Showing papers on "Order (ring theory) published in 1987"

Dynamics of the dissipative two-state system

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.

Quantum Theory of Gravity. I. The Canonical Theory

Abstract: Following an historical introduction, the conventional canonical formulation of general relativity theory is presented. The canonical Lagrangian is expressed in terms of the extrinsic and intrinsic curvatures of the hypersurface ${x}^{0}=\mathrm{constant}$, and its relation to the asymptotic field energy in an infinite world is noted. The distinction between finite and infinite worlds is emphasized. In the quantum theory the primary and secondary constraints become conditions on the state vector, and in the case of finite worlds these conditions alone govern the dynamics. A resolution of the factor-ordering problem is proposed, and the consistency of the constraints is demonstrated. A 6-dimensional hyperbolic Riemannian manifold is introduced which takes for its metric the coefficient of the momenta in the Hamiltonian constraint. The geodesic incompletability of this manifold, owing to the existence of a frontier of infinite curvature, is demonstrated. The possibility is explored of relating this manifold to an infinite-dimensional manifold of 3-geometries, and of relating the structure of the latter manifold in turn to the dynamical behavior of space-time. The problem is approached through the WKB approximation and Hamilton-Jacobi theory. Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a "force term." The classical phenomenon of gravitational collapse shows that the force term is not powerful enough to prevent the trajectory of space-time from running into the frontier. The as-yet unresolved problem of determining when the collapse phenomenon represents a real barrier to the quantum-state functional is briefly discussed, and a boundary condition at the barrier is proposed. The state functional of a finite world can depend only on the 3-geometry of the hypersurface ${x}^{0}=\mathrm{constant}$. The label ${x}^{0}$ itself is irrelevant, and "time" must be determined intrinsically. A natural definition for the inner product of two such state functionals is introduced which, however, encounters difficulties with negative probabilities owing to the barrier boundary condition. In order to resolve these difficulties, a simplified model, the quantized Friedmann universe, is studied in detail. In order to obtain nonstatic wave functions which resemble a universe evolving, it is necessary to introduce a clock. In order that the combined wave functions of universe-cum-clock be normalizable, it turns out that the periods of universe and clock must be commensurable. Wave packets exhibiting quasiclassical behavior are constructed, and attention is called to the phenomenological character of "time." The innerproduct definition is rescued from its negative-probability difficulties by making use of the fact that probability flows in a closed finite circuit in configuration space. The article ends with some speculations on the uniqueness of the state functional of the actual universe. It is suggested that a viewpoint due to Everett should be adopted in its interpretation.

Power diagrams: properties, algorithms and applications

Abstract: The power pow $(x,s)$ of a point x with respect to a sphere s in Euclidean d-space $E^d $ is given by $d^2 (x,z) - r^2 $, where d denotes the Euclidean distance function, and z and r are the center and the radius of s. The power diagram of a finite set S of spheres in $E^d $ is a cell complex that associates each $s \in S$ with the convex domain $\{ x \in E^d | {\operatorname{pow}} (x,s) < {\operatorname{pow}} (x,t), {\text{ for all }} t \in S - \{ s\} \}$.The close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited. Efficient algorithms that compute the power diagram and its order-k modifications are obtained. Among the applications of these results are algorithms for detecting k-sets, for union and intersection problems for cones and paraboloids, and for constructing weighted Voronoi diagrams and Voronoi diagrams for spheres. Upper space bounds for these geometric problems are derived.

Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field.

