Framed vertex operator algebras, codes and the moonshine module

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.Comment: Latex, 54 page

Downscaling soil moisture over regions that include multiple coarse-resolution grid cells

2016 Fall.Includes bibliographical references.Many applications require soil moisture estimates over large spatial extents (30-300 km) and at fine-resolutions (10-30 m). Remote-sensing methods can provide soil moisture estimates over very large spatial extents (continental to global) at coarse resolutions (10-40 km), but their output must be downscaled to reach fine resolutions. When large spatial extents are considered, the downscaling procedure must consider multiple coarse-resolution grid cells, yet little attention has been given to the treatment of multiple grid cells. The objective of this paper is to compare the performance of different methods for addressing multiple coarse grid cells. To accomplish this goal, the Equilibrium Moisture from Topography, Vegetation, and Soil (EMT+VS) downscaling model is generalized to accept multiple coarse grid cells, and two methods for their treatment are implemented and compared. The first method (fixed window) is a direct extension of the original EMT+VS model and downscales each coarse grid cell independently. The second method (shifting window) replaces the coarse grid cell values with values that are calculated from windows that are centered on each fine grid cell. The window values are weighted averages of the coarse grid values within the window extent, and three weighting methods are considered (box, disk, and Gaussian). The methods are applied to three small catchments with detailed soil moisture observations and one large region. The fixed window typically provides more accurate estimates of soil moisture than the shifting window, but it produces abrupt changes in soil moisture at the coarse grid boundaries, which may be problematic for some applications. The three weighting methods produce similar results

Radial departures and plane embeddings of arc-like continua

We study the problem of Nadler and Quinn from 1972, which asks whether, given an arc-like continuum $X$ and a point $x \in X$, there exists an embedding of $X$ in $\mathbb{R}^2$ for which $x$ is an accessible point. We develop the notion of a radial departure of a map $f \colon [-1,1] \to [-1,1]$, and establish a simple criterion in terms of the bonding maps in an inverse system on intervals to show that there is an embedding of the inverse limit for which a given point is accessible. Using this criterion, we give a partial affirmative answer to the problem of Nadler and Quinn, under some technical assumptions on the bonding maps of the inverse system.Comment: 23 pages, 8 figure

Lelek's problem is not a metric problem

We show that Lelek's problem on the chainability of continua with span zero is not a metric problem: from a non-metric counterexample one can construct a metric one.Comment: Final version as sent to edito