Entropy Loss is Maximal for Uniform Inputs

A secure sketch (defined by Dodis et al.) is an algorithm that on an input w produces an output s such that w can be reconstructed given its noisy version w' and s. Security is defined in terms of two parameters m and m˜ : if w comes from a distribution of entropy m, then a secure sketch guarantees that the distribution of w conditioned on s has entropy m˜ , where λ = m−m˜ is called the entropy loss. In this note we show that the entropy loss of any secure sketch (or, more generally, any randomized algorithm) on any distribution is no more than it is on the uniform distribution.

Differential and numerical models of hysteretic systems with stochastic and deterministic inputs

Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.

Estimation of atmospheric nutrient inputs to the Atlantic Ocean from 50°N to 50°S based on large-scale field sampling: Iron and other dust-associated elements

Atmospheric inputs of mineral dust supply iron and other trace metals to the remote ocean and can influence the marine carbon cycle due to iron's role as a potentially limiting micronutrient. Dust generation, transport, and deposition are highly heterogeneous, and there are very few remote marine locations where dust concentrations and chemistry (e.g., iron solubility) are routinely monitored. Here we use aerosol and rainwater samples collected during 10 large-scale research cruises to estimate the atmospheric input of iron, aluminum, and manganese to four broad regions of the Atlantic Ocean over two 3 month periods for the years 2001–2005. We estimate total inputs of these metals to our study regions to be 4.2, 17, and 0.27 Gmol in April–June and 4.9, 14, and 0.19 Gmol in September–November, respectively. Inputs were highest in regions of high rainfall (the intertropical convergence zone and South Atlantic storm track), and rainfall contributed higher proportions of total input to wetter regions. By combining input estimates for total and soluble metals for these time periods, we calculated overall percentage solubilities for each metal that account for the contributions from both wet and dry depositions and the relative contributions from different aerosol types. Calculated solubilities were in the range 2.4%–9.1% for iron, 6.1%–15% for aluminum, and 54%–73% for manganese. We discuss sources of uncertainty in our estimates and compare our results to some recent estimates of atmospheric iron input to the Atlantic.