Abstract: The conductance of any metallic sample has been shown to fluctuate as a function of chemical potential, magnetic field, or impurity configuration by an amount of order ${e}^{2}$/h independent of sample size and degree of disorder at zero temperature. We discuss the relationship of these results to other results in the theory of weak and strong localization, and discuss its physical implications. We discuss the physical assumptions underlying the ergodic hypothesis used to relate theory to experiment. We review the zero-temperature theory and provide a detailed discussion of the conductance correlation functions in magnetic field and Fermi energy. We show that the zero-temperature amplitude of the fluctuations is unaffected by electron-electron interactions to lowest order in (${k}_{f}$l${)}^{\mathrm{\ensuremath{-}}1}$, and at finite temperature interactions only enter insofar as they contribute to the inelastic scattering rate. We calculate the effects of finite temperature on both the amplitude of the fluctuations and their scale. We discuss the conditions for dimensional crossover at finite temperature, and the behavior of different experimental measures of the fluctuation amplitude, in order to facilitate quantitative comparisons of experiment and theory.

New high-precision comparison of electron and positron g factors.

Abstract: Single electrons and positrons have been alternately isolated in the same compensated Penning trap in order to form the geonium pseudoatom under nearly identical conditions. For each, the g-factor anomaly is obtained by measurement of both the spin-cyclotron difference frequency and the cyclotron frequency. A search for systematic effects uncovered a small (but common) residual shift due to the cyclotron excitation field. Extrapolation to zero power yields ${e}^{+}$ and ${e}^{\mathrm{\ensuremath{-}}}$ g factors with a smaller statistical error and a new particle-antiparticle comparison: g(${e}^{\mathrm{\ensuremath{-}}}$)/ g(${e}^{+}$)=1+(0.5\ifmmode\pm\else\textpm\fi{}2.1)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$. . AE

Magnetic excitations and ordering in the heavy-electron superconductor URu2Si2.

Abstract: Neutron scattering demonstrates the coexistence of antiferromagnetic order and superconductivity below 1 K in the heavy-electron system ${\mathrm{URu}}_{2}$${\mathrm{Si}}_{2}$. It is found that the upper (17.5 K) transition is to an antiferromagnetic phase with a (100) modulation wave vector and spins along the tetragonal c-axis. The ordered moment is unusually small, (0.03\ifmmode\pm\else\textpm\fi{}0.01)${\mathrm{\ensuremath{\mu}}}_{\mathrm{B}}$. However, spin waves develop from damped, finite-gap excitations above ${\mathrm{T}}_{\mathrm{N}}$, and they are intense, propagating, and longitudinal with a zone centre gap of 1.8 meV.

p -spin-interaction spin-glass models: Connections with the structural glass problem

Abstract: The static and the dynamical theories for the mean-field p-spin (pg2) interaction spin-glass model are studied. A broken-replica-symmetric equilibrium solution leads to a glass transition at a temperature ${T}_{g}^{\mathcal{'}}$ where the Edwards-Anderson order parameter is discontinuous but where there is no latent heat and there is a discontinuous specific heat. The dynamical theory leads to a continuous slowing down and predicts a glass transition at ${T}_{g}$g${T}_{g}^{\mathcal{'}}$. A reinterpretation of the equilibrium solution allows us to relate the dynamical transition to the equilibrium theory. The mathematical structure of the mean-field dynamical theory is closely related to certain recent dynamical theories of the structural glass transition.

Ordering in lipid monolayers studied by synchrotron x-ray diffraction and fluorescence microscopy.

Abstract: Synchrotron x-ray diffraction results of phospholipid monolayers at an air/water interface are reported and compared with fluorescence microscopic observations of the fluid-gel lipid phase transition. The pressure-induced transition to an orientationally ordered phase occurs at a pressure ${\ensuremath{\pi}}_{c}$ which is by more than 15 mN/m below a pressure ${\ensuremath{\pi}}_{s}$ where a transition to a positionally ordered phase occurs. Hence there exists an intermediate phase with long-range orientational and short-range positional order.

