## Congruence of one polynomial modulo another...

Definition Let "F"  be a field with f(x), g(x), p(x) being polynomials in F[x] and "p(x)" nonzero. Then if "p(x)|f(x)-g(x)" we say "f(x)" is congruent to "g(x)" modulo "p(x)".

For example, in Q[x], x^2+x+1 is congruent to x+2 modulo x+1 because ï¼ˆx^2+x+1)-(x+2) = x^2-1=(x+1)(x-1).

This congruence among polynomials is similar, but not quite the same as congruence among integers.

Is there a way, in Maple, to solve problems like this? Could Maple tell me, for example, what polynomials are congruent to x^2+x+1 modulo x+1? An answer might be in the form x^2 +x+1 belongs to [x-1], an  equivalence class of polynomials.

I have not been able to find any such function in Maple, nor any calculator for it on the web. Maybe I'm not using the right search terms.

## how to draw this region...

Hello, consider 0<x<1, 0<y<1 and t >0 (t: fixed)

Let the region:
R={(x,y) | x*(1-y)/y*(1-x) < t}

How can I draw the region R in Maple. thanks.

## Unable to solve for w?...

I am attaching the sheet, I want to solve for w or find range of w.
q1.mw

## why Maple sometimes solves an ode using method dif...

I noticed sometimes Maple dsolve solves an ode using a method different than what odeadvisor says it is.

In this example ode, advisor said it is separable. But when solving it, dsolve actually solved it as dAlembert.

Why is that? Should not these be the same?

I also noticed Maple does not verify the solution of the ode when asked to solve it as separable, which is what the advisor says. But it does verify the solution using dAlembert.

So in this example, should not advisor have said this ode is dAlembert and not separable then?

 > interface(version);

 > Physics:-Version();

 > ode:=y(x) = x + 3*ln(diff(y(x), x)); DEtools:-odeadvisor(ode);

 > infolevel[dsolve]:=5: sol_1:=dsolve(ode);

Methods for first order ODEs:

-> Solving 1st order ODE of high degree, 1st attempt

trying 1st order WeierstrassP solution for high degree ODE

trying 1st order WeierstrassPPrime solution for high degree ODE

trying 1st order JacobiSN solution for high degree ODE

trying 1st order ODE linearizable_by_differentiation

trying differential order: 1; missing variables

trying d'Alembert

<- d'Alembert successful

 > map(X->odetest(X,ode),[sol_1]);

 > sol_2:=dsolve(ode,[separable]);

Classification methods on request

Methods to be used are: [separable]

----------------------------

* Tackling ODE using method: separable

--- Trying classification methods ---

trying separable

<- separable successful

 > #notice, solution does not verify. odetest(sol_2,ode);

## print content of overloaded proc from a package...

I use a procedure called Isee  to print to screen procedures from my packages.

However if the procedure is overloaded it Isee doesn't print it. Is there a way around this.
I have inserted screen shots to show the outputs for the package.

 > restart
 > Test:=proc(a::{vector}) print(a); end proc;
 (1)
 (2)
 > Isee := proc(a) interface(verboseproc = 3); printf("%P", eval(a)); end proc;
 (3)
 > Isee(Test)
 proc(a::{vector})     print(a); end proc
 > Isee(Test1)
 proc()     option overload;     [proc(a::{vector}) option overload; print(a); end proc,         proc(a::{Matrix}) option overload; print(a); end proc]; end proc
 > #with(Routines); #
 > #An non overloaded procedure in a package
 > Isee(ConicMatrix); #
 >
 ConicMatrix
 > #An overloaded procecure in a package
 > Isee(FactReduce); #
 FactReduce

## Rossler Attractor with animations...

I was looking at the application center about attractors and found the Rossler attractor app that illustrates the Rossler Attractor with animations, as you can see below. But when I try to run it on my laptop  the two last plots remain empty. Why is this happening?

Rossler Flow System - Rossler Attractor

by Yufang Hao, <yhao@student.math.uwaterloo.ca>

This worksheet contains the images of the Rossler Attractor and the animations that follow the trajectory.

 > restart; with(DEtools): with(plots):
 Warning, the name changecoords has been redefined

The Rossler attractor is defined by a set of three Differential equations:

x' =

y' =

z' = b +  -

where the coefficients a, b, and c are adjustable constants.

