Order Statistics and Benford's Law

Fix a base B and let zeta have the standard exponential distribution; the distribution of digits of zeta base B is known to be very close to Benford's Law. If there exists a C such that the distribution of digits of C times the elements of some set is the same as that of zeta, we say that set exhibits shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1, >..., X_N be independent identically distributed random variables. If the X_i's are drawn from the uniform distribution on [0,L], then as N\to\infty the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior (with a shift of \log_B L/N \bmod 1). By differentiating the cumulative distribution function of the logarithms modulo 1, applying Poisson Summation and then integrating the resulting expression, we derive rapidly converging explicit formulas measuring the deviations from Benford's Law. Fix a delta in (0,1) and choose N independent random variables from any compactly supported distribution with uniformly bounded first and second derivatives and a second order Taylor series expansion at each point. The distribution of digits of any N^\delta consecutive differences \emph{and} all N-1 normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the un-normalized differences converges to Benford's Law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.Comment: 14 pages, 2 figures, version 4: Version 3: most of the numerical simulations on shifted exponential behavior have been suppressed (though are available from the authors upon request). Version 4: a referee pointed out that we need epsilon > 1/3 - delta/2 in the proof of Theorem 1.5; this has now been adde

Theory of Analogous Force on Number Sets

A general statistical thermodynamic theory that considers given sequences of x-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions p_{x} for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering p_{x} to describe complex phenomena, the theory leads to derive a distinct analogous force f_{x} on number sets proportional to $(\frac{\partial p_{x}}{\partial x} )_{T}$ at an analogous system temperature T. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed.Comment: RevTeX, PostScript Fig, To Appear Phys.

First Digit Distribution of Hadron Full Width

A phenomenological law, called Benford's law, states that the occurrence of the first digit, i.e., $1,2,...,9$, of numbers from many real world sources is not uniformly distributed, but instead favors smaller ones according to a logarithmic distribution. We investigate, for the first time, the first digit distribution of the full widths of mesons and baryons in the well defined science domain of particle physics systematically, and find that they agree excellently with the Benford distribution. We also discuss several general properties of Benford's law, i.e., the law is scale-invariant, base-invariant, and power-invariant. This means that the lifetimes of hadrons follow also Benford's law.Comment: 8 latex pages, 4 figures, final version in journal publicatio

Tracking the Equator Into the Paleogene (abstract of paper presented at AGU Fall Meeting, San Francisco, 8-12 Dec 2003)

Stratigraphy has been compiled for 63 tropical Pacific drill sites that sample lower Neogene and Paleogene sediments. These Sites derive from drilling on DSDP Leg 5 through ODP Leg 199. All Sites have been put on the biostratigraphic and paleomagnetic timescale refined by Leg 199 scientists. Sediment accumulation rates have been calculated for ten intervals ranging in age from 10 Ma to 56 Ma. A simple fixed hotspot model was used for Pacific lithospheric plate rotation in reconstructing the position of the selected sites for each of these ten intervals. The reconstruction of all intervals show the development of a tongue of relatively high accumulation rates associated with the oceanographic divergence at the geographic equator. The estimated position of the geographic equator based on these reconstructions lies consistently south of the position of the equator based on the rotation model used. However, the southward displacement is generally 2 degrees of latitude or less. We believe that this relatively small disagreement between the two estimates of equatorial position back to 56 Ma indicates: 1) Whatever hotspot movement that may have occurred in the interval between 40 and 56 Ma did not affect the motion of the Pacific plate; its motion after 40 Ma appears to have been approximately the same as before 40 Ma. 2) The estimated rate of true polar wander during the interval of 40 - 56 Ma must be very small (~0.125$\deg$/m.y.) and is probably not significant (i.e., well within the error of these reconstructions)

Los egresados de la UAM en el mercado de trabajo : investigación evaluativa sobre la calidad de la oferta de servicios educativos

1 archivo PDF (155 páginas)El libro está constituido en dos secciones: La primera comprende una breve introducción del proyecto, así como los principales criterios metodológicos que guiaron la investigación. La segunda comprende particularidades de la UAM, los resultados de la investigación, descripción de los egresados, análisis de los ritmos de inserción laboral, un esbozo del mercado laboral al cual se incorporan los egresados y finalmente se analizan algunos rasgos generales del desempeño profesional

Pliocene-Pleistocene radiolarian events and magnetostratigraphic calibrations for the tropical Indian Ocean