Magnetostatic trapping fields for neutral atoms

Abstract: In view of the recent successful confinement of decelerated sodium atoms in a magnetostatic trap, it is of interest to evaluate possible trap-field configurations. Neutral atoms in a Zeeman sublevel whose energy increases with field can be confined by a field whose magnitude \ensuremath{\Vert}B\ensuremath{\Vert} increases with distance from the center. Because this same basic requirement applies also to traps for neutrons and for plasmas (in the guiding-center approximation), trap configurations developed previously for these purposes are of interest for neutral atoms. However, the desired properties differ considerably because of very different objectives and different behavior of very cold atoms as compared with hot plasmas. We characterize basic trap configurations using both the exact expressions for the field, and a multipole polynomial expansion that facilitates studies of symmetry properties and classical or quantum orbits. Polynomial terms for the field components are derived and coefficients obtained by comparison with Taylor-series expansions and by global fit. Contours of \ensuremath{\Vert}B\ensuremath{\Vert} for various trap configurations are also presented. Under certain restrictive conditions, \ensuremath{\Vert}B\ensuremath{\Vert}, and hence the effective potential, can be made isotropic to second order.

Nature of the Griffiths phase.

Abstract: Arguments are given that, for random spin systems, the density of states \ensuremath{\rho}(\ensuremath{\mu}) of the inverse of the susceptibility matrix vanishes as \ensuremath{\rho}(\ensuremath{\mu})\ensuremath{\sim}exp(-A/\ensuremath{\mu}), for \ensuremath{\mu}\ensuremath{\rightarrow}0, throughout the ``Griffiths phase.'' The amplitude A vanishes at the onset of magnetic long-range order, and diverges at the transition between ``Griffiths'' and ``paramagnetic'' phases. For an O(m) spin system, with m\ensuremath{\rightarrow}\ensuremath{\infty}, the spin autocorrelation function C(t) is found to have the ``stretched-exponential'' form, lnC(t)\ensuremath{\sim}-(At${)}^{1/2}$, in the Griffiths phase.

Neutron spin echo study of dynamic correlations near the liquid-glass transition

Abstract: Neutron spin-echo measurements have been performed on the ionic system ${\mathrm{Ca}}_{0.4}$${\mathrm{K}}_{0.6}$(${\mathrm{NO}}_{3}$${)}_{1.4}$ around the glass transition temperature ${\mathrm{T}}_{0}$, in order to determine the time dependence of the density correlation function ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{q}}$(t), which plays a central role in recent theories. Above ${\mathrm{T}}_{0}$ the results reveal that ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{q}}$(t) contains two distinct ``slow'' components, the slower of which (a) has a stretched exponential form exp[-(t/\ensuremath{\tau}${)}^{\mathrm{\ensuremath{\beta}}}$], (b) shows a slowing down as T approaches ${\mathrm{T}}_{0}$ which scales with the Stokes-Einstein diffusion constant, and (c) tends to become the nonergodic fraction of the structure factor at ${\mathrm{T}}_{0}$. .AE

Crystallographic description of phases in the Y-Ba-Cu-O superconductor

Abstract: Single-crystal x-ray techniques were used to determine two structures that comprise the Y-Ba-Cu-O superconductor. A green phase has the orthorhombic ${\mathrm{Y}}_{2}$${\mathrm{BaCuO}}_{5}$ structure. An opaque phase (${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6+\mathrm{x}}$, 0\ensuremath{\le}x\ensuremath{\le}1.0) is tetragonal (a=3.859 A\r{}, c=11.71 A\r{}; space group P4\ifmmode\bar\else\textasciimacron\fi{}m2;Z=1). The structure is related to a 1:1:3 AB${\mathrm{O}}_{3}$ perovskite: Ba and Y order in A sites and Cu is in B sites. Systematic oxygen vacancies between adjacent copper atoms lead to a fivefold-coordinate Cu in proximity to Y and a twofold-coordinate Cu in the vicinity of the Ba site.

