The E8 geometry from a Clifford perspective

This paper considers the geometry of $E_8$ from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system $H_3$ gives rise to the largest (and therefore exceptional) non-crystallographic root system $H_4$. Arnold's trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and $E_8$. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the $120$ elements of the icosahedral group $H_3$ are doubly covered by $240$ 8-component objects, which endowed with a `reduced inner product' are exactly the $E_8$ root system. It was previously known that $E_8$ splits into $H_4$-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, $D_6$ as well as $E_8$, whose Coxeter versor factorises as $W=\exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)$. This explicitly describes 30-fold rotations in 4 orthogonal planes with the correct exponents $\{1, 7, 11, 13, 17, 19, 23, 29\}$ arising completely algebraically from the factorisation

Mandelbrot's 1/f fractional renewal models of 1963-67: The non-ergodic missing link between change points and long range dependence

The problem of 1/f noise has been with us for about a century. Because it is so often framed in Fourier spectral language, the most famous solutions have tended to be the stationary long range dependent (LRD) models such as Mandelbrot's fractional Gaussian noise. In view of the increasing importance to physics of non-ergodic fractional renewal models, I present preliminary results of my research into the history of Mandelbrot's very little known work in that area from 1963-67. I speculate about how the lack of awareness of this work in the physics and statistics communities may have affected the development of complexity science, and I discuss the differences between the Hurst effect, 1/f noise and LRD, concepts which are often treated as equivalent.Comment: 11 pages. Corrected and improved version of a manuscript submitted to ITISE 2016 meeting in Granada, Spai

Anomalous spatial diffusion and multifractality in optical lattices

Transport of cold atoms in shallow optical lattices is characterized by slow, nonstationary momentum relaxation. We here develop a projector operator method able to derive in this case a generalized Smoluchowski equation for the position variable. We show that this explicitly non-Markovian equation can be written as a systematic expansion involving higher-order derivatives. We use the latter to compute arbitrary moments of the spatial distribution and analyze their multifractal properties.Comment: 5 pages, 3 figure

Gaussian white noise as a resource for work extraction

11 pages, 5 figuresThe present study was supported by KAKENHI (Nos. 25103002 and 26610115). A. D. was employed as an International Research Fellow of the Japan Society for the Promotion of Science

Comparison of predicted and actual over-refractions when using the Mastervue\u27s RGP fitting recommendations

The use of computerized corneal videokeratography is one of the most exciting advances in the fitting of rigid contact lenses today. Traditional RGP fitting methods and our understanding of physiological and contact lens optics dictates that power should be determined by comparing lens lens base curve to the central three millimeter zone flat keratometric reading, and apply SAMFAP to compensate for lacrimal lens effects. This study addresses the Issue that since the Mastervue lens fitting software selects lens base curve determined using a more peripheral corneal zone, clinically, how and where should lens power be determined? Twenty-eight subjects were fit with rigid gas permeable lenses generated by the Mastervue corneal topographer fitting software. Accurate monocular over-refractions were performed and compared to the predicted over-refractions determined using the simulated keratometric values at each the three millimeter and six millimeter corneal zones. From this data it can be shown which simulated keratometric readings should be used to determine the clinically most accurate contact lens power. The data showed that it did not matter which corneal zone SAMFAP is applied to when determining contact lens power. Through the interpretation of these results, it can be deduced that the Mastervue fitting software is effective in suggesting an initial RGP to be dispensed, however, trial fitting with an over-refraction is still the most accurate way of determining final lens power

Machine-learning a virus assembly fitness landscape

Realistic evolutionary fitness landscapes are notoriously difficult to construct. A recent cutting-edge model of virus assembly consists of a dodecahedral capsid with 12 corresponding packaging signals in three affinity bands. This whole genome/phenotype space consisting of 312 genomes has been explored via computationally expensive stochastic assembly models, giving a fitness landscape in terms of the assembly efficiency. Using latest machine-learning techniques by establishing a neural network, we show that the intensive computation can be short-circuited in a matter of minutes to astounding accuracy

Machine Learning Clifford Invariants of ADE Coxeter Elements

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A8, D8 and E8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results

Effects of Management Practices on Grassland Birds: Grasshopper Sparrow

Grasshopper Sparrow (Ammodramus savannarum): Breeding range Suitable habitat Area requirements Brown-headed Cowbird brood parasitism Breeding-season phenology and site fidelity Species’ response to management Management Recommendations Habitat Characteristic

Effects of Management Practices on Grassland Birds: Sprague’s Pipit

Sprague’s Pipit (Anthus spragueii): Breeding range Suitable habitat Area requirements Brown-headed Cowbird brood parasitism Breeding-season phenology and site fidelity Species’ response to management Management Recommendations Habitat Characteristic

Effects of Management Practices on Grassland Birds: Sprague’s Pipit

Sprague’s Pipit (Anthus spragueii): Breeding range Suitable habitat Area requirements Brown-headed Cowbird brood parasitism Breeding-season phenology and site fidelity Species’ response to management Management Recommendations Habitat Characteristic