 > rosslerEqns := [ diff(x(t),t) = -(y(t)+z(t)), diff(y(t),t) = x(t) + a*y(t), diff(z(t),t) = b + x(t)*z(t) - c*z(t) ];
 (1)
 > a:=0.17: b:=0.4: c:=8.5: DEplot3d(rosslerEqns, {x(t),y(t),z(t)}, t=0..300,          [[x(0)=0, y(0)=0, z(0)=0]],          x =-15..15, y=-15..15,z=-5..25,          stepsize=0.05, linecolour=1+sin(t*Pi/3)/2,          thickness=1, orientation = [-110,71]);
 > a:=0.17: b:=0.4: c:=8.5: display(   [seq(     DEplot3d(rosslerEqns, {x(t),y(t),z(t)}, t=0..4*i,          [[x(0)=0, y(0)=0, z(0)=0]],          x =-15..15, y=-15..15,z=-5..25,          stepsize=0.05, linecolour=1+sin((i-t)*Pi/5)/2,          thickness=2, orientation = [-110,71]),     i=1..25) # end seq   ], # end DEplot3d list insequence=true);
 > a:=0.17: b:=0.4: c:=8.5: display(   [seq(     DEplot3d(rosslerEqns, {x(t),y(t),z(t)}, t=0..4*i,          [[x(0)=0, y(0)=0, z(0)=0]],          x =-15..15, y=-15..15,z=-5..25,          stepsize=0.05, linecolour=1+sin((i-t)*Pi/5)/2,          thickness=2, orientation = [-110,71]),     i=1..25) # end seq   ], # end DEplot3d list insequence=true);
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## From radians to degrees & degrees, minutes and sec...

1) Maple gives me the result in radians and I want it in degrees, example:

evalf(17*sin(34)/sin(115)) = 9.513506993

The result in degrees should be 10.48901874

2) How do I transform degrees into degrees, minutes and seconds?

Example: 15.925º

= 15º 55' 30''

## simplify equation befor convert the math equatio...

when i do the convert in maple to latex  is do but not fully simplify and some kind of clearer must write for paper and i must do this case by case by hand but how i can simplify before i convert to latex and remove all extra thing like multiply between two squar root

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 (1)
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 \left[A_{1} = 0, A_{0} = 0, B_{1} = \mp \frac{\sqrt{2}\, \sqrt{a_{5}}}{\sqrt{a_{4}}}, k = k, a_{2} = -a_{5}, w = -\frac{2 a_{5} a_{3} \left(4 k^{2}-1\right)}{3 a_{4}}, a_{1} = \frac{8 a_{5} a_{3}}{3 a_{4}}, v = 2 a_{1} k\right]
 >

## Is it possible to animate spacecurves that fade wi...

In this post about a vibrating T-shaped structure, the ends of the T are traced over time.
The trace of the end encircled below in yellow fades with time

How to do the same with Maple? For example, can an attractor be animated this way?

## Problem finding the center of the circumcircle to ...

restart;
_local(D, O);
with(Student:-MultivariateCalculus);
A := [0, 0, 0];
B := [a, 0, 0];
C := [a, b, 0];
D := [0, b, 0];
S := [0, 0, h];
O := [x, y, z];
lineSC := Line(S, C);
lineSD := Line(S, D);
H := Projection(A, lineSC);
K := Projection(A, lineSD);
OH := H - O;
OK := K - O;
OC := C - O;
M := Matrix([OH, OK, OC]);
O := eval(O, %);
simplify(Distance(O, H));
O

Error, invalid input: eval received Matrix(3, 3, {(1, 1) = -x+h^2*a/(a^2+b^2+h^2), (1, 2) = -y+h^2*b/(a^2+b^2+h^2), (1, 3) = -z+h*(a^2+b^2)/(a^2+b^2+h^2), (2, 1) = -x, (2, 2) = -y+h^2*b/(b^2+h^2), (2, 3) = -z+h*b^2/(b^2+h^2), (3, 1) = -x+a, (3, 2) = -y+b, (3, 3) = -z}), which is not valid for its 2nd argument, eqns
How to correct this error ? Thank you.

## why am i facing problem while using two loops in ...

i am writing code for an iterative process at the end i want to evaluate the summation expression with two loops but it is not evaluating kindly help me out here automatic_differentiation.mw