Transition to turbulence via spatio-temporal intermittency

Abstract: The study of low-dimensional dissipative dynamical systems has provided a reasonable understanding of the transition to temporal chaos in strongly confined systems for which the spatial structure can be considered as frozen. The situation is still less advanced for weakly confined systems where chaos has both a spatial and a temporal meaning. In order to approach the specificities of the latter, we have chosen to study first a partial differential equation (PDE) displaying a convective-type nonlinear term, steady cellular solutions as in convection, and a transition to spatiotemporal chaos, namely the damped Kuramoto-Sivashinsky (KS) equation: $${\partial _t}\phi + \eta \phi + {\partial _{{x^2}}}\phi + {\partial _{{x^4}}}\phi + 2\phi {\partial _x}\phi = 0.$$ (1)

Stochastic random network model in Ge and Si chalcogenide glasses.

Abstract: A stochastic random network model is proposed for the structure of ${\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{S}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$, ${\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$, ${\mathrm{Si}}_{\mathrm{x}}$${\mathrm{S}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$, and ${\mathrm{Si}}_{\mathrm{x}}$${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ (x\ensuremath{\le}0.33) glasses. This model is constructed to explain the existence of two types of microcrystalline states induced by photoirradiation above or below the threshold intensity. This model characterizes the glass structure by one parameter P which is related to the existing probability of the edge-sharing bonds between the tetrahedral ${\mathrm{MX}}_{4}$ molecules relative to the corner-sharing bonds. P depends only on the species of atoms forming the glass and not on x. In order to prove the validity of the present model, Raman scattering experiments were made and the x dependence of the intensity ratio of the ${A}_{1}^{c}$ companion peak to the ${A}_{1}$ peak, I(${A}_{1}^{c}$)/I(${A}_{1}$), was obtained. From the viewpoint of phonon localization, the ${A}_{1}$ mode is assigned to the breathing mode of ${\mathrm{MX}}_{4}$ molecules and the ${A}_{1}^{c}$ mode to the vibration of chalcogen atoms on the edge-sharing double bonds. The x dependence of the intensity ratio I(${A}_{1}^{c}$)/I(${A}_{1}$) calculated by the present model is in good agreement with the experimentally obtained ratio. The P obtained increases in order from ${\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{S}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$, ${\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$, ${\mathrm{Si}}_{\mathrm{x}}$${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ to ${\mathrm{Si}}_{\mathrm{x}}$${\mathrm{S}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ with the same order of tendency of getting edge-sharing bonds in the crystals. The value of P is independent of the method of making the amorphous but it can be changed by photoirradiation. P decreases with irradiation below the threshold intensity, but it increases with irradiation above the threshold. The local energy in the glass is lower in the corner-sharing bonds, but the total energy is lowest in the same structure as the crystal. The threshold irradiation intensity for Se glass is less than one-hundredth of that for ${\mathrm{GeSe}}_{2}$ glass.

Intersubband relaxation in GaAs- Al x Ga 1 − x As quantum well structures observed directly by an infrared bleaching technique

Abstract: Direct intersubband excitations in narrow, modulation doped GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As quantum well structures are studied by picosecond infrared spectroscopy. The bleaching of the intersubband absorption induced by a high-intensity picosecond pump pulse, is studied by a delayed probe pulse. Typical intersubband relaxation times are of the order of 10 ps at 300 K for a subband splitting of about 150 meV. Polar LO-phonon scattering is discussed as the most relevant relaxation mechanism.

Giant oscillator strength of free excitons in GaAs

Abstract: The radiative lifetime of free excitons in GaAs is unambiguously determined for the first time using ultrapure material grown by molecular-beam epitaxy. Its value amounts to 3.3\ifmmode\pm\else\textpm\fi{}0.5 ns in the limit of very low temperatures. Consequently, the oscillator strength of free excitons is on the order of unity, in contrast to what commonly is deduced from the total absorption cross section. At excitation densities above 0.5 W/${\mathrm{cm}}^{2}$ we have observed the radiative recombination of excitonic molecules (biexcitons) at 1.5146 eV in accordance with theoretical considerations.

