, method='Cholesky')^*;

Warning, Matrix is not positive-definite

  

_rtable[18446744074366462782]

 (1)
> 

A - L^* . L; # check

_rtable[18446744074366456886]

 (2)

f(x)(x)...

vv 14112
February 10 2018
0 3

f := (x-2)^2+1:
ff := f(x);
    (x(x)-2)^2+1

So, ff is a mathematical nonsense but syntactically correct.
Because  const(x)  simplifies to  const  we have e.g.
eval(ff, x=12);
    101

So, actually ff will work in numerical computations if eval is used.

Minimize(g, {x >= -2, x <= 2});
gives a syntax error. Use:

Minimize(g, -2..1);
          [2., Vector[column](1, [1.])]


About the piecewise stuff.

The Optimization package assumes that the objective function and constraints are twice continuously differentiable.
You cannot expect correct results if they are not so.
 

ImportMatrix...

vv 14112
February 09 2018
0 2

Use

A:=ImportMatrix("yourpath\\-Validierung-Stahl-AI-.txt");

and you will have a 1214 x 1356  Matrix.

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