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Home : Support : Online Help : System : Information : Updates : Maple 18 : Advanced Mathematics

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Advanced Math

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

Fractals

Graph Theory

Group Theory

Numerical Integration with Cuba Library

QDifferenceEquations

RootOf

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

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 .

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.

$G ≔ GroupTheory :- Group (Perm ([[1, 2]]), Perm ([[2, 3, 4]])) &semi;$

$G := ⟨(1, 2), (2, 3, 4)⟩$

(3.1)

$GroupTheory :- CycleIndexPolynomial (G, [a, b, c, d]) &semi;$

$\frac{1}{24} a^{4} + \frac{1}{8} b^{2} + \frac{1}{4} b + \frac{1}{3} c + \frac{1}{4} d$

(3.2)

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 multidimensional numerical integration. For more details, see the help page for the Cuba library .

Example

$spikes ≔ Statistics :- Sample (Uniform (0, 1), [6, 4]) &semi;$

$integrand := add (mul (\ln (1.7 |x_{i} - {spikes}_{i, j}|), i = 1 .. 6), j = 1 .. 4)$

$integrand := \ln (1.7 |x_{1} - 0.814723686393179|) \ln (1.7 |x_{2} - 0.905791937075619|) \ln (1.7 |x_{3} - 0.126986816293506|) \ln (1.7 |x_{4} - 0.913375856139019|) \ln (1.7 |x_{5} - 0.632359246225410|) \ln (1.7 |x_{6} - 0.0975404049994095|) + \ln (1.7 |x_{1} - 0.278498218867048|) \ln (1.7 |x_{2} - 0.546881519204984|) \ln (1.7 |x_{3} - 0.957506835434298|) \ln (1.7 |x_{4} - 0.964888535199277|) \ln (1.7 |x_{5} - 0.157613081677548|) \ln (1.7 |x_{6} - 0.970592781760616|) + \ln (1.7 |x_{1} - 0.957166948242946|) \ln (1.7 |x_{2} - 0.485375648722841|) \ln (1.7 |x_{3} - 0.800280468888800|) \ln (1.7 |x_{4} - 0.141886338627215|) \ln (1.7 |x_{5} - 0.421761282626275|) \ln (1.7 |x_{6} - 0.915735525189067|) + \ln (1.7 |x_{1} - 0.792207329559554|) \ln (1.7 |x_{2} - 0.959492426392903|) \ln (1.7 |x_{3} - 0.655740699156587|) \ln (1.7 |x_{4} - 0.0357116785741896|) \ln (1.7 |x_{5} - 0.849129305868777|) \ln (1.7 |x_{6} - 0.933993247757551|)$

(4.1)

$region ≔ [seq (x_{i} = 0 .. 1, i = 1 .. 6)] &semi;$

$region := [x_{1} = 0 .. 1, x_{2} = 0 .. 1, x_{3} = 0 .. 1, x_{4} = 0 .. 1, x_{5} = 0 .. 1, x_{6} = 0 .. 1]$

(4.2)

$int (integrand, region,' numeric',' epsilon' = .001,' method =_CubaSuave',' methodoptions = [flatness = 1, nnew = 10000]') &semi;$

$1.52943702315794$

(4.3)

QDifferenceEquations

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

$L ≔ (x^{2} - 1) \cdot Q - (q^{2} \cdot x^{2} - 1) &colon;$

•

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

$C ≔ QDifferenceEquations :- Closure (L, Q, x, q)$

$C := [(x^{2} - 1) Q - q^{2} x^{2} + 1, (- q x - 1) Q^{2} + (q^{3} x + q^{2} + q x + 1) Q - (q x + 1) q^{2}, (q x - 1) Q^{2} + (- q^{3} x + q^{2} - q x + 1) Q + (q x - 1) q^{2}]$

(5.1)

•

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

$M ≔ QDifferenceEquations :- Desingularize (L, Q, x, q)$

$M := Q^{2} + (- q^{2} - 1) Q + q^{2}$

(5.2)

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.

$allvalues (RootOf (x &Hat; 2 - x - 1, 1 / 2))$

$\frac{1}{2} \sqrt{5} + \frac{1}{2}, \frac{1}{2} - \frac{1}{2} \sqrt{5}$

(6.1)

$RootOf (x &Hat; 2 - x - 1, 1 .. 2)$

$RootOf ({_Z}^{2} -_Z - 1, 1 .. 2)$

(6.2)

$evalf ()$

$1.618033989$

(6.3)

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

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