Imaginary eigenvalues and complex eigenvectors explained by real geometry

This paper first reviews how anti-symmetric matrices in two dimensions yield imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to rotations by means of the Cayley transformation. Then a real geometric interpretation is given to the eigenvalues and eigenvectors by means of real geometric algebra. The eigenvectors are seen to be \textit{two component eigenspinors} which can be further reduced to underlying vector duplets. The eigenvalues are interpreted as rotation operators, which rotate the underlying vector duplets. The second part of this paper extends and generalizes the treatment to three dimensions. Finally the four-dimensional problem is stated.Comment: 11 page

New views of crystal symmetry guided by profound admiration of the extraordinary works of Grassmann and Clifford

This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis of geometric algebras. Geometric algebras have been expanded to conformal geometric algebras, which provide an ideal framework for modern computer graphics. Within this framework a new visualization of three-dimensional crystallographic space groups has been created. The complex beauty of this new visualization is shown by a range of images of a diamond cell. Mathematical details are given in an appendix.Comment: 11 pages, 8 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1306.159

Quaternion Fourier Transform on Quaternion Fields and Generalizations

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.Comment: 21 page

1-factor and cycle covers of cubic graphs

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if $\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is an upper bound for the girth of $G$. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with $\mu_4(G) = 0$ have a 4-cycle cover of length $\frac{4}{3} |E(G)|$ and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang. We also give a negative answer to a problem of Zhang.Comment: final versio

The Geometry of Light Paths for Equiangular Spirals

First geometric calculus alongside its description of equiangular spirals, reflections and rotations is introduced briefly. Then single and double reflections at such a spiral are investigated. It proves suitable to distinguish incidence from the \textit{right} and \textit{left} relative to the radial direction. The properties of geometric light propagation inside the equiangular spiral are discussed, as well as escape conditions and characteristics. Finally the dependence of right and left incidence from the source locations are examined, revealing a well defined inner \textit{critical} curve, which delimits the area of purely right incident propagation. This critical curve is self similar to the original equiangular spiral.Comment: 24 pages, 15 figure

Lectures on pure spinors and moment maps

The goal of this article is to give an elementary introduction to Dirac geometry and group-valued moment maps, via pure spinors. The material is based on my lectures at the summer school on 'Poisson geometry in Mathematics and Physics' at Keio University, June 2006

On Riemann-Roch Formulas for Multiplicities

A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments involved, the result also extends to non Kaehlerian settings.Comment: 21 pages, AMS-LaTe

Asymptotic fitness distribution in the Bak-Sneppen model of biological evolution with four species

We suggest a new method to compute the asymptotic fitness distribution in the Bak-Sneppen model of biological evolution. As applications we derive the full asymptotic distribution in the four-species model, and give an explicit linear recurrence relation for a set of coefficients determining the asymptotic distribution in the five-species model.Comment: 10 pages, one figure; to appear in Journal of Statistical Physic

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Keywords: Quaternionic adaptive filtering, Hypercomplex adaptive filtering, Nonlinear adaptive filtering, Hypercomplex Multilayer Perceptron, Clifford geometric algebraComment: 14 pages, 1 figur

Directional Uncertainty Principle for Quaternion Fourier Transform

This paper derives a new directional uncertainty principle for quaternion valued functions subject to the quaternion Fourier transformation. This can be generalized to establish directional uncertainty principles in Clifford geometric algebras with quaternion subalgebras. We demonstrate this with the example of a directional spacetime algebra function uncertainty principle related to multivector wave packets.Comment: 14 page