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by transform...

vv 13992
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 13992
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 13992
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 13992
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 13992
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 13992
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 13992
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 13992
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 13992
February 04 2018
0 0

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

MultiplicativeOrder...

vv 13992
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 13992
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);

 

solve...

vv 13992
February 02 2018
2 2

solve(y>0);

or

Y:=numer(y)*denom(y);
SolveTools:-SemiAlgebraic([Y>0]);

obtain the answer in a few seconds (Maple 2017).

But there are 56 regions and it depends on your final aim to use them.

Edit. You may try to change the order of the variables and hope for simpler regions.

simple...

vv 13992
February 01 2018
0 0

The Maple implementation NumberTheory:-Totient is of course faster (for large numbers), but the simplest one is probably:

eulerphi:=(n::posint) -> add(`if`(igcd(k,n)=1,1,0),k=1..n);

 

typo...

vv 13992
January 30 2018
0 0

Should be

solve(SysEqE,{diff(phi[1](t),t),diff(phi[3](t),t)});

op...

vv 13992
January 29 2018
0 0

op(0,a);

But you should try to see how this appeared, because it is a nonsense (probably from a programming error).

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