Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page

Derivatives of meromorphic functions of finite order

A result is proved concerning meromorphic functions of finite order in the plane such that all but finitely many zeros of the second derivative are zeros of the first derivative

A 4D Light-Field Dataset and CNN Architectures for Material Recognition

We introduce a new light-field dataset of materials, and take advantage of the recent success of deep learning to perform material recognition on the 4D light-field. Our dataset contains 12 material categories, each with 100 images taken with a Lytro Illum, from which we extract about 30,000 patches in total. To the best of our knowledge, this is the first mid-size dataset for light-field images. Our main goal is to investigate whether the additional information in a light-field (such as multiple sub-aperture views and view-dependent reflectance effects) can aid material recognition. Since recognition networks have not been trained on 4D images before, we propose and compare several novel CNN architectures to train on light-field images. In our experiments, the best performing CNN architecture achieves a 7% boost compared with 2D image classification (70% to 77%). These results constitute important baselines that can spur further research in the use of CNNs for light-field applications. Upon publication, our dataset also enables other novel applications of light-fields, including object detection, image segmentation and view interpolation.Comment: European Conference on Computer Vision (ECCV) 201

Flexible parameter-sparse global temperature time-profiles that stabilise at 1.5C and 2.0C

The meeting of the United Nations Framework Convention on Climate Change (UNFCCC) in December 2015 committed parties to the Convention to hold the rise in global average temperature to well below 2.0 C above pre-industrial levels. It also committed the parties to pursue efforts to limit warming to 1.5 C. This leads to two key questions. First,what extent of emission reductions will achieve either target? Second, what is the benefit of the reduced climate impacts by keeping warming at or below 1.5 C? To provide answers, climate model simulations need to follow trajectories consistent with these global temperature limits. It is useful to operate models in an inverse mode to make model-specific estimates of greenhouse gas (GHG) concentration pathways consistent with the prescribed temperature profiles. Further inversion derives related emissions pathways for these concentrations. For this to happen, and to enable climate research centres to compare GHG concentrations and emissions estimates, common temperature trajectory scenarios are required. Here we define algebraic curves which asymptote to a stabilised limit, while also matching the magnitude and gradient of recent warming levels. The curves are deliberately parameter-sparse, needing prescription of just two parameters plus the final temperature. Yet despite this simplicity, they can allow for temperature overshoot and for generational changes where more effort to decelerate warming change is needed by future generations. The curves capture temperature profiles from the existing Representative Concentration Pathway (RCP2.6) scenario projections by a range of different earth system models (ESMs), which have warming amounts towards the lower levels of those that society is discussing

Role of OH variability in the stalling of the global atmospheric CH4 growth rate from 1999 to 2006

The growth in atmospheric methane (CH4) concentrations over the past two decades has shown large variability on a timescale of several years. Prior to 1999 the globally averaged CH4 concentration was increasing at a rate of 6.0 ppb/yr, but during a stagnation period from 1999 to 2006 this growth rate slowed to 0.6 ppb/yr. From 2007 to 2009 the growth rate again increased to 4.9 ppb/yr. These changes in growth rate are usually ascribed to variations in CH4 emissions. We have used a 3-D global chemical transport model, driven by meteorological reanalyses and variations in global mean hydroxyl (OH) concentrations derived from CH3CCl3 observations from two independent networks, to investigate these CH4 growth variations. The model shows that between 1999 and 2006, changes in the CH4 atmospheric loss contributed significantly to the suppression in global CH4 concentrations relative to the pre-1999 trend. The largest factor in this is relatively small variations in global mean OH on a timescale of a few years, with minor contributions of atmospheric transport of CH4 to its sink region and of atmospheric temperature. Although changes in emissions may be important during the stagnation period, these results imply a smaller variation is required to explain the observed CH4 trends. The contribution of OH variations to the renewed CH4 growth after 2007 cannot be determined with data currently available

Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie's lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica

Proof of a conjecture of Polya on the zeros of successive derivatives of real entire functions

We prove Polya's conjecture of 1943: For a real entire function of order greater than 2, with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity with n. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.Comment: 26 page