New existence results for the mean field equation on compact surfaces via degree theory

We consider a class of equations with exponential non-linearities on a compact surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We prove an existence result via degree theory. This yields new existence results in case of a topological sphere. The proof is carried out by considering the parity of the Leray-Schauder degree associated to the problem. With this method we recover also some known previous results

An Existence Result for the Mean Field Equation on Compact Surfaces in a Doubly Supercritical Regime

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulance with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing this, we present a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of both e^u and e^{-u}, where u is the unknown function in the equation.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1105.3701 by other authors. The proof of Lemma 3.9 has been fixe

Interactive Vegetation Rendering with Slicing and Blending

Detailed and interactive 3D rendering of vegetation is one of the challenges of traditional polygon-oriented computer graphics, due to large geometric complexity even of simple plants. In this paper we introduce a simplified image-based rendering approach based solely on alpha-blended textured polygons. The simplification is based on the limitations of human perception of complex geometry. Our approach renders dozens of detailed trees in real-time with off-the-shelf hardware, while providing significantly improved image quality over existing real-time techniques. The method is based on using ordinary mesh-based rendering for the solid parts of a tree, its trunk and limbs. The sparse parts of a tree, its twigs and leaves, are instead represented with a set of slices, an image-based representation. A slice is a planar layer, represented with an ordinary alpha or color-keyed texture; a set of parallel slices is a slicing. Rendering from an arbitrary viewpoint in a 360 degree circle around the center of a tree is achieved by blending between the nearest two slicings. In our implementation, only 6 slicings with 5 slices each are sufficient to visualize a tree for a moving or stationary observer with the perceptually similar quality as the original model

Tensors, !-graphs, and non-commutative quantum structures

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entire families of diagrams. One candidate is the language of !-graphs, which consist of graphs with certain subgraphs marked with boxes (called !-boxes) that can be repeated any number of times. New !-graph equations can then be proved using a powerful technique called !-box induction. However, previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.Comment: In Proceedings QPL 2014, arXiv:1412.810

Analytic aspects of the Tzitz\'eica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities: \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitz\'eica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result

Dusty disks at the bottom of the IMF

'Isolated planetary mass objects' (IPMOs) have masses close to or below the Deuterium-burning mass limit (~15 Jupiter masses) -- at the bottom of the stellar initial mass function. We present an exploratory survey for disks in this mass regime, based on a dedicated observing campaign with the Spitzer Space Telescope. Our targets include the full sample of spectroscopically confirmed IPMOs in the Sigma Orionis cluster, a total of 18 sources. In the mass range 8... 20 MJup, we identify 4 objects with >3sigma colour excess at a wavelength of 8.0mu, interpreted as emission from dusty disks. We thus establish that a substantial fraction of IPMOs harbour disks with lifetimes of at least 2-4 Myr (the likely age of the cluster), indicating an origin from core collapse and fragmentation processes. The disk frequency in the IPMO sample is 29% (16-45%) at 8.0mu, very similar to what has been found for stars and brown dwarfs (~30%). The object SOri70, a candidate 3 MJup object in this cluster, shows IRAC colours in excess of the typical values for field T dwarfs (on a 2sigma level), possibly due to disk emission or low gravity. This is a new indication for youth and thus an extremely low mass for SOri70.Comment: 12 pages, 2 figures, accepted for publication in ApJ