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Open Newton–Cotes...

vv 13837
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 13837
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 13837
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 13837
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 13837
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 13837
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 13837
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 13837
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 13837
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 13837
February 09 2018
0 2

Use

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

and you will have a 1214 x 1356  Matrix.

Numeric solution...

vv 13837
February 09 2018
1 2

It's a specially cooked system.

 

> 

restart;
> 

sys:={
21000 = ((10/1000)/60)*4181*983*(x-y),
H = -((10/1000)/60)*4181*ln(1-((x-y)/(x-40))),
H = 124+102*ln(x-40)+(7+2*ln(x-40)) };

{21000 = (4109923/6000)*x-(4109923/6000)*y, H = -(4181/6000)*ln(1-(x-y)/(x-40)), H = 131+104*ln(x-40)}

 (1)
> 

evalf[1000]( fsolve(sys, {x,y,H}, {x=50..100,y=30..100,H=400..500} )   );

{H = 486.9792717948501357867918018059386067855156547114174326869481418810967243566285398975787784552235204706669347559475195400290118640303716801287372634993661346148619910232495800569296330392394830627549743601915872902357242427148731055409553705001658423528217321565927918891190491572966479397271232198426513612569355038417041808873881442623841262343853418832298327693593790610240451083265130917877844785166083247626127110701898215838963336513184408835894194697247808253085787062731557015693465725077077852896999228843274976631083570662482036300138497124715790019203338925750433405104219573337388245857036034857797986214888125128213468426540193206945839888044251684815660805344557214322036920099275934107239459489909624991382336777779198068497280647366466732283885118035797277802575588920390167295368604982886417919721008912823848353938665076022764471765288194352037535498835268165004232995241166817494393305369865683989720565493276815812710671414695060596389471870171101177308498370799633961387014797870, x = 70.65750866865388962274962328977939489377294903091858411945917234945764190715981783600325358893585110961932863462405500054380580852731304211782069883061069513954397685796059926183531905585579097223962590053390294660021611110475792368859465250322207982971943756610525306678494949905387521858682023969792135750167009335860887007621303198544321368035359069175536312026362350448543477228656103669649753257153387272484737179648441014252898713320281206372610926492497110938583549273159942143012717266939216706865489265272371561718110362755576369288566379279748261854103496737662967801146998815897672787725783471077732933162054089997390996937068458728288379330756605842335927722582205633540658909087373621564984005454626854462329726540402400771083530825578130314424765827873541484037247233111699579267679369342370990364580403457147196305388562501611449040359242137916479015561482141032930179069021129023984141254333721654117596419374749676562545752857082962606395129368846731712926167576535994552285681007055, y = 40.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000958347308577476849155817266967208600479833803954882640800358625509636683984970528639967775046602065201922053597715478431016498682880079609787467682537323162308315760688601560985179585421913717270699604525982018646370272917837250840635796722371496634076989812851876433207535565380863444548970077618286474517739106107056439827996365038277140944772163420380808436265855437736016956601251559483196367236991702787708084161571078609077635832578435472524929603687671421461968192303179415316459681254027553729208024619289830444214278717989841247739063311010334060299714878719957897139070573801145583905093544680767085382367073228848646299934993482758918092665796726848329309766789651450780060432670622190}

 (2)
> 

evalf(%);

{H = 486.9792718, x = 70.65750867, y = 40.00000000}

 (3)
> 

 

approx...

vv 13837
February 04 2018
0 3

The exact answer should be Pr(infinity), where

Pr := proc(n::posint) local k; 
add((2/3)^k/3*`if`(is(sin(k)<=1/2),1,0),k=0..n)
end:

which of course cannot be computed by Maple.
Maple answers:
Pr(13); evalf(%);
     2839595/4782969
     0.5936887736
which is an approximation (not too good).

evalf[50](Pr(300));
    .59502775989667091140797025425657732088366035178640


 

 

Not periodic...

vv 13837
February 04 2018
0 0

You cannot because the function is not periodic (for any values of the parameters).

MultiplicativeOrder...

vv 13837
February 03 2018
0 0 
findOrderOf:= proc(g::posint, p::prime)
  local G,i;
  G:=1;
  for i from 1 to p-1 do
    G:=g*G mod p;
    if(G = 1) then return i end if;
  end do;  
end proc;

Maple has it builtin:  NumberTheory:-MultiplicativeOrder

You should read showstat(NumberTheory:-MultiplicativeOrder)
and try to understand the algorithm for an efficient implementation.

 

typo...

vv 13837
February 03 2018
1 2 
f := n -> sum(i^3, i=1..n);

But note that f(...) will fail if i is assigned, so a better definition is:

f := n -> sum('i'^3, 'i'=1..n);

 

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