Abstract
Let us now consider a more interesting example of a graph G, one whose eigenvalues have come up in a variety of applications. Let \(\mathbb {Z}_2\) denote the cyclic group of order 2, with elements 0 and 1 and group operation being addition modulo 2.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For abelian groups other than \(\mathbb {Z}_2^n\) it is necessary to use complex numbers rather than real numbers. We could use complex numbers here, but there is no need to do so.
- 2.
Recall from linear algebra that nonzero orthogonal vectors in a real vector space are linearly independent.
References
E.D. Bolker, The finite Radon transform, in Integral Geometry, Brunswick, Maine, 1984. Contemporary Mathematics, vol. 63 (American Mathematical Society, Providence, 1987), pp. 27–50
M.R. DeDeo, E. Velasquez, The Radon transform on \(\mathbb {Z}_n^k\). SIAM J. Discrete Math. 18, 472–478 (electronic) (2004/2005)
P. Diaconis, R.L. Graham, The Radon transform on \(\mathbb {Z}_2^k\). Pac. J. Math. 118, 323–345 (1985)
T.W. Hawkins, The origins of the theory of group characters. Arch. Hist. Exact Sci. 7, 142–170 (1970/1971)
T.W. Hawkins, New light on Frobenius’ creation of the theory of group characters. Arch. Hist. Exact Sci. 12, 217–243 (1974)
J.P.S. Kung, Radon transforms in combinatorics and lattice theory, in Combinatorics and Ordered Sets, Arcata, CA, 1985. Contemporary Mathematics, vol. 57 (American Mathematical Society, Providence, 1986), pp. 33–74
J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig 69, 262–277 (1917); Translation by P.C. Parks, On the determination of functions from their integral values along certain manifolds. IEEE Trans. Med. Imaging 5, 170–176 (1986)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Stanley, R.P. (2018). Cubes and the Radon Transform. In: Algebraic Combinatorics. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77173-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-77173-1_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77172-4
Online ISBN: 978-3-319-77173-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)