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# Order (ring theory)

About: Order (ring theory) is a research topic. Over the lifetime, 41815 publications have been published within this topic receiving 960524 citations.

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01 Jan 1997TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.

Abstract: We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by
$$ Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u $$
is elliptic on Ω ⊂ ℝ n if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q,
$$ Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) $$
[D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝ n × ℝ × ℝ n ] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝ n for Ω a domain in ℝ n . The function u will be in C 2(Ω) unless explicitly stated otherwise.

8,299 citations

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TL;DR: Numerical evidence is presented that this model results in a kinetic phase transition from no transport to finite net transport through spontaneous symmetry breaking of the rotational symmetry.

Abstract: A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation $(\ensuremath{\eta})$ added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $|{\mathbf{v}}_{a}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since $|{\mathbf{v}}_{a}|$ is found to scale as $({\ensuremath{\eta}}_{c}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}0.45$.

6,514 citations

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TL;DR: A form for the exchange-correlation potential in local-density band theory, appropriate for Mott insulators, and finds that all late-3d-transition-metal monoxides, as well as the parent compounds of the high-${\mathit{T}$ compounds, are large-gap magnetic insulators of the charge-transfer type.

Abstract: We propose a form for the exchange-correlation potential in local-density band theory, appropriate for Mott insulators The idea is to use the ``constrained-local-density-approximation'' Hubbard parameter U as the quantity relating the single-particle potentials to the magnetic- (and orbital-) order parameters Our energy functional is that of the local-density approximation plus the mean-field approximation to the remaining part of the U term We argue that such a method should make sense, if one accepts the Hubbard model and the success of constrained-local-density-approximation parameter calculations Using this ab initio scheme, we find that all late-3d-transition-metal monoxides, as well as the parent compounds of the high-${\mathit{T}}_{\mathit{c}}$ compounds, are large-gap magnetic insulators of the charge-transfer type Further, the method predicts that ${\mathrm{LiNiO}}_{2}$ is a low-spin ferromagnet and NiS a local-moment p-type metal The present version of the scheme fails for the early-3d-transition-metal monoxides and for the late 3d transition metals

5,481 citations

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Bell Labs

^{1}TL;DR: In this paper, the Anderson theory of superexchange was extended to include spin-orbit coupling and the antisymmetric spin coupling suggested by Dzialoshinski from purely symmetry grounds and the symmetric pseudodipolar interaction were derived.

Abstract: A theory of anisotropic superexchange interaction is developed by extending the Anderson theory of superexchange to include spin-orbit coupling. The antisymmetric spin coupling suggested by Dzialoshinski from purely symmetry grounds and the symmetric pseudodipolar interaction are derived. Their orders of magnitudes are estimated to be ($\frac{\ensuremath{\Delta}g}{g}$) and ${(\frac{\ensuremath{\Delta}g}{g})}^{2}$ times the isotropic superexchange energy, respectively. Higher order spin couplings are also discussed. As an example of antisymmetric spin coupling the case of Cu${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O is illustrated. In Cu${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O, a spin arrangement which is different from one accepted so far is proposed. This antisymmetric interaction is shown to be responsible for weak ferromagnetism in $\ensuremath{\alpha}$-${\mathrm{Fe}}_{2}$${\mathrm{O}}_{3}$, MnC${\mathrm{O}}_{3}$, and Cr${\mathrm{F}}_{3}$. The paramagnetic susceptibility perpendicular to the trigonal axis is expected to increase very sharply near the N\'eel temperature as the temperature is lowered, as was actually observed in Cr${\mathrm{F}}_{3}$.

5,322 citations

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TL;DR: In this article, a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment is presented, and an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of dissipative environments to the spin-boson problem is given.

Abstract: This paper presents the results of a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment. It is primarily an extended account of results obtained over the last four years by the authors; while they try to provide some background for orientation, it is emphatically not intended as a comprehensive review of the literature on the subject. Its contents include (1) an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of a dissipative environment to the "spin-boson" problem; (2) the derivation of an exact formula for the dynamics of the latter problem; (3) the demonstration that there exists a simple approximation to this exact formula which is controlled, in the sense that we can put explicit bounds on the errors incurred in it, and that for almost all regions of the parameter space these errors are either very small in the limit of interest to us (the "slow-tunneling" limit) or can themselves be evaluated with satisfactory accuracy; (4) use of these results to obtain quantitative expressions for the dynamics of the system as a function of the spectral density $J(\ensuremath{\omega})$ of its coupling to the environment. If $J(\ensuremath{\omega})$ behaves as ${\ensuremath{\omega}}^{s}$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For $sl1$ the system is localized at zero temperature, and at finite $T$ relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. For $sg2$ it undergoes underdamped coherent oscillations for all relevant temperatures, while for $1lsl2$ there is a crossover from coherent oscillation to overdamped relaxation as $T$ increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, $s=1$, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\ensuremath{\alpha}$ as well as the temperature $T$: over most of the ($\ensuremath{\alpha}$,$T$) plane (including the whole region $\ensuremath{\alpha}g1$) the behavior is an incoherent relaxation at a rate proportional to ${T}^{2\ensuremath{\alpha}\ensuremath{-}1}$, but for low $T$ and $0l\ensuremath{\alpha}l\frac{1}{2}$ the authors predict a combination of damped coherent oscillation and incoherent background which appears to disagree with the results of all previous approximations. The case of finite bias is also discussed.

4,047 citations