Bounds on hadronic axions from stellar evolution

Abstract: We consider in detail the effect of the emission of ``hadronic'' invisible axions (which do not couple to electrons) from the interior of stars on stellar evolution. To this end we calculate plasma emission rates for axions due to the Primakoff process for the full range of conditions encountered in a giant star. Much attention is paid to plasma, degeneracy, and screening effects. We reconsider the solar bound by evolving a 1.0 ${M}_{\ensuremath{\bigodot}}$ star to solar age and lowering the presolar helium abundance so as to obtain the correct present-day luminosity of the Sun. The previous bound on the axion-photon coupling of ${G}_{9}$\ensuremath{\lesssim}2.5 (corresponding to ${m}_{a}$\ensuremath{\lesssim}17 eV R where R is a model-dependent factor of order unity) is confirmed, where ${G}_{9}$ is the coupling constant G in units of ${10}^{\mathrm{\ensuremath{-}}9}$ ${\mathrm{GeV}}^{\mathrm{\ensuremath{-}}1}$. We then follow the evolution of a 1.3${M}_{\ensuremath{\bigodot}}$ star from zero age to the top of the giant branch.Helium ignites for all values of G consistent with the solar bound; however, the core mass, surface temperature, and luminosity at the helium flash exceed the standard values. The luminosity at the helium flash is larger than about twice the standard value unless ${G}_{9}$\ensuremath{\lesssim}0.3 (corresponding to ${m}_{a}$\ensuremath{\lesssim}2 eV R), in conflict with observational data, which are statistically weak, however. We find our most stringent limits from the helium-burning lifetime. In the absence of axion cooling we calculate a lifetime of 1.2\ifmmode\times\else\texttimes\fi{}${10}^{8}$ yr which corresponds well with the value 1.5\ifmmode\times\else\texttimes\fi{}${10}^{8}$ yr derived from the number of red giants in the ``clump'' of the open cluster M67 and with the value 1.3\ifmmode\times\else\texttimes\fi{}${10}^{8}$ yr derived from the number of such stars in the old galactic disk population. We obtain a conservative limit of ${G}_{9}$0.3 which, at saturation, results in a helium-burning lifetime an order of magnitude low. We believe that ${G}_{9}$\ensuremath{\lesssim}0.1 (${m}_{a}$\ensuremath{\lesssim}0.7 eV R) is a reasonably safe limit which, if saturated, leads to a calculated helium-burning lifetime a factor of 2 below the observed value. Our results exclude the recently suggested possibility of detecting cosmic axions through their 2\ensuremath{\gamma} decay and probably the possibility of measuring the solar hadronic axion flux which, according to our bounds, must be less than 2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$ of the solar luminosity. There remains a narrow range of parameters (0.01\ensuremath{\lesssim}${G}_{9}$\ensuremath{\lesssim}0.1, ${m}_{a}$\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}4}$ eV) in which a recently proposed laboratory experiment might still measure axionlike particles.

Multipolar response of small metallic spheres: Nonlocal theory

Abstract: The multipolar response of a small metallic sphere is studied with use of a nonlocal dielectric function. Results obtained with the hydrodynamic and Lindhard-Mermin models are presented and compared to those given by the local Drude model. We find an enhancement of the imaginary part of the multipole polarizabilities at low frequencies and pole order l${l}_{c}$, where ${l}_{c}$ is a cutoff order that corresponds to excitations at the high-wave-vector edge of the electron-hole pair continuum. The absorption coefficient for two very close spheres is calculated and the effect of nonlocality on the number and position of the multipolar absorption peaks is discussed.

Structure of the Ag/Si(111) surface by scanning tunneling microscopy.

