abelian 1.0.1

Classes and methods

The most important classes are listed below. The software contains many other functions and methods not listed.

• The LCA class represents elementary LCAs, i.e. R, Z, T = R/Z, Z_n and direct sums of these groups.

  • Fundamental methods: identity LCA, direct sums, equality, isomorphic, element projection, Pontryagin dual.

• The HomLCA class represents homomorphisms between LCAs.

  • Fundamental methods: identity morphism, zero morphism, equality, composition, evaluation, stacking, element-wise operations, kernel, cokernel, image, coimage, dual (adjoint) morphism.

• The LCAFunc class represents functions from LCAs to complex numbers.

  • Fundamental methods: evaluation, composition, shift (translation), pullback, pushforward, point-wise operators (i.e. addition).

Example

The following example shows Fourier analysis on a hexagonal lattice.

We create a Gaussian on R^2 and a homomorphism for sampling.

from abelian import LCA, HomLCA, LCAFunc, voronoi
from math import exp, pi, sqrt
Z = LCA(orders = [0], discrete = [True])
R = LCA(orders = [0], discrete = [False])

# Create the Gaussian function on R^2
function = LCAFunc(lambda x: exp(-pi*sum(j**2 for j in x)), domain = R**2)

# Create an hexagonal sampling homomorphism (lattice on R^2)
phi = HomLCA([[1, 1/2], [0, sqrt(3)/2]], source = Z**2, target = R**2)
phi = phi * (1/7) # Downcale the hexagon
function_sampled = function.pullback(phi)

Next we approximate the two-dimensional integral of the Gaussian.

# Approximate the two dimensional integral of the Gaussian
scaling_factor = phi.A.det()
integral_sum = 0
for element in phi.source.elements_by_maxnorm(list(range(20))):
    integral_sum += function_sampled(element)
print(integral_sum * scaling_factor) # 0.999999997457763

We use the FFT to move approximate the Fourier transform of the Gaussian.

# Sample, periodize and take DFT of the Gaussian
phi_p = HomLCA([[10, 0], [0, 10]], source = Z**2, target = Z**2)
periodized = function_sampled.pushforward(phi_p.cokernel())
dual_func = periodized.dft()

# Interpret the output of the DFT on R^2
phi_periodize_ann = phi_p.annihilator()

# Compute a Voronoi transversal function, interpret on R^2
sigma = voronoi(phi.dual(), norm_p=2)
factor = phi_p.A.det() * scaling_factor
total_error = 0
for element in dual_func.domain.elements_by_maxnorm():
    value = dual_func(element)
    coords_on_R = sigma(phi_periodize_ann(element))

    # The Gaussian is invariant under Fourier transformation, so we can
    # compare the error using the analytical expression
    true_val = function(coords_on_R)
    approximated_val = abs(value)
    total_error += abs(true_val - approximated_val*factor)

assert total_error < 10e-15

Please see the documentation for more examples and information.