Advanced Math

Maple 18 includes numerous cutting-edge updates in a variety of branches of mathematics:

 

Fractals

Maple 18 features a new package for generating Fractals. This includes various fractal generators, such as BurningShip, Julia, Lyapunov, Mandelbrot, and Newton. For more, see Fractals in Maple 18.

BurningShip 

Julia

Lyapunov  

Mandelbrot

Newton

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Graph Theory

Several improvements and enhancements have been made to the GraphTheory package including the new Latex command for generating code for displaying a graph using the LaTeX picture environment. 

 

For more, see Updates to Graph Theory

 Image 

 

Group Theory

There are numerous improvements for Group Theory, including a new library of Perfect Groups. Other new commands include: 

  • AbelianInvariants: Compute the Abelian invariants of a finitely presented group.
  • CycleIndexPolynomial: Return the degree of a permutation group.  
  • PresentationComplexity: Return a measure of the complexity of a presentation of a finitely presented group.  
  • Simplify: Simplify the presentation for a group.
> G := GroupTheory:-Group(Perm([[1, 2]]), Perm([[2, 3, 4]])); 1
 
PermutationGroup({thismodule, object}, )
 
> GroupTheory:-CycleIndexPolynomial(G, [a, b, c, d]); 1
 
`+`(`*`(`/`(1, 24), `*`(`^`(a, 4))), `*`(`/`(1, 8), `*`(`^`(b, 2))), `*`(`/`(1, 4), `*`(b)), `*`(`/`(1, 3), `*`(c)), `*`(`/`(1, 4), `*`(d)))
 

For more details, see Updates to Group Theory.

 

Numerical Integration with Cuba Library

Maple 18 provides more methods for numerical integration, adding four routines for high-dimensional numerical integration that rely on the Cuba library for mutlidimensional numerical integration.

 Example

> spikes := Statistics:-Sample(Uniform(0, 1), [6, 4]); 1
 
spikes := Matrix(%id = 18446744078088539662)
 
> integrand := add(mul(ln(`+`(`*`(1.7, `*`(abs(`+`(x[i], `-`(spikes[i, j]))))))), i = 1 .. 6), j = 1 .. 4)
 
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
`+`(`*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[1], `-`(HFloat(0.8147236863931789)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[2], `-`(HFloat(0.9057919370756192)))))))), `*`(ln(`+`(`*`(1.7, `*`(abs(`+`(x[3], `-`(...
 
> region := [seq(x[i] = 0 .. 1, i = 1 .. 6)]; 1
 
[x[1] = 0 .. 1, x[2] = 0 .. 1, x[3] = 0 .. 1, x[4] = 0 .. 1, x[5] = 0 .. 1, x[6] = 0 .. 1]
 
> int(integrand, region, 'numeric', 'epsilon' = 0.1e-2, 'method = _CubaSuave', 'methodoptions = [flatness = 1, nnew = 10000]'); 1
 
HFloat(1.5294370231579368)

 

QDifferenceEquations

The QDifferenceEquations package includes two new commands for working with q-difference operators. 

> L := `+`(`*`(`+`(`*`(`^`(x, 2)), `-`(1)), `*`(Q)), `+`(`-`(`*`(`^`(q, 2), `*`(`^`(x, 2)))), 1)); -1
 

Closure computes the closure in the ring of linear q-difference operators with polynomial coefficients.

> C := QDifferenceEquations:-Closure(L, Q, x, q)
 
[`+`(`*`(`+`(`*`(`^`(x, 2)), `-`(1)), `*`(Q)), `-`(`*`(`^`(q, 2), `*`(`^`(x, 2)))), 1), `+`(`*`(`+`(`-`(`*`(q, `*`(x))), `-`(1)), `*`(`^`(Q, 2))), `*`(`+`(`*`(`^`(q, 3), `*`(x)), `*`(`^`(q, 2)), `*`(q...
[`+`(`*`(`+`(`*`(`^`(x, 2)), `-`(1)), `*`(Q)), `-`(`*`(`^`(q, 2), `*`(`^`(x, 2)))), 1), `+`(`*`(`+`(`-`(`*`(q, `*`(x))), `-`(1)), `*`(`^`(Q, 2))), `*`(`+`(`*`(`^`(q, 3), `*`(x)), `*`(`^`(q, 2)), `*`(q...

Desingularize computes a multiple of a given q-difference operator with fewer singularities.

> M := QDifferenceEquations:-Desingularize(L, Q, x, q)
 
`+`(`*`(`^`(Q, 2)), `*`(`+`(`-`(`*`(`^`(q, 2))), `-`(1)), `*`(Q)), `*`(`^`(q, 2)))
 

For details, see Q-Difference Equation in Maple 18.

 

RootOf

There have been several ease of use enhancements made to the function RootOf, including with numeric, interval, and index selectors. 

> allvalues(RootOf(`+`(`*`(`^`(x, 2)), `-`(x), `-`(1)), `/`(1, 2)))
 
`+`(`*`(`/`(1, 2), `*`(`^`(5, `/`(1, 2)))), `/`(1, 2)), `+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(`^`(5, `/`(1, 2))))))
 
> RootOf(`+`(`*`(`^`(x, 2)), `-`(x), `-`(1)), 1 .. 2)
 
RootOf(`+`(`*`(`^`(_Z, 2)), `-`(_Z), `-`(1)), 1 .. 2)
 
> evalf(RootOf(`+`(`*`(`^`(_Z, 2)), `-`(_Z), `-`(1)), 1 .. 2))
 
1.618033989
 

For more on improvements to RootOf in Maple 18, see the RootOf updates page.