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 u_1 = -alpha*u[1]*(20*u[2]-20*u[0])+1600*u[0]-3200*u[1]+1600*u[2] u_2 = -alpha*u[2]*(20*u[3]-20*u[1])+1600*u[1]-3200*u[2]+1600*u[3] u_3 = -alpha*u[3]*(20*u[4]-20*u[2])+1600*u[2]-3200*u[3]+1600*u[4] u_4 = -alpha*u[4]*(20*u[5]-20*u[3])+1600*u[3]-3200*u[4]+1600*u[5] u_5 = -alpha*u[5]*(20*u[6]-20*u[4])+1600*u[4]-3200*u[5]+1600*u[6] u_6 = -alpha*u[6]*(20*u[7]-20*u[5])+1600*u[5]-3200*u[6]+1600*u[7] u_7 = -alpha*u[7]*(20*u[8]-20*u[6])+1600*u[6]-3200*u[7]+1600*u[8] u_8 = -alpha*u[8]*(20*u[9]-20*u[7])+1600*u[7]-3200*u[8]+1600*u[9] u_9 = -alpha*u[9]*(20*u[10]-20*u[8])+1600*u[8]-3200*u[9]+1600*u[10] u_10 = -alpha*u[10]*(20*u[11]-20*u[9])+1600*u[9]-3200*u[10]+1600*u[11] u_11 = -alpha*u[11]*(20*u[12]-20*u[10])+1600*u[10]-3200*u[11]+1600*u[12] u_12 = -alpha*u[12]*(20*u[13]-20*u[11])+1600*u[11]-3200*u[12]+1600*u[13] u_13 = -alpha*u[13]*(20*u[14]-20*u[12])+1600*u[12]-3200*u[13]+1600*u[14] u_14 = -alpha*u[14]*(20*u[15]-20*u[13])+1600*u[13]-3200*u[14]+1600*u[15] u_15 = -alpha*u[15]*(20*u[16]-20*u[14])+1600*u[14]-3200*u[15]+1600*u[16] u_16 = -alpha*u[16]*(20*u[17]-20*u[15])+1600*u[15]-3200*u[16]+1600*u[17] u_17 = -alpha*u[17]*(20*u[18]-20*u[16])+1600*u[16]-3200*u[17]+1600*u[18] u_18 = -alpha*u[18]*(20*u[19]-20*u[17])+1600*u[17]-3200*u[18]+1600*u[19] u_19 = -alpha*u[19]*(20*u[20]-20*u[18])+1600*u[18]-3200*u[19]+1600*u[20] u_20 = -alpha*u[20]*(20*u[21]-20*u[19])+1600*u[19]-3200*u[20]+1600*u[21] u_21 = -alpha*u[21]*(20*u[22]-20*u[20])+1600*u[20]-3200*u[21]+1600*u[22] u_22 = -alpha*u[22]*(20*u[23]-20*u[21])+1600*u[21]-3200*u[22]+1600*u[23] u_23 = -alpha*u[23]*(20*u[24]-20*u[22])+1600*u[22]-3200*u[23]+1600*u[24] u_24 = -alpha*u[24]*(20*u[25]-20*u[23])+1600*u[23]-3200*u[24]+1600*u[25] u_25 = -alpha*u[25]*(20*u[26]-20*u[24])+1600*u[24]-3200*u[25]+1600*u[26] u_26 = -alpha*u[26]*(20*u[27]-20*u[25])+1600*u[25]-3200*u[26]+1600*u[27] u_27 = -alpha*u[27]*(20*u[28]-20*u[26])+1600*u[26]-3200*u[27]+1600*u[28] u_28 = -alpha*u[28]*(20*u[29]-20*u[27])+1600*u[27]-3200*u[28]+1600*u[29] u_29 = -alpha*u[29]*(20*u[30]-20*u[28])+1600*u[28]-3200*u[29]+1600*u[30] u_30 = -alpha*u[30]*(20*u[31]-20*u[29])+1600*u[29]-3200*u[30]+1600*u[31] u_31 = -alpha*u[31]*(20*u[32]-20*u[30])+1600*u[30]-3200*u[31]+1600*u[32] u_32 = -alpha*u[32]*(20*u[33]-20*u[31])+1600*u[31]-3200*u[32]+1600*u[33] u_33 = -alpha*u[33]*(20*u[34]-20*u[32])+1600*u[32]-3200*u[33]+1600*u[34] u_34 = -alpha*u[34]*(20*u[35]-20*u[33])+1600*u[33]-3200*u[34]+1600*u[35] u_35 = -alpha*u[35]*(20*u[36]-20*u[34])+1600*u[34]-3200*u[35]+1600*u[36] u_36 = -alpha*u[36]*(20*u[37]-20*u[35])+1600*u[35]-3200*u[36]+1600*u[37] u_37 = -alpha*u[37]*(20*u[38]-20*u[36])+1600*u[36]-3200*u[37]+1600*u[38] u_38 = -alpha*u[38]*(20*u[39]-20*u[37])+1600*u[37]-3200*u[38]+1600*u[39] u_39 = -alpha*u[39]*(20*u[40]-20*u[38])+1600*u[38]-3200*u[39]+1600*u[40]
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## integral of 1/ln(x) in Maple...

How to get Li(x) to display as result of int(1/ln(x),x) in Maple instead of -Ei(1,-ln(x))

I'd like to match result of Maple with another software I use and it is also simpler to look at

int(1/ln(x),x)

Now gives

-Ei(1,-ln(x))

How to make it show  Li(x) instead?

The Maple help page for Li is this

And the other software Li help page is this