Going beyond variation of sets

We study integralgeometric representations of variations of general sets A ⊂ Rn without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function χA is a signed Borel measure on R n with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss-Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a ‘measure-theoretic boundary’ plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of ‘measure theoretic boundary’ and one can address the question to find notions of measure-theoretic boundary that are as fine as possible

Constancy results for special families of projections

Let {\mathbb{V} = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let \pi_{\mathbb{V}} : R^n --> \mathbb{V} be the orthogonal projection onto \mathbb{V}. We prove that the mapping V \mapsto Dim \pi_{\mathbb{V}}(B) is almost surely constant for any analytic set B \subset R^n, where Dim denotes either Hausdorff or packing dimension.Comment: 22 pages. v2: corrected typos and improved readability throughout the paper, to appear in Math. Proc. Cambridge Philos. So

Simulating spatial and temporal variation of corn canopy temperature during an irrigation cycle

The canopy air temperature difference (delta T) which provides an index for scheduling irrigation was examined. The Monteith transpiration equation was combined with both uptake from a single layered root zone and change in internal storage of the plant and the continuity equation for water flux in the soil plant atmosphere system was solved. The model indicates that both daily total transpiration and soil induced depression of plant water potential may be inferred from mid-day delta T. It is suggested that for the soil plant weather data used in the simulation, either a mid day spatial variability of about 0.8K in canopy temperatures or a field averaged delta T of 2 to 4K may be a suitable criterion for irrigation scheduling

Testing surface area with arbitrary accuracy

Recently, Kothari et al.\ gave an algorithm for testing the surface area of an arbitrary set $A \subset [0, 1]^n$. Specifically, they gave a randomized algorithm such that if $A$'s surface area is less than $S$ then the algorithm will accept with high probability, and if the algorithm accepts with high probability then there is some perturbation of $A$ with surface area at most $\kappa_n S$. Here, $\kappa_n$ is a dimension-dependent constant which is strictly larger than 1 if $n \ge 2$, and grows to $4/\pi$ as $n \to \infty$. We give an improved analysis of Kothari et al.'s algorithm. In doing so, we replace the constant $\kappa_n$ with $1 + \eta$ for $\eta > 0$ arbitrary. We also extend the algorithm to more general measures on Riemannian manifolds.Comment: 5 page