Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

Feinsilver, Philip; McSorley, John P.

doi:10.1155/2011/539030

Cited by 8 publications

(6 citation statements)

References 3 publications

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“…Proposition 1.2. Let u ∈ CZ n , and let κ denote the index of nilpotency 4 of Du. It follows that u is uniquely invertible if and only if Cu = 0, and the inverse is given by…”

Section: Multiplicative Properties Of Zeonsmentioning

confidence: 99%

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A Spectral Theorem for Zeon Matrices

Staples¹

2022

Preprint

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In this paper, a spectral theorem for matrices with (complex) zeon entries is established. In particular, it is shown that when A is an m × m self-adjoint matrix whose characteristic polynomial χA(u) splits over the zeon algebra CZ n , there exist m spectrally simple eigenvalues λ1, . . . , λm and m linearly independent normalized zeon eigenvectors v1, . . . , vm such that A = m j=1 λjπj, where πj = vjvj † is a rank-one projection onto the zeon submodule span{vj} for j = 1, . . . , m.

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“…Proposition 1.2. Let u ∈ CZ n , and let κ denote the index of nilpotency 4 of Du. It follows that u is uniquely invertible if and only if Cu = 0, and the inverse is given by…”

Section: Multiplicative Properties Of Zeonsmentioning

confidence: 99%

“…A permanent trace formula analogous to MacMahon's Master Theorem was presented and applied by Feinsilver and McSorley in [4], where the connections of zeons with permutation groups acting on sets and the Johnson association scheme were illustrated.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Theorem for Zeon Matrices

Staples¹

2022

Preprint

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“…3 The term "dual" here is motivated by regarding zeons as higher-dimensional dual numbers. 4 In particular, κ is the least positive integer such that (Du) κ = 0.…”

Section: Multiplicative Properties Of Zeonsmentioning

confidence: 99%

Spectrally Simple Zeros of Zeon Polynomials

Staples

2021

Preprint

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Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons.In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods.

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“…I⊆[n]u I ζ I . Combinatorial properties of zeons have been developed in a number of works in recent years[3,4,14,16].…”

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confidence: 99%

Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution

Staples

2017

Int J Theor Phys

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Operators are induced on fermion and zeon algebras by the action of adjacency matrices and combinatorial Laplacians on the vector spaces spanned by the graph's vertices. Properties of the algebras automatically give information about the graph's spanning trees and vertex coverings by cycles & matchings. Combining the properties of operators induced on fermions and zeons gives a fermion-zeon convolution that recovers the number of Hamiltonian cycles in an arbitrary graph. The mathematics underlying the graph-theoretic interpretation of these operators is provided by Kirchhoff's theorem and by the seminal works of Goulden and Jackson and Liu, who established formulas for enumeration of Hamiltonian cycles and paths using determinants and permanents of adjacency matrices.

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Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

Cited by 8 publications

References 3 publications

A Spectral Theorem for Zeon Matrices

A Spectral Theorem for Zeon Matrices

Spectrally Simple Zeros of Zeon Polynomials

Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution

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