A Mathematical Analysis of Benford's Law and its Generalization

We explain Kossovsky's generalization of Benford's law which is a formula that approximates the distribution of leftmost digits in finite sequences of natural data and apply it to six sequences of data including populations of US cities and towns and times between earthquakes. We model the natural logarithms of these two data sequences as samples of random variables having normal and reflected Gumbel densities respectively. We show that compliance with the general law depends on how nearly constant the periodized density functions are and that the models are generally more compliant than the natural data. This surprising result suggests that the generalized law might be used to improve density estimation which is the basis of statistical pattern recognition, machine learning and data science.Comment: 15 pages, 8 figure

Multidimensional Fourier Quasicrystals I. Sufficient Conditions

We derive sufficient conditions for an atomic measure $\sum_{\lambda \in \Lambda} m_\lambda\, \delta_\lambda,$ where $\Lambda \subset \mathbb R^n,$ $m_\lambda$ are positive integers, and $\delta_\lambda$ is the point measure at $\lambda,$ to be a Fourier quasicrystal, and suggest why they may also be necessary. These conditions extend the necessary and sufficient conditions derived by Lev, Olevskii, and Ulanovskii for $n = 1.$ Our methods exploit the toric geometry relation between Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii.Comment: 17 page

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics and Fourier analysis.Comment: 25 page

Spectral Relationships Between Kicked Harper and On-Resonance Double Kicked Rotor Operators

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these operators. In this paper we apply C*-algebra methods to explain this resemblance. We show that each pair of corresponding operators belong to a common rotation C*-algebra B_\alpha, prove that their spectrums are equal if \alpha is irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if \alpha = p/q with p and q coprime integers. Moreover, we show that corresponding operators in B_\alpha are homomorphic images of mother operators in the universal rotation C*-algebra A_\alpha that are unitarily equivalent and hence have identical spectrums. These results extend analogous results for almost Mathieu operators. We also utilize the C*-algebraic framework to develop efficient algorithms to compute the spectrums of these mother operators for rational \alpha and present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if \alpha is irrational. This conjecture for almost Mathieu operators, called the Ten Martini Problem, was recently proved after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators.Comment: 26 pages, 8 figure

Decline of Birds in a Human Modified Coastal Dune Forest Landscape in South Africa

Previous studies demonstrate that old-growth forest remnants and vegetation regenerating after anthropogenic disturbance provide habitat for birds in a human modified coastal dune forest landscape in northern KwaZulu-Natal, South Africa. However, occurrence does not ensure persistence. Based on a 13-year monitoring database we calculated population trends for 37 bird species and general trends in overall bird density in different vegetation types. We evaluated species' characteristics as covariates of population trend and assessed changes in rainfall and proportional area and survey coverage per vegetation type. 76% of species assessed have declined, 57% significantly so at an average rate of 13.9% per year. Overall, bird density has fallen at 12.2% per year across old-growth forest and woody regenerating vegetation types. Changes in proportional area and coverage per vegetation type may partly explain trends for a few species but are unlikely to account for most. Below average rainfall may have contributed to bird declines. However, other possibilities warrant further investigation. Species with larger range extents tended to decline more sharply than did others, and these species may be responding to environmental changes on a broader geographical scale. Our results cast doubt on the future persistence of birds in this human modified landscape. More research is needed to elucidate the mechanisms driving population decline in the study area and to investigate whether the declines identified here are more widespread across the region and perhaps the continent

The retrospective analysis of Antarctic tracking data project

The Retrospective Analysis of Antarctic Tracking Data (RAATD) is a Scientific Committee for Antarctic Research project led jointly by the Expert Groups on Birds and Marine Mammals and Antarctic Biodiversity Informatics, and endorsed by the Commission for the Conservation of Antarctic Marine Living Resources. RAATD consolidated tracking data for multiple species of Antarctic meso- and top-predators to identify Areas of Ecological Significance. These datasets and accompanying syntheses provide a greater understanding of fundamental ecosystem processes in the Southern Ocean, support modelling of predator distributions under future climate scenarios and create inputs that can be incorporated into decision making processes by management authorities. In this data paper, we present the compiled tracking data from research groups that have worked in the Antarctic since the 1990s. The data are publicly available through biodiversity.aq and the Ocean Biogeographic Information System. The archive includes tracking data from over 70 contributors across 12 national Antarctic programs, and includes data from 17 predator species, 4060 individual animals, and over 2.9 million observed locations