Abstract: We present the first real-space images, obtained by scanning tunneling microscopy, which show the ordered atomic structures of the Ag/Si(111) surface for several submonolayer coverages. The (\ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 )R30\ifmmode^\circ\else\textdegree\fi{} surface is shown to be of the honeycomb structure, and the room-temperature coverage is determined to be theta=(2/3). The (1\ifmmode\times\else\texttimes\fi{}3) overlayer consists predominantly of rows of Ag atoms separated by two empty rows. At low coverage the surface shows long-range 7\ifmmode\times\else\texttimes\fi{}7 order, but patches occur within the 7\ifmmode\times\else\texttimes\fi{}7 cells whree Si adatoms are absent.

Estimating a Density under Order Restrictions: Nonasymptotic Minimax Risk

Abstract: Let us consider the class of all unimodal densities defined on some interval of length $L$ and bounded by $H$; we shall study the minimax risk over this class, when we estimate using $n$ i.i.d. observations, the loss being measured by the $\mathbb{L}^1$ distance between the estimator and the true density. We shall prove that if $S = \operatorname{Log}(HL + 1)$, upper and lower bounds for the risk are of the form $C(S/n)^{1/3}$ and the ratio between those bounds is smaller than 44 when $S/n$ is smaller than 220$^{-1}$.

Resistance of a one-atom contact in the scanning tunneling microscope

Abstract: The resistance as a function of tip-sample separation in the scanning tunneling microscope is calculated for distances in the transition region between tunneling and point contact. A resistance plateau appears near point contact with value $\frac{A\ensuremath{\pi}\ensuremath{\hbar}}{{e}^{2}}$, where $A$ is of order unity, its exact value depending on the identity of the tip atom. Good agreement is found with the recent experimental data of Gimzewski and M\"oller.

Morita equivalence for rings withot identity

Abstract: In the paper [1] Abrams made a first step in extending the theory of Morita equivalence to rings without identity. He considered rings in which a set of commuting idempotents is given such that every element of the ring admits one of these idempotents as a two-sided unit, and the categories of all left modules over these rings which are unitary in a natural sense. He proved that two such module categories over the rings R and S, say, are equivalent if and only if there exists a unitary left i?-module P which is a generator, the direct limit of a given kind of system of finitelygenerated projective modules, and such that S is isomorphic to the ring of certain endomorphisms of P. The aim of the present paper is to extend this theory in two ways: to cover a wider range of rings, and to transfer more of the classical Morita theory. Firstly, one can weaken the condition of commutativity of the idempotents in question: it sufficesto require that any two of them have a common upper bound under the natural partial order (i.e., any two elements of the ring admit a common two-sided identity), a condition which is fulfilledby all regular rings (regular in the sense of Neumann). Whenever one has such a system of idempotents, then any larger system, in particular, the set of all idempotents, is also such, which is not the case for the systems of Abrams. Secondly, by a suitable modification of some homological lemmas we obtain also the two-sided characterizations of Morita equivalence, arriving thus at a complete analogy to the classicalcase of rings with identity. Our presentation is a combination of those in Anderson-Fuller [2], §§21-22,and Bass [5] (see also [6], Chapter II). This machinery allows us to avoid the elaborate construction of Abrams. As examples we describe, among others, those rings with local units which are Morita equivalent to division rings and primary rings, respectively. The Rees matrix rings studied in [4] turn out to have a natural place in this theory. The theory we present here is a counterpart of the theory of Morita duality developed by Yamagata [10]. On the one hand, we shall use the same modified Hom-functors but for projective and not injective modules, and on the

Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations

Abstract: The Arbitrary Lagrangian-Eulerian (ALE) method in computational fluid dynamics requires the periodic remapping of conserved quantities such as mass, momentum, and energy from a Lagrangian mesh to some other arbitrarily defined mesh. This procedure is a type of interpolation which is usually constrained to be conservative and monotone. It is typically carried out by solving a partial differential equation analogous to the continuity equation. Alternatively, the remapping may be carried out using an integral formulation which is (for the remapping of mass) \[ m_k = \iiint_{V_k } {\rho ({\bf r}) dV,} \] where $m_k $ is the mass of a cell k of the new mesh whose volume is $V_k $, and $\rho (\bf {r})$ is the known density distribution on the old (Lagrangian) mesh. Remapping using this integral method avoids many drawbacks of the continuous method but the evaluation of such integrals is costly and difficult for arbitrary meshes. Recently, very efficient methods [6], [7] have been demonstrated for the case of constant cell density in two dimensions. We now extend these methods to the much more accurate case in which the density distribution in linear within each cell. This results in second order accuracy. However, to preserve monotonicity the gradients are limited using the option of two simple local limiters. To the extent that the limiting is applied the method locally reverts to first order accuracy.