A composite of four piston cores from the Central Indian Basin and adjacent Ninetyeast Ridge has yielded a continuous magnetobiostratigraphic reference section for most of the Pliocene and the Pleistocene (0.0–4.5 Ma). We identified thirty-three radiolarian events (first- or last-occurrences), and precisely correlated each event to the Neogene geomagnetic polarity time scale. Thirteen of these events are based on revised taxonomic studies of the genera Anthocyrtidium and Pterocorys. Some events show significant departures from synchroneity: five of the radiolarian first-appearances and seven of the last-appearances are time-transgressive by 0.4 m.y. or greater. We here propose a revised, eleven-fold radiolarian zonation for the Pliocene-Pleistocene, using zonal boundaries defined by events which are easily recognized and are demonstrably synchronous in the tropical Indo-Pacific. The sequence of faunal and floral events reported in this paper will allow high-resolution biostratigraphic correlations within the tropical Indian Ocean; however, the same sequence of events is not necessarily applicable to other tropical or extratropical regions

The Significant Digit Law in Statistical Physics

The occurrence of the nonzero leftmost digit, i.e., 1, 2, ..., 9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic distribution, named Benford's law. We investigate three kinds of widely used physical statistics, i.e., the Boltzmann-Gibbs (BG) distribution, the Fermi-Dirac (FD) distribution, and the Bose-Einstein (BE) distribution, and find that the BG and FD distributions both fluctuate slightly in a periodic manner around the Benford distribution with respect to the temperature of the system, while the BE distribution conforms to it exactly whatever the temperature is. Thus the Benford's law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature. Furthermore, various elegant properties of Benford's law, especially the mantissa distribution of data sets, are discussed.Comment: 21 latex pages, 5 figures, final version in journal publicatio

Site 1217

Site 1217 (16°52.02´N, 138°06.00´W; 5342 meters below sea level [mbsl]; Fig. F1) is one of seven sites drilled to target upper Paleocene crust along a latitudinal transect during Leg 199 and will be used to investigate paleoceanographic processes in the northern tropical early Eocene Pacific Ocean. Site 1217 is situated ~1° north of the Clarion Fracture Zone on abyssal hill topography typical of the central Pacific. Based on magnetic lineations, basement age at Site 1217 should be in magnetic Anomaly C25r or ~57 Ma (Cande et al., 1989; timescale of Cande and Kent, 1995). The Cenozoic history of sedimentation in this region was poorly constrained prior to Leg 199 drilling because the nearest drill site (Deep Sea Drilling Project [DSDP] Site 162) is situated ~300 km south and west on 48-Ma crust. Based on data from this early rotary-cored hole, magnetic anomaly maps, a shallow-penetration piston core near Site 1217 (EW9709-4PC), and seismic profiling (Fig. F2), we expected the sedimentary sequence at Site 1217 to comprise a relatively thick (25 to 35 m thick) section of red clays overlying a radiolarian ooze and a basal carbonate section with possible chert near basement (estimated total depth ~125-150 meters below seafloor [mbsf]) deposited when the site was near the ridge crest in the late Paleocene and early Eocene. Site 1217 was chosen because it is anticipated to have been located just outside of the equatorial region at 56 Ma, ~5°N, 106°W based upon a fixed hotspot model (Gripp and Gordon, 1990, for 0- to 5-Ma Pacific hotspot rotation pole; Engebretson et al., 1985, for older poles). On the same basis at 40 Ma, the site was located at ~8°N, 111°W. Thus, Site 1217 should help define the paleoceanography of the northern tropical Pacific, in particular locating the ancient North Equatorial Countercurrent (NECC) region. General circulation-model experiments for the early Eocene (see Huber, this volume) suggest that the NECC was a well-developed current during this time period. Other paleoceanographic and paleoclimatic objectives of drilling the sedimentary sequence anticipated at Site 1217 are as follows: (1) to help define the shift in the Intertropical Convergence Zone through the Paleogene by following the change in eolian dust composition and flux through time (red clays); (2) to help constrain the middle-late Eocene calcite compensation depth (CCD); and (3) to sample the Paleocene/Eocene (P/E) boundary, one of the most climatologically critical intervals of Cenozoic time. Recovery of deep-sea sediments from this time interval during Leg 199 is a high priority because the P/E boundary has never before been sampled in the central tropical Pacific Ocean. Results from Site 1217 will also provide important information to test whether there was significant motion of the Hawaiian hotspot, with respect to the Earth's spin axis during the early Cenozoic. At 56 Ma, the backtracked location based upon a hotspot reference frame is ~5°N, 106°W, and at 40 Ma is ~8°N, 106°W. If significant hotspot motion or true polar wander occurred since 57 Ma (Petronotis et al., 1994), this drill site could have been much nearer to the equator