Random arrangements for lattice designs

From 1936 onwards Yates (11, 12, 13, 14, 15, 16)has introduced a series of incomplete block designs; the purpose of the designs was to segregate the variation within the complete blocks into that due to differences between and that within the incomplete blocks and at the same time to retain the advantages of the randomized complete blocks. The complete block was divided into incomplete blocks, each containing a portion of the total number of the treatments compared, in such a way that the differences between incomplete blocks could be eliminated from the treatment comparisons.https://lib.dr.iastate.edu/specialreports/1003/thumbnail.jp

A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (p.d.f.) of the random shear tensor at a general point in the lens plane due to point masses in the limit of an infinite number of stars. Up to this order, the p.d.f. depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the stars' masses. As a consequence, the p.d.f.s of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic p.d.f. of the shear magnitude in the limit of an infinite number of stars is also presented. Extending to general random distributions of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of {\it global} expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.Comment: To appear in JM

Effectiveness of selection based on variability uncomplicated by heterotic effects

Effectiveness of selection based on variability uncomplicated by heterotic effect

Regularity of higher codimension area minimizing integral currents

This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher codimension, due originally to F. Almgren, which is contained in a series of papers in collaboration with C. De Lellis (University of Zurich).Comment: This text will appear in "Geometric Measure Theory and Real Analysis", pp. 131--192, Proceedings of the ERC school in Pisa (2013), L. Ambrosio Ed., Edizioni SNS (CRM Series

Predicting the effects of climate change on water yield and forest production in the northeastern United States

Rapid and simultaneous changes in temperature, precipitation and the atmospheric concentration of CO2 are predicted to occur over the next century. Simple, well-validated models of ecosystem function are required to predict the effects of these changes. This paper describes an improved version of a forest carbon and water balance model (PnET-II) and the application of the model to predict stand- and regional-level effects of changes in temperature, precipitation and atmospheric CO2 concentration. PnET-II is a simple, generalized, monthly time-step model of water and carbon balances (gross and net) driven by nitrogen availability as expressed through foliar N concentration. Improvements from the original model include a complete carbon balance and improvements in the prediction of canopy phenology, as well as in the computation of canopy structure and photosynthesis. The model was parameterized and run for 4 forest/site combinations and validated against available data for water yield, gross and net carbon exchange and biomass production. The validation exercise suggests that the determination of actual water availability to stands and the occurrence or non-occurrence of soil-based water stress are critical to accurate modeling of forest net primary production (NPP) and net ecosystem production (NEP). The model was then run for the entire NewEngland/New York (USA) region using a 1 km resolution geographic information system. Predicted long-term NEP ranged from -85 to +275 g C m-2 yr-1 for the 4 forest/site combinations, and from -150 to 350 g C m-2 yr-1 for the region, with a regional average of 76 g C m-2 yr-1. A combination of increased temperature (+6*C), decreased precipitation (-15%) and increased water use efficiency (2x, due to doubling of CO2) resulted generally in increases in NPP and decreases in water yield over the region

On the Conduct of an Investigation

12 pages, 1 article*On the Conduct of an Investigation* (Federer, E. H.; Federer, W. T.) 12 page

The enclosure method for the heat equation

This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary value problems whose governing equation is the heat equation is considered. An explicit method to extract an approximation of the value of the support function at a given direction of unknown discontinuity embedded in a heat conductive body from the temperature for a suitable heat flux on the lateral boundary for a fixed observation time is given.Comment: 12pages. This is the final versio

A Theorem on the origin of Phase Transitions

For physical systems described by smooth, finite-range and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that unless the equipotential hypersurfaces of configuration space \Sigma_v ={(q_1,...,q_N)\in R^N | V(q_1,...,q_N) = v}, v \in R, change topology at some v_c in a given interval [v_0, v_1] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (\beta(v_0), \beta(v_1)) also in the N -> \infty$limit. Thus the occurrence of a phase transition at some \beta_c =\beta(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {\Sigma_v}_{v < v_c}$ and the {\Sigma_v}_{v > v_c}, which is the consequence of the existence of critical points of V on \Sigma_{v=v_c}, that is points where \nabla V=0.Comment: 10 pages, Statistical Mechanics, Phase Transitions, General Theory. Phys. Rev. Lett., in pres