The retrospective analysis of Antarctic tracking data project

The Retrospective Analysis of Antarctic Tracking Data (RAATD) is a Scientific Committee for Antarctic Research project led jointly by the Expert Groups on Birds and Marine Mammals and Antarctic Biodiversity Informatics, and endorsed by the Commission for the Conservation of Antarctic Marine Living Resources. RAATD consolidated tracking data for multiple species of Antarctic meso- and top-predators to identify Areas of Ecological Significance. These datasets and accompanying syntheses provide a greater understanding of fundamental ecosystem processes in the Southern Ocean, support modelling of predator distributions under future climate scenarios and create inputs that can be incorporated into decision making processes by management authorities. In this data paper, we present the compiled tracking data from research groups that have worked in the Antarctic since the 1990s. The data are publicly available through biodiversity.aq and the Ocean Biogeographic Information System. The archive includes tracking data from over 70 contributors across 12 national Antarctic programs, and includes data from 17 predator species, 4060 individual animals, and over 2.9 million observed locations.Supplementary Figure S1: Filtered location data (black) and tag deployment locations (red) for each species. Maps are Lambert Azimuthal projections extending from 90° S to 20° S.Supplementary Table S1: Names and coordinates of the major study sites in the Southern Ocean and on the Antarctic Continent where tracking devices were deployed on the selected species (indicated by their 4-letter codes in the last column).Online Table 1: Description of fields (column names) in the metadata and data files.Supranational committees and organisations including the Scientific Committee on Antarctic Research Life Science Group and BirdLife International. National institutions and foundations, including but not limited to Argentina (Dirección Nacional del Antártico), Australia (Australian Antarctic program; Australian Research Council; Sea World Research and Rescue Foundation Inc., IMOS is a national collaborative research infrastructure, supported by the Australian Government and operated by a consortium of institutions as an unincorporated joint venture, with the University of Tasmania as Lead Agent), Belgium (Belgian Science Policy Office, EU Lifewatch ERIC), Brazil (Brazilian Antarctic Programme; Brazilian National Research Council (CNPq/MCTI) and CAPES), France (Agence Nationale de la Recherche; Centre National d’Etudes Spatiales; Centre National de la Recherche Scientifique; the French Foundation for Research on Biodiversity (FRB; www.fondationbiodiversite.fr) in the context of the CESAB project “RAATD”; Fondation Total; Institut Paul-Emile Victor; Programme Zone Atelier de Recherches sur l’Environnement Antarctique et Subantarctique; Terres Australes et Antarctiques Françaises), Germany (Deutsche Forschungsgemeinschaft, Hanse-Wissenschaftskolleg - Institute for Advanced Study), Italy (Italian National Antarctic Research Program; Ministry for Education University and Research), Japan (Japanese Antarctic Research Expedition; JSPS Kakenhi grant), Monaco (Fondation Prince Albert II de Monaco), New Zealand (Ministry for Primary Industries - BRAG; Pew Charitable Trusts), Norway (Norwegian Antarctic Research Expeditions; Norwegian Research Council), Portugal (Foundation for Science and Technology), South Africa (Department of Environmental Affairs; National Research Foundation; South African National Antarctic Programme), UK (Darwin Plus; Ecosystems Programme at the British Antarctic Survey; Natural Environment Research Council; WWF), and USA (U.S. AMLR Program of NOAA Fisheries; US Office of Polar Programs).http://www.nature.com/sdataam2021Mammal Research Institut