Successive Standardization of Rectangular Arrays

In this note we illustrate and develop further with mathematics and examples, the work on successive standardization (or normalization) that is studied earlier by the same authors in Olshen and Rajaratnam (2010) and Olshen and Rajaratnam (2011). Thus, we deal with successive iterations applied to rectangular arrays of numbers, where to avoid technical difficulties an array has at least three rows and at least three columns. Without loss, an iteration begins with operations on columns: first subtract the mean of each column; then divide by its standard deviation. The iteration continues with the same two operations done successively for rows. These four operations applied in sequence completes one iteration. One then iterates again, and again, and again,.... In Olshen and Rajaratnam (2010) it was argued that if arrays are made up of real numbers, then the set for which convergence of these successive iterations fails has Lebesgue measure 0. The limiting array has row and column means 0, row and column standard deviations 1. A basic result on convergence given in Olshen and Rajaratnam (2010) is true, though the argument in Olshen and Rajaratnam (2010) is faulty. The result is stated in the form of a theorem here, and the argument for the theorem is correct. Moreover, many graphics given in Olshen and Rajaratnam (2010) suggest that but for a set of entries of any array with Lebesgue measure 0, convergence is very rapid, eventually exponentially fast in the number of iterations. Because we learned this set of rules from Bradley Efron, we call it "Efron's algorithm". More importantly, the rapidity of convergence is illustrated by numerical examples

Successive normalization of rectangular arrays

Standard statistical techniques often require transforming data to have mean $0$ and standard deviation $1$. Typically, this process of "standardization" or "normalization" is applied across subjects when each subject produces a single number. High throughput genomic and financial data often come as rectangular arrays where each coordinate in one direction concerns subjects who might have different status (case or control, say), and each coordinate in the other designates "outcome" for a specific feature, for example, "gene," "polymorphic site" or some aspect of financial profile. It may happen, when analyzing data that arrive as a rectangular array, that one requires BOTH the subjects and the features to be "on the same footing." Thus there may be a need to standardize across rows and columns of the rectangular matrix. There arises the question as to how to achieve this double normalization. We propose and investigate the convergence of what seems to us a natural approach to successive normalization which we learned from our colleague Bradley Efron. We also study the implementation of the method on simulated data and also on data that arose from scientific experimentation.Comment: Published in at http://dx.doi.org/10.1214/09-AOS743 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). With Correction

Localization and Fluctuations in Quantum Kicked Rotors

We address the issue of fluctuations, about an exponential lineshape, in a pair of one-dimensional kicked quantum systems exhibiting dynamical localization. An exact renormalization scheme establishes the fractal character of the fluctuations and provides a new method to compute the localization length in terms of the fluctuations. In the case of a linear rotor, the fluctuations are independent of the kicking parameter $k$ and exhibit self-similarity for certain values of the quasienergy. For given $k$, the asymptotic localization length is a good characteristic of the localized lineshapes for all quasienergies. This is in stark contrast to the quadratic rotor, where the fluctuations depend upon the strength of the kicking and exhibit local "resonances". These resonances result in strong deviations of the localization length from the asymptotic value. The consequences are particularly pronounced when considering the time evolution of a packet made up of several quasienergy states.Comment: REVTEV Document. 9 pages, 4 figures submitted to PR

FDM preparation of bio-compatible UHMWPE polymer for artificial implant

Due to its properties of high wear, creep resistance, high stiffness and strength, Ultra-High Molecular Weight Polyethylene (UHMWPE) was developed to eliminate most metallic wear in artificial implant, which conventionally found in stainless steel, Cobalt Chromium (Co-Cr) and Titanium (Ti) alloys. UHMWPE has an ultra-high viscosity that renders continuous melt-state processes including one of the additive manufacturing processes, Fused Deposition Modeling (FDM) ineffective for making UHMWPE implant. Attempt to overcome this problem and adapting this material to FDM is by blending UHMWPE with other polyethylene including High Density Polyethylene (HDPE) and Polyethylene-Glycol (PEG) which provide adequate mechanical properties for biomedical application along with the improvement in extrudability. It was demonstrated that the inclusion of 60% HDPE fraction has improved the flowability of UHMWPE in MFI test and showing adequate thermal stability in TGA

