Nonisomorphic curves that become isomorphic over extensions of coprime degrees

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K_0 be an arbitrary prime field and let r and s be integers greater than 1 that are coprime to one another. We show that one can find a finite extension K of K_0, a degree-r extension L of K, a degree-s extension M of K, and two curves C and D over K such that C and D become isomorphic to one another over L and over M, but not over any proper subextensions of L/K or M/K. We show that such C and D can never have genus 0, and that if K is finite, C and D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degree extension of F_3. On the other hand, when {r,s}={2,3} we show that genus-2 examples occur in every characteristic other than 3. Our detailed analysis of the case {r,s} = {2,3} shows that over every finite field K there exist nonisomorphic curves C and D that become isomorphic to one another over the quadratic and cubic extensions of K. Most of our proofs rely on Galois cohomology. Without using Galois cohomology, we show that two nonisomorphic genus-0 curves over an arbitrary field remain nonisomorphic over every odd-degree extension of the base field.Comment: LaTeX, 32 pages. Further references added to the discussion in Section 1

Seven Steps Towards the Classical World

Classical physics is about real objects, like apples falling from trees, whose motion is governed by Newtonian laws. In standard Quantum Mechanics only the wave function or the results of measurements exist, and to answer the question of how the classical world can be part of the quantum world is a rather formidable task. However, this is not the case for Bohmian mechanics, which, like classical mechanics, is a theory about real objects. In Bohmian terms, the problem of the classical limit becomes very simple: when do the Bohmian trajectories look Newtonian?Comment: 16 pages, LaTeX, uses latexsy

Scaling of the electron dissipation range of solar wind turbulence

Electron scale solar wind turbulence has attracted great interest in recent years. Clear evidences have been given from the Cluster data that turbulence is not fully dissipated near the proton scale but continues cascading down to the electron scales. However, the scaling of the energy spectra as well as the nature of the plasma modes involved at those small scales are still not fully determined. Here we survey 10 years of the Cluster search-coil magnetometer (SCM) waveforms measured in the solar wind and perform a statistical study of the magnetic energy spectra in the frequency range [$1, 180$]Hz. We show that a large fraction of the spectra exhibit clear breakpoints near the electon gyroscale $\rho_e$, followed by steeper power-law like spectra. We show that the scaling below the electron breakpoint cannot be determined unambiguously due to instrumental limitations that will be discussed in detail. We compare our results to recent ones reported in other studies and discuss their implication on the physical mechanisms and the theoretical modeling of energy dissipation in the SW.Comment: 10 pages, submitte

Ecosystem Services in Decision Making: Time to Deliver

Over the past decade, efforts to value and protect ecosystem services have been promoted by many as the last, best hope for making conservation mainstream – attractive and commonplace worldwide. In theory, if we can help individuals and institutions to recognize the value of nature, then this should greatly increase investments in conservation, while at the same time fostering human well-being. In practice, however, we have not yet developed the scientific basis, nor the policy and finance mechanisms, for incorporating natural capital into resource- and land-use decisions on a large scale. Here, we propose a conceptual framework and sketch out a strategic plan for delivering on the promise of ecosystem services, drawing on emerging examples from Hawai‘i. We describe key advances in the science and practice of accounting for natural capital in the decisions of individuals, communities, corporations, and governments

Ecosystem Services in Decision Making: Time to Deliver

Over the past decade, efforts to value and protect ecosystem services have been promoted by many as the last, best hope for making conservation mainstream – attractive and commonplace worldwide. In theory, if we can help individuals and institutions to recognize the value of nature, then this should greatly increase investments in conservation, while at the same time fostering human well-being. In practice, however, we have not yet developed the scientific basis, nor the policy and finance mechanisms, for incorporating natural capital into resource- and land-use decisions on a large scale. Here, we propose a conceptual framework and sketch out a strategic plan for delivering on the promise of ecosystem services, drawing on emerging examples from Hawai‘i. We describe key advances in the science and practice of accounting for natural capital in the decisions of individuals, communities, corporations, and governments