New sum-product type estimates over finite fields

Let $F$ be a field with positive odd characteristic $p$. We prove a variety of new sum-product type estimates over $F$. They are derived from the theorem that the number of incidences between $m$ points and $n$ planes in the projective three-space $PG(3,F)$, with $m\geq n=O(p^2)$, is $$O( m\sqrt{n} + km ),$$ where $k$ denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that \begin{equation}\label{mres} |A\pm A|+|A\cdot A| =\Omega \left(|A|^{1+\frac{1}{5}}\right), \end{equation} for any $A$ such that $|A|<p^{\frac{5}{8}}.$Comment: This is a revised version: Theorem 1 was incorrect as stated. We give its correct statement; this does not seriously affect the main arguments throughout the paper. Also added is a seres of remarks, placing the result in the context of the current state of the ar

Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable singularity theorem of Rado type for J-holomorphic maps. Let $\Omega$ be an open subset in $\mathbf C$ and let $E$ be a closed polar subset of $\Omega$. Let $u$ be a continuous map from $\Omega$ into an almost complex manifold $(M,J)$ with $J$ of class $C^1$. We show that if $u$ is J-holomorphic on $\Omega\setminus E$ then it is J-holomorphic on $\Omega$

Random Forests: some methodological insights

This paper examines from an experimental perspective random forests, the increasingly used statistical method for classification and regression problems introduced by Leo Breiman in 2001. It first aims at confirming, known but sparse, advice for using random forests and at proposing some complementary remarks for both standard problems as well as high dimensional ones for which the number of variables hugely exceeds the sample size. But the main contribution of this paper is twofold: to provide some insights about the behavior of the variable importance index based on random forests and in addition, to propose to investigate two classical issues of variable selection. The first one is to find important variables for interpretation and the second one is more restrictive and try to design a good prediction model. The strategy involves a ranking of explanatory variables using the random forests score of importance and a stepwise ascending variable introduction strategy

On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure

Weak convergence of CD kernels and applications

We prove a general result on equality of the weak limits of the zero counting measure, $d\nu_n$, of orthogonal polynomials (defined by a measure $d\mu$) and $\frac{1}{n} K_n(x,x) d\mu(x)$. By combining this with Mate--Nevai and Totik upper bounds on $n\lambda_n(x)$, we prove some general results on $\int_I \frac{1}{n} K_n(x,x) d\mu_s\to 0$ for the singular part of $d\mu$ and $\int_I |\rho_E(x) - \frac{w(x)}{n} K_n(x,x)| dx\to 0$, where $\rho_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$

A Proof of the Isoenergetic KAM-Theorem from the “Ordinary” One

A proof is given of the isoenergetic KAM-theorem for Hamiltonian systems, using the “ordinary” KAM-theorem and a transversality argument.

Advanced Transport Operating System (ATOPS) color displays software description microprocessor system

This document describes the software created for the Sperry Microprocessor Color Display System used for the Advanced Transport Operating Systems (ATOPS) project on the Transport Systems Research Vehicle (TSRV). The software delivery known as the 'baseline display system', is the one described in this document. Throughout this publication, module descriptions are presented in a standardized format which contains module purpose, calling sequence, detailed description, and global references. The global reference section includes procedures and common variables referenced by a particular module. The system described supports the Research Flight Deck (RFD) of the TSRV. The RFD contains eight cathode ray tubes (CRTs) which depict a Primary Flight Display, Navigation Display, System Warning Display, Takeoff Performance Monitoring System Display, and Engine Display