Second post-Newtonian gravitational radiation reaction for two-body systems: Nonspinning bodies

Starting from the recently obtained 2PN accurate forms of the energy and angular momentum fluxes from inspiralling compact binaries, we deduce the gravitational radiation reaction to 2PN order beyond the quadrupole approximation - 4.5PN terms in the equation of motion - using the refined balance method proposed by Iyer and Will. We explore critically the features of their construction and illustrate them by contrast to other possible variants. The equations of motion are valid for general binary orbits and for a class of coordinate gauges. The limiting cases of circular orbits and radial infall are also discussed.Comment: 38 pages, REVTeX, no figures, to appear in Phys. Rev.

Pad\'e approximants for truncated post-Newtonian neutron star models

Pad\'e approximants to truncated post-Newtonian neutron star models are constructed. The Pad\'e models converge faster to the general relativistic (GR) solution than the truncated post-Newtonian ones. The evolution of initial data using the Pad\'e models approximates better the evolution of full GR initial data than the truncated Taylor models. In the absence of full GR initial data (e.g., for neutron star binaries or black hole binary systems), Pad\'e initial data could be a better option than the straightforward truncated post-Newtonian (Taylor) initial data.Comment: 19 pages (RevTeX), 9 eps figures. Three new figures and additional discussion on 1-parameter Pad\'e expansion. Accepted for publication in Physical Review

Strategic Network Formation with Attack and Immunization

Strategic network formation arises where agents receive benefit from connections to other agents, but also incur costs for forming links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization against attack. An agent's benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework is a stylized model of settings where reachability rather than centrality is the primary concern and vertices vulnerable to attacks may reduce risk via costly measures. In the reachability benefit model without attack or immunization, the set of equilibria is the empty graph and any tree. The introduction of attack and immunization changes the game dramatically; new equilibrium topologies emerge, some more sparse and some more dense than trees. We show that, under a mild assumption on the adversary, every equilibrium network with $n$ agents contains at most $2n-4$ edges for $n\geq 4$. So despite permitting topologies denser than trees, the amount of overbuilding is limited. We also show that attack and immunization don't significantly erode social welfare: every non-trivial equilibrium with respect to several adversaries has welfare at least as that of any equilibrium in the attack-free model. We complement our theory with simulations demonstrating fast convergence of a new bounded rationality dynamic which generalizes linkstable best response but is considerably more powerful in our game. The simulations further elucidate the wide variety of asymmetric equilibria and demonstrate topological consequences of the dynamics e.g. heavy-tailed degree distributions. Finally, we report on a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium.Comment: The short version of this paper appears in the proceedings of WINE-1

The transient response of global-mean precipitation to increasing carbon dioxide levels

The transient response of global-mean precipitation to an increase in atmospheric carbon dioxide levels of 1% yr(-1) is investigated in 13 fully coupled atmosphere-ocean general circulation models (AOGCMs) and compared to a period of stabilization. During the period of stabilization, when carbon dioxide levels are held constant at twice their unperturbed level and the climate left to warm, precipitation increases at a rate of similar to 2.4% per unit of global-mean surface-air-temperature change in the AOGCMs. However, when carbon dioxide levels are increasing, precipitation increases at a smaller rate of similar to 1.5% per unit of global-mean surface-air-temperature change. This difference can be understood by decomposing the precipitation response into an increase from the response to the global surface-temperature increase (and the climate feedbacks it induces), and a fast atmospheric response to the carbon dioxide radiative forcing that acts to decrease precipitation. According to the multi-model mean, stabilizing atmospheric levels of carbon dioxide would lead to a greater rate of precipitation change per unit of global surface-temperature change