A new distributed algorithm to find breadth first search trees

Abstract: A new distributed algorithm is presented for constructing breadth first search (BFS) trees. A BFS tree is a tree of shortest paths from a given root node to all other nodes of a network under the assumption of unit edge weights; such trees provide useful building blocks for a number of routing and control functions in communication networks. The order of communication complexity for the new algorithm is O(V^{1.6} + E) where V is the number of nodes and E the number of edges. For dense networks with E \geq V^{1.6} this order of complexity is optimum.

Solving tridiagonal systems on ensemble architectures

Abstract: The concurrent solution of tridiagonal systems on linear and 2-dimensional arrays, complete binary trees, shuffle-exchange and perfect shuffle networks, and boolean cubes by elimination methods are devised and analyzed The methods can be obtained by symmetric permutations of some rows and columns, and amounts to cyclic reduction or a combination of Gaussian elimination and cyclic reduction (GECR) The ensembles have only local storage and no global control Synchronization is accomplished via message passing to neighboring processorsThe parallel arithmetic complexity of GECR for N equations on a K processor ensemble is $O({ N / K } + \log _2 K)$, and the communication complexity is $O(K)$ for the linear array, $O(\sqrt K )$ for the 2-dimensional mesh, and $O(\log _2 K)$ for the networks of diameter $O(\log _2 K)$ The maximum speed-up for the linear array is attained at $K \approx {({ N / \alpha })}^{1/2} $ and for the 2-d mesh at $K \approx ({N / 2\alpha })^{2/3} $, where $\alpha $ (the time to communicate one floating-point number)/(the time for a floating-point arithmetic operation) For the binary tree the maximum speed-up is attained at $K = N$, and for the perfect shuffle and boolean k-cube networks, $K = N/(1 + \alpha )$ yields the maximum speed-up The minimum time complexity is of order $O(N^{1/2} )$ for the linear array, of order $O(N^{1/3} )$ for the mesh, and of order $O(\log _2 N)$ for the binary tree, the shuffle-exchange, the perfect shuffle and the boolean k-cubeThe relative decrease in computational complexity due to a truncation of the reduction process in a highly concurrent system is much greater than on a uniprocessor The reduction in the arithmetic complexity is proportional to the number of steps avoided, if the number of processing elements equals the number of equations So also is the reduction in the communication complexity for ensembles configured as binary trees, shuffle-exchange and perfect shuffle networks, and boolean cubesPartitioning the ensemble into subsets of processors is shown to be more efficient for the solution of multiple independent problems than pipelining the solutions over the entire ensemble A balanced cyclic reduction algorithm is presented for the case where each system is spread uniformly over the processing elements, and its complexity is compared with Gaussian elimination

Adaptive Prediction by Least Squares Predictors in Stochastic Regression Models with Applications to Time Series

Abstract: Herein we consider the asymptotic performance of the least squares predictors $\hat{y}_n$ of the stochastic regression model $y_n = \beta_1 x_{n1} + \cdots + \beta_p x_{np} + \varepsilon_n$. In particular, the accumulated cost function $\sum^n_{k=1} (y_k - \hat{y}_k - \varepsilon_k)^2$ is studied. The results are then applied to nonstationary autoregressive time series. A statistic is also constructed to show how many times one should difference a nonstationary time series in order to obtain a stationary series.

