Zero-Temperature Configurations of Short Odd-Numbered Classical Spin Chains with Bilinear and Biquadratic Exchange Interactions

The lowest energy configurations of short odd open chains with classical spins are determined for antiferromagnetic bilinear and biquadratic nearest-neighbor exchange interactions. The zero field residual magnetization generates differences with the magnetic behavior of even chains, as the odd chain is like a small magnet for weak magnetic fields. The lowest energy configuration is calculated as a function of the total magnetization M, even for M less than the zero field residual magnetization. Analytic expressions and their proofs are provided for the threshold magnetic field needed to drive the system away from the antiferromagnetic configuration and the spin polar angles in its vicinity, when the biquadratic interaction is relatively weak. They are also given for the saturation magnetic field and the spin polar angles close to it. Finally, an analytic expression along with its proof is given for the maximum magnetization in zero magnetic field for stronger biquadratic interaction, where the lowest energy configuration is highly degenerate.Comment: 17 pages, 9 figure

Thermalization away from Integrability and the Role of Operator Off-Diagonal Elements

We investigate the rate of thermalization of local operators in the one-dimensional anisotropic antiferromagnetic Heisenberg model with next-nearest neighbor interactions that break integrability. This is done by calculating the scaling of the difference of the diagonal and canonical thermal ensemble values as function of system size, and by directly calculating the time evolution of the expectation values of the operators with the Chebyshev polynomial expansion. Spatial and spin symmetry is exploited and the Hamiltonian is divided in subsectors according to their symmetry. The rate of thermalization depends on the proximity to the integrable limit. When integrability is weakly broken thermalization is slow, and becomes faster the stronger the next-nearest neighbor interaction is. Three different regimes for the rate of thermalization with respect to the strength of the integrability breaking parameter are identified. These are shown to be directly connected with the relative strength of the low and higher energy difference off-diagonal operator matrix elements in the symmetry eigenbasis of the Hamiltonian. Close to the integrable limit the off-diagonal matrix elements peak at higher energies and high frequency fluctuations are important and slow down thermalization. Away from the integrable limit a strong low energy peak gradually develops that takes over the higher frequency fluctuations and leads to quicker thermalization.Comment: 11 pages, 9 figure

Leveraging Coding Techniques for Speeding up Distributed Computing

Large scale clusters leveraging distributed computing frameworks such as MapReduce routinely process data that are on the orders of petabytes or more. The sheer size of the data precludes the processing of the data on a single computer. The philosophy in these methods is to partition the overall job into smaller tasks that are executed on different servers; this is called the map phase. This is followed by a data shuffling phase where appropriate data is exchanged between the servers. The final so-called reduce phase, completes the computation. One potential approach, explored in prior work for reducing the overall execution time is to operate on a natural tradeoff between computation and communication. Specifically, the idea is to run redundant copies of map tasks that are placed on judiciously chosen servers. The shuffle phase exploits the location of the nodes and utilizes coded transmission. The main drawback of this approach is that it requires the original job to be split into a number of map tasks that grows exponentially in the system parameters. This is problematic, as we demonstrate that splitting jobs too finely can in fact adversely affect the overall execution time. In this work we show that one can simultaneously obtain low communication loads while ensuring that jobs do not need to be split too finely. Our approach uncovers a deep relationship between this problem and a class of combinatorial structures called resolvable designs. Appropriate interpretation of resolvable designs can allow for the development of coded distributed computing schemes where the splitting levels are exponentially lower than prior work. We present experimental results obtained on Amazon EC2 clusters for a widely known distributed algorithm, namely TeraSort. We obtain over 4.69$\times$ improvement in speedup over the baseline approach and more than 2.6$\times$ over current state of the art

Assessing the Rise of Organic Farming in the European Union: Environmental and Socio-economic Consequences

Although organic farming is considered the poster child of rural development in Europe, there is little empirical evidence assessing its success in achieving the ambitious environmental and socio-economic objectives that it is purported to assist. This paper presents empirical evidence from the growth of organic farming in Europe over the past two decades that questions the highly optimistic claims of policy makers. Although policies in support of organic impact have had an overall positive environmental impact, their social impact is ambiguous, as organic farming appears to have grown more in areas with larger average farm sizes. Additionally, contrary to what is often assumed, organic farms in Europe display larger average sizes and lower rates of labor intensity than their conventional counterparts, casting doubts on the efficacy of organic farms to allow family farmers to remain in the countryside as high-value producers. I assert that this development should be viewed as evidence of the "conventionalization" of organic farming, and suggest that policy makers take into account the transformations of the structures of production, which benefit from the support for organic farming. Treating the experience of organic farmers in the EU as a lesson for schemes paying for environmental services, I suggest that the success of organic farming should be evaluated by the numbers of participating farmers, rather than by area covered, as has been the predominant approach so far. Finally, I assert that strong agricultural cooperatives are necessary to secure a long-lasting passage of small farmers to organic methods of production. JEL Categories: Q1, Q58, O52Organic Farming, European Union, Environment