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

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the super-aging behavior.Comment: 16 pages, 6 figure

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

The p75 receptor mediates axon growth inhibition through an association with PIR-B

The Nogo receptor and paired immunoglobulin-like receptor B (PIR-B) are receptors for three myelin-derived axon-growth inhibitors, including myelin-associated glycoprotein (MAG). In this study, we report that the p75 receptor is required for the signal transduction of PIR-B, which interacted with p75 upon ligand binding. In addition, p75 was required for activation of Src homology 2-containing protein tyrosine phosphatase (SHP), which is induced by MAG binding to PIR-B. Mice carrying a mutation in the p75 gene showed promotion of axonal regeneration after optic nerve injury. Thus, our results indicate that p75 has a critical role in axon growth inhibition in specific neuronal tracts

Using Web Search Query Data to Monitor Dengue Epidemics: A New Model for Neglected Tropical Disease Surveillance

A variety of obstacles, including bureaucracy and lack of resources, delay detection and reporting of dengue and exist in many countries where the disease is a major public health threat. Surveillance efforts have turned to modern data sources such as Internet usage data. People often seek health-related information online and it has been found that the frequency of, for example, influenza-related web searches as a whole rises as the number of people sick with influenza rises. Tools have been developed to help track influenza epidemics by finding patterns in certain web search activity. However, few have evaluated whether this approach would also be effective for other diseases, especially those that affect many people, that have severe consequences, or for which there is no vaccine. In this study, we found that aggregated, anonymized Google search query data were also capable of tracking dengue activity in Bolivia, Brazil, India, Indonesia and Singapore. Whereas traditional dengue data from official sources are often not available until after a long delay, web search query data is available for analysis within a day. Therefore, because it could potentially provide earlier warnings, these data represent a valuable complement to traditional dengue surveillance

On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967

The problem of “1∕ƒ” 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, and their links to the CTRW, I present preliminary results of my research into the history of Mandelbrot’s very little known work in that area from 1963 to 1967. 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∕ƒ” noise and LRD, concepts which are often treated as equivalent