Entropy accumulation

We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an $n$-partite system $A = (A_1, \ldots A_n)$ corresponds to the sum of the entropies of its parts $A_i$. The Asymptotic Equipartition Property implies that this is indeed the case to first order in $n$, under the assumption that the parts $A_i$ are identical and independent of each other. Here we show that entropy accumulation occurs more generally, i.e., without an independence assumption, provided one quantifies the uncertainty about the individual systems $A_i$ by the von Neumann entropy of suitably chosen conditional states. The analysis of a large system can hence be reduced to the study of its parts. This is relevant for applications. In device-independent cryptography, for instance, the approach yields essentially optimal security bounds valid for general attacks, as shown by Arnon-Friedman et al.Comment: 44 pages; expandable to 48 page

Pipelining Saturated Accumulation

Aggressive pipelining and spatial parallelism allow integrated circuits (e.g., custom VLSI, ASICs, and FPGAs) to achieve high throughput on many Digital Signal Processing applications. However, cyclic data dependencies in the computation can limit parallelism and reduce the efficiency and speed of an implementation. Saturated accumulation is an important example where such a cycle limits the throughput of signal processing applications. We show how to reformulate saturated addition as an associative operation so that we can use a parallel-prefix calculation to perform saturated accumulation at any data rate supported by the device. This allows us, for example, to design a 16-bit saturated accumulator which can operate at 280 MHz on a Xilinx Spartan-3(XC3S-5000-4) FPGA, the maximum frequency supported by the component's DCM

Temporal and spatial variability of snow accumulation in central Greenland

Snow accumulation records from central Greenland are explored to improve the understanding of the accumulation signal in Greenland ice core records. Results from a “forest” of 100 bamboo poles and automated accumulation monitors in the vicinity of Summit as well as shallow cores collected in the Summit and Crete areas are presented. Based on these accumulation data, a regression has been calculated to quantify the signal-to-noise variance ratio of ice core accumulation signals on a variety of temporal (1 week to 2 years) and spatial (20 m to 200 km) scales. Results are consistent with data obtained from year-round automated accumulation measurements deployed at Summit which suggest that it is impossible to obtain regional snow accumulation data with seasonal resolution using four accumulation monitors positioned over a length scale of ∼30 km. Given this understanding of the temporal and spatial dependence of noise in the ice core accumulation signal, the accumulation records from 17 shallow cores are revisited. Each core spans the time period from 1964 to 1983. By combining the accumulation records, the regional snow accumulation record has been obtained for this period. The results show that 9 of the 20 years can be identified as having an accumulation different from the 20 year mean with 99% confidence. The signal-to-noise variance ratio for the average accumulation signal sampled at annual intervals is 5.8±0.5. The averaged accumulation time series may be useful to climate modelers attempting to validate their models with accurate regional hydrologic data sets

Simple threshold rules solve explore/exploit trade‐offs in a resource accumulation search task

How, and how well, do people switch between exploration and exploitation to search for and accumulate resources? We study the decision processes underlying such exploration/exploitation trade‐offs using a novel card selection task that captures the common situation of searching among multiple resources (e.g., jobs) that can be exploited without depleting. With experience, participants learn to switch appropriately between exploration and exploitation and approach optimal performance. We model participants' behavior on this task with random, threshold, and sampling strategies, and find that a linear decreasing threshold rule best fits participants' results. Further evidence that participants use decreasing threshold‐based strategies comes from reaction time differences between exploration and exploitation; however, participants themselves report non‐decreasing thresholds. Decreasing threshold strategies that “front‐load” exploration and switch quickly to exploitation are particularly effective in resource accumulation tasks, in contrast to optimal stopping problems like the Secretary Problem requiring longer exploration

Orders of accumulation of entropy

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if $M$ is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to $M$. These bounds are given in terms of the Cantor-Bendixson rank of \overline{\ex(M)}, the closure of the extreme points of $M$, and the relative Cantor-Bendixson rank of \overline{\ex(M)} with respect to \ex(M). We also address the optimality of these bounds.Comment: 48 page

Reconstructing ice-sheet accumulation rates at ridge B, East Antarctica

Understanding how ice sheets responded to past climate change is fundamental to forecasting how they will respond in the future. Numerical models calculating the evolution of ice sheets depend upon accumulation data, which are principally available from ice cores. Here, we calculate past rates of ice accumulation using internal layering. The englacial structure of the East Antarctic ice divide at ridge B is extracted from airborne ice-penetrating radar. The isochronous surfaces are dated at their intersection with the Vostok ice-core site, where the depth–age relationship is known. The dated isochrons are used as input to a one-dimensional ice-flow model to investigate the spatial accumulation distribution. The calculations show that ice-accumulation rates generally increase from Vostok lake towards ridge B. The western flank of the ice divide experiences markedly more accumulation than in the east. Further, ice accumulation increases northwards along the ice divide. The results also show the variability of accumulation in time and space around the ridge B ice divide over the last 124 000 years

Differential mortality and wealth accumulation

The issue of asset accumulation and decumulation is central to the life cycle theory of consumer behavior and to many policy questions. One of the main implications of the life cycle model is that assets are decumulated in the last part of life. Most empirical studies in this area use cross-sectional data of estimate mean or median wealth-age profiles. The use of cross-sections to estimate the age profile of assets is full of pitfalls. For example, if wealth and mortality are related, in that poorer individuals die younger, one overestimates the last part of the wealth-age profile when using cross-sectional data because means (or other measures of location) are taken over a population which becomes 'richer' as it ages. This paper examines the effect of differential mortality on cross-sectional estimates of wealth-age profiles. Our approach is to quantify the dependence of mortality rates on wealth and use these estimates to 'correct' wealth-age profiles for sample selection due to differential mortality. We estimate mortality rates as a function of wealth and age for a sample of married couples drawn from the Survey of Income and Program Participation (SIPP). Our results show that accounting for differential mortality produces wealth profiles with significantly more dissaving among the elderly