Generation of random orthogonal matrices

Abstract: In order to generate a random orthogonal matrix distributed according to Haar measure over the orthogonal group it is natural to start with a matrix of normal random variables and then factor it by the singular value decomposition. A more efficient method is obtained by using Householder transformations. We propose another alternative based on the product of ${{n(n - 1)}/2}$ orthogonal matrices, each of which represents an angle of rotation. Some numerical comparisons of alternative methods are made.

Random-field smearing of the proton-glass transition

Abstract: Within a replica-symmetric mean-field theory we have studied the proton pseudo-spin-glass behavior of a random-bond classical Ising system in a homogeneous transverse field \ensuremath{\Omega} and a random longitudinal field. This model is expected to describe some properties of the mixed hydrogen-bonded ferro- and antiferroelectric crystals such as ${\mathrm{Rb}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$(${\mathrm{NH}}_{4}$${)}_{\mathit{x}}$${\mathrm{H}}_{2}$${\mathrm{PO}}_{4}$ which have recently been investigated experimentally. It is shown that in the presence of Gaussian random fields with zero mean and variance \ensuremath{\Delta} the proton-glass transition is smeared out, i.e., the cusp in the dielectric susceptibility is rounded off and the proton-glass order parameter remains finite at temperatures above the nominal freezing temperature. However, the average dielectric polarization is strictly zero for a symmetric bond distribution. We have also determined the limits of stability of the replica-symmetric solution for the case of a deuterated system (\ensuremath{\Omega}=0). The replica-symmetric proton-glass phase is separated from the phase with broken replica symmetry by a line of instability in the (T,\ensuremath{\Delta}) plane. The crossing of this line is thus connected with a phase transition which persists in the presence of random fields. Finally, the distribution function of local parallel fields P(h) determining the magnetic resonance line shape has been calculated within the random-field model and the results applied to interpret some recent experimental data.

Degrees of freedom of turbulence

Abstract: We compute in the framework of a multifractal model for three-dimensional fully developed turbulence the number of degrees of freedom N as function of the Reynolds number R. $N--- depends on the whole spectrum of singularities h related to the anomalous scaling of the velocity differences. On the other hand, we have also considered what is the total number ${N}_{T}^{\mathrm{*}}$ of equations needed in a computer simulation since N just has theoretical relevance. We stress, however, that the main features of intermittency can be described by an effective number N${\ifmmode \tilde{}\else \~{}\fi{}}^{\mathrm{*}}$ which is much smaller than ${N}_{T}^{\mathrm{*}}$ because N${\ifmmode \tilde{}\else \~{}\fi{}}^{\mathrm{*}}$ neglects very improbable events. We show that ${N}_{T}^{\mathrm{*}}$\ensuremath{\propto}${R}^{3}$ while we get from a fit of experimental data that N${\ifmmode \tilde{}\else \~{}\fi{}}^{\mathrm{*}}$\ensuremath{\propto}${R}^{2.3}$ is of the same order of N\ensuremath{\propto}${R}^{2.2}$. .AE

Pressure-induced bond-angle variation in amorphous SiO 2

Abstract: Magic-angle spinning nuclear magnetic resonance measurements have been performed on $^{29}\mathrm{Si}$ nuclei in amorphous silica before and after densification \ensuremath{\sim}16% by pressure compaction at 50 kbar and 600?deC. The pressure-induced shift in the Si---O---Si bond-angle distribution maximum from 143\ifmmode^\circ\else\textdegree\fi{} to 138\ifmmode^\circ\else\textdegree\fi{} is consistent with results obtained from x-ray scattering and Raman spectroscopy. Variation in the bond-angle distribution is discussed in terms of recent models for the structure of amorphous ${\mathrm{SiO}}_{2}$. It is concluded that the important parameter modified by densification is the oxygen--second-nearest-neighbor-oxygen smallest distance of approach. This in turn leads to a variation in the nature of the short-range order.