Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.Comment: 25 page

Thermal transport in granular metals

We study the electron thermal transport in granular metals at large tunnel conductance between the grains, $g_T \gg 1$ and not too low a temperature $T > g_T\delta$, where $\delta$ is the mean energy level spacing for a single grain. Taking into account the electron-electron interaction effects we calculate the thermal conductivity and show that the Wiedemann-Franz law is violated for granular metals. We find that interaction effects suppress the thermal conductivity less than the electrical conductivity.Comment: Replaced with published versio

Approach to a rational rotation number in a piecewise isometric system

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.Comment: LaTeX, 57 pages with 13 figure

Quenching across quantum critical points in periodic systems: dependence of scaling laws on periodicity

We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-1/2 chain with equal xx and yy couplings and subject to a periodically varying magnetic field in the z direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period 2q, where q is a natural number. For a linear quench of the magnetic field strength (or potential strength) at rate 1/\tau across a quantum critical point, we find that the density of defects thereby produced scales as 1/\tau^{q/(q+1)}, deviating from the 1/\sqrt{\tau} scaling that is ubiquitous to a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum k undergoing a non-linear quench as well as by performing numerical simulations. We also find that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of \tau although it may exhibit a cross-over at intermediate values of \tau. Finally, for the case where a zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of q and interaction strength.Comment: 13 pages including 11 figure

Consequences of converting graded to action potentials upon neural information coding and energy efficiency

Information is encoded in neural circuits using both graded and action potentials, converting between them within single neurons and successive processing layers. This conversion is accompanied by information loss and a drop in energy efficiency. We investigate the biophysical causes of this loss of information and efficiency by comparing spiking neuron models, containing stochastic voltage-gated Na+ and K+ channels, with generator potential and graded potential models lacking voltage-gated Na+ channels. We identify three causes of information loss in the generator potential that are the by-product of action potential generation: (1) the voltage-gated Na+ channels necessary for action potential generation increase intrinsic noise and (2) introduce non-linearities, and (3) the finite duration of the action potential creates a ‘footprint’ in the generator potential that obscures incoming signals. These three processes reduce information rates by ~50% in generator potentials, to ~3 times that of spike trains. Both generator potentials and graded potentials consume almost an order of magnitude less energy per second than spike trains. Because of the lower information rates of generator potentials they are substantially less energy efficient than graded potentials. However, both are an order of magnitude more efficient than spike trains due to the higher energy costs and low information content of spikes, emphasizing that there is a two-fold cost of converting analogue to digital; information loss and cost inflation

The Future of U.S. Natural Gas Production, Use, and Trade

Abstract and PDF report are also available on the MIT Joint Program on the Science and Policy of Global Change website (http://globalchange.mit.edu/)Two computable general equilibrium models, one global and the other providing U.S. regional detail, are applied to analysis of the future of U.S. natural gas as an input to an MIT study of the topic. The focus is on uncertainties including the scale and cost of gas resources, the costs of competing technologies, the pattern of greenhouse gas mitigation, and the evolution of global natural gas markets. Results show that the outlook for gas over the next several decades is very favorable. In electric generation, given the unproven and relatively high cost of other low-carbon generation alternatives, gas likely is the preferred alternative to coal. A broad GHG pricing policy would increase gas use in generation but reduce use in other sectors, on a balance increasing its role from present levels. The shale gas resource is a major contributor to this optimistic view of the future of gas, but it is far from a panacea over the longer term. Gas can be an effective bridge to a lower emissions future, but investment in the development of still lower CO2 technologies remains an important priority. Also, international gas resources may well prove to be less costly than those in the U.S., except for the lowest-cost domestic shale resources, and the emergence of an integrated global gas market could result in significant U.S. gas imports.American Clean Skies Foundation, with additional support from the Hess Corporation, the Agencia Nacional de Hidrocarburos (Columbia), the Energy Futures Coalition, and the MIT Energy Initiative

Factorizations and Physical Representations

A Hilbert space in M dimensions is shown explicitly to accommodate representations that reflect the prime numbers decomposition of M. Representations that exhibit the factorization of M into two relatively prime numbers: the kq representation (J. Zak, Phys. Today, {\bf 23} (2), 51 (1970)), and related representations termed $q_{1}q_{2}$ representations (together with their conjugates) are analysed, as well as a representation that exhibits the complete factorization of M. In this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up M

Plasmon oscillations in ellipsoid nanoparticles: beyond dipole approximation

The plasmon oscillations of a metallic triaxial ellipsoid nanoparticle have been studied within the framework of the quasistatic approximation. A general method has been proposed for finding the analytical expressions describing the potential and frequencies of the plasmon oscillations of an arbitrary multipolarity order. The analytical expressions have been derived for an electric potential and plasmon oscillation frequencies of the first 24 modes. Other higher orders plasmon modes are investigated numerically.Comment: 33 pages, 12 figure

Polynomials over quaternions and coquaternions: a unified approach

This paper aims to present, in a unified manner, results which are valid on both the algebras of quaternions and coquaternions and, simultaneously, call the attention to the main differences between these two algebras. The rings of one-sided polynomials over each of these algebras are studied and some important differences in what concerns the structure of the set of their zeros are remarked. Examples illustrating this different behavior of the zero-sets of quaternionic and coquaternionic polynomials are also presented.(undefined)info:eu-repo/semantics/publishedVersio