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

Fractal dimension crossovers in turbulent passive scalar signals

The fractal dimension $\delta_g^{(1)}$ of turbulent passive scalar signals is calculated from the fluid dynamical equation. $\delta_g^{(1)}$ depends on the scale. For small Prandtl (or Schmidt) number $Pr<10^{-2}$ one gets two ranges, $\delta_g^{(1)}=1$ for small scale r and $\delta_g^{(1)}$=5/3 for large r, both as expected. But for large $Pr> 1$ one gets a third, intermediate range in which the signal is extremely wrinkled and has $\delta_g^{(1)}=2$. In that range the passive scalar structure function $D_\theta(r)$ has a plateau. We calculate the $Pr$-dependence of the crossovers. Comparison with a numerical reduced wave vector set calculation gives good agreement with our predictions.Comment: 7 pages, Revtex, 3 figures (postscript file on request

Modeling physical and chemical climate of the northeastern United States for a geographic information system

A model of physical and chemical climate was developed for New York and New England that can be used in a GIs for integration with ecosystem models. The variables included are monthly average maximum and minimum daily temperatures, precipitation, humidity, and solar radiation, as well as annual atmospheric deposition of sulfur and nitrogen. Equations generated from regional data bases were combined with a digital elevation model of the region to generate digital coverages of each variable

The Arabidopsis mutant feronia disrupts the female gametophytic control of pollen tube reception

Reproduction in angiosperms depends on communication processes of the male gametophyte (pollen) with the female floral organs (pistil, transmitting tissue) and the female gametophyte (embryo sac). Pollen-pistil interactions control pollen hydration, germination and growth through the stylar tissue. The female gametophyte is involved in guiding the growing pollen tube towards the micropyle and embryo sac. One of the two synergids flanking the egg cell starts to degenerate and becomes receptive for pollen tube entry. Pollen tube growth arrests and the tip of the pollen tube ruptures to release the sperm cells. Failures in the mutual interaction between the synergid and the pollen tube necessarily impair fertility. But the control of pollen tube reception is not understood. We isolated a semisterile, female gametophytic mutant from Arabidopsis thaliana, named feronia after the Etruscan goddess of fertility, which impairs this process. In the feronia mutant, embryo sac development and pollen tube guidance were unaffected in all ovules, although one half of the ovules bore mutant female gametophytes. However, when the pollen tube entered the receptive synergid of a feronia mutant female gametophyte, it continued to grow, failed to rupture and release the sperm cells, and invaded the embryo sac. Thus, the feronia mutation disrupts the interaction between the male and female gametophyte required to elicit these processes. Frequently, mutant embryo sacs received supernumerary pollen tubes. We analysed feronia with synergid-specific GUS marker lines, which demonstrated that the specification and differentiation of the synergids was normal. However, GUS expression in mutant gametophytes persisted after pollen tube entry, in contrast to wild-type embryo sacs where it rapidly decreased. Apparently, the failure in pollen tube reception results in the continued expression of synergid-specific genes, probably leading to an extended expression of a potential pollen tube attractant

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

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

On a microcanonical relation between continuous and discrete spin models

A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of a Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a O(n) model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the n = 1 case, i.e., a system of Ising spins with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for n=2 and n=3 in three dimensions. For n=2 in two dimensions (XY model) the conjecture yields a prediction for the critical energy of the Berezinskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.Comment: 5 pages, no figure

The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector fields have been now extended and moved in a self-contained paper available at: arXiv:1708.0139

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

A refined TALDICE-1a age scale from 55 to 112 ka before present for the Talos Dome ice core based on high-resolution methane measurements

A precise synchronization of different climate records is indispensable for a correct dynamical interpretation of paleoclimatic data. A chronology for the TALDICE ice core from the Ross Sea sector of East Antarctica has recently been presented based on methane synchronization with Greenland and the EDC ice cores and &amp;delta;&lt;sup&gt;18&lt;/sup&gt;O&lt;sub&gt;ice&lt;/sub&gt; synchronization with EDC in the bottom part (TALDICE-1). Using new high-resolution methane data obtained with a continuous flow analysis technique, we present a refined age scale for the age interval from 55–112 thousand years (ka) before present, where TALDICE is synchronized with EDC. New and more precise tie points reduce the uncertainties of the age scale from up to 1900 yr in TALDICE-1 to below 1100 yr over most of the refined interval and shift the Talos Dome dating to significantly younger ages during the onset of Marine Isotope Stage 3. Thus, discussions of climate dynamics at sub-millennial time scales are now possible back to 110 ka, in particular during the inception of the last ice age. Calcium data of EDC and TALDICE are compared to show the impact of the refinement to the synchronization of the two ice cores not only for the gas but also for the ice age scale