Heuristic Description Research Articles

Recoiling from a kick in the head-on collision of spinning black holes

Recoil ``kicks'' induced by gravitational radiation are expected in the inspiral and merger of black holes. Recently the numerical relativity community has begun to measure the significant kicks found when both unequal masses and spins are considered. Because understanding the cause and magnitude of each component of this kick may be complicated in inspiral simulations, we consider these effects in the context of a simple test problem. We study recoils from collisions of binaries with initially head-on trajectories, starting with the simplest case of equal masses with no spin and then adding spin and varying the mass ratio, both separately and jointly. We find spin-induced recoils to be significant relative to unequal-mass recoils even in head-on configurations. Additionally, it appears that the scaling of transverse kicks with spins is consistent with post-Newtonian theory, even though the kick is generated in the nonlinear merger interaction, where post-Newtonian theory should not apply. This suggests that a simple heuristic description might be effective in the estimation of spin kicks.

The effect of pH on the kinetics of spontaneous Fe(II) oxidation by O 2 in aqueous solution – basic principles and a simple heuristic description

The spontaneous chemical oxidation of Fe(II) to Fe(III) by O 2 is a complex process involving meta-stable partially oxidized intermediate species such as green rusts, which ultimately transform into a variety of stable iron oxide end-products such as hematite, magnetite, goethite and lepidocrocite. Although in many practical situations the nature of the end-products is of less interest than the oxidation kinetics, it is difficult to find in the literature a description of all the basic steps and principles governing the kinetics of these reactions. This paper uses basic aquatic-chemistry equilibrium theory as the framework upon which to present a heuristic model of the oxidation kinetics of Fe(II) species to ferric iron by O 2. The oxidation rate can be described by the equation (in units of mol Fe(II)/(l min)): - d [ Fe 2 + ] / d t = 6 × 10 - 5 [ Fe 2 + ] + 1.7 [ Fe ( OH ) + ] + 4.3 × 10 5 [ Fe ( OH ) 2 0 ] . This rate equation yields a sigmoid-shaped curve as a function of pH; at pH values below ∼4, the Fe 2+ concentration dominates and the rate is independent of pH. At pH > ∼5, [ Fe ( OH ) 2 0 ] determines the rate because it is far more readily oxidized than both Fe 2+ and FeOH +. Between pH 5 and 8 the Fe ( OH ) 2 0 concentration rises steeply with pH and the overall oxidation rate increases accordingly. At pH values > ∼8 [ Fe ( OH ) 2 0 ] no longer varies with pH and the oxidation rate is again independent of pH. The paper presents a heuristic overview of the pH dependent kinetics of aqueous ferrous oxidation by O 2(aq) which we believe will be useful to professionals at both research and technical levels.

Multiscale Hall‐Magnetohydrodynamic Turbulence in the Solar Wind

The spectra of solar wind magnetic fluctuations exhibit a significant power-law steepening at frequencies f > 1 Hz. The origin of this multiple scaling is investigated through dispersive Hall magnetohydrodynamics. We perform three-dimensional numerical simulations in the framework of a highly turbulent shell model and show that the large-scale magnetic fluctuations are characterized by a k-5/3-type spectrum that steepens at scales smaller than the ion inertial length di, to k-7/3 if the magnetic energy overtakes the kinetic energy, or to k-11/3 in the opposite case. These results are in agreement both with a heuristic description a la Kolmogorov and with the range of power-law indices found in the solar wind.

LargeNBPS states and emergent quantum gravity

This paper provides a heuristic derivation of how classical gravitational physics in the AdS/CFT correspondence appears from the strong dynamics of the N=4 SYM theory in a systematic way. We do this in a minisuperspace approximation by studying 1/8 BPS configurations. We show that this is related to a gauged matrix quantum mechanics of commuting matrices. We can show that our description matches the semiclassical physics of 1/8 BPS states in supergravity. We also provide a heuristic description of how massive strings appear in the geometry, and how at strong 't Hooft coupling they become local on the five sphere suggesting that they can be realized as a sigma model on a weakly curved background. In the process we also clarify some aspects of 1/2 BPS states. We also have a conjectured realization of some 1/8 BPS giant graviton wave functions in the dynamics, which captures all 1/8 BPS giant gravitons constructed by Mikhailov. This leads to a lot of different topology changes which can be treated heuristically.

The nine-point conic: a rediscovery and proof by computer

A heuristic description is given of the rediscovery with Sketchpad of a less-well-known, but beautiful, generalization of the nine-point circle to a nine-point conic, as well as an associated generalization of the Euler line. The author's initial analytic geometry proofs, which made use of the symbolic algebra facility of the TI-92 calculator, are also presented.

Physics of the Resonating Valence Bond (Pseudogap) State of the Doped Mott Insulator: Spin-Charge Locking

The properties of the pseudogap phase above T(C) of the high-T(C) cuprate superconductors are described by showing that the Anderson-Nambu SU(2) spinors of a resonating valence bond spin gap "lock" to those of the electron charge system because of the resulting improvement of kinetic energy. This enormously extends the range of the vortex liquid state in these materials. A heuristic description of the nonlocal electrodynamics of this pseudogap-vortex liquid state is proposed.

Rejection of two-photon fluorescence background in thick tissue by differential aberration imaging

We present a simple and robust way to reject out-of-focus background when performing deep two-photon excited fluorescence (TPEF) imaging in thick tissue. The technique is based on the use of a deformable mirror (DM) to introduce illumination aberrations that preferentially degrade TPEF signal while leaving TPEF background relatively unchanged. A subtraction of aberrated from unaberrated images leads to background rejection. We present a heuristic description of our technique, which we corroborate with experiment. An added benefit of our technique is that it leads to somewhat improved image resolution.

A Heuristic Description of Internal Wave Dynamics

The oceanic internal wave spectrum has long been interpreted as being shaped by nonlinear processes. The empirical Garrett and Munk synthesis of oceanic observations is E(m, v) } Nm22v 22 at high wavenumber and frequency. Results of both approximate analytic and numerical estimates of weak nonlinear interactions under the resonant interaction approximation have previously been interpreted as implying the dominance of scaleseparated interactions and that the Garrett and Munk spectrum is stationary with respect to the nonlinear interactions. However, dimensional analysis cobbled together with several basic observational constraints requires a stationary spectrum of E(m, v) } Nm22v 23/2 at high vertical wavenumber and frequency. Given this stationary spectrum, dimensional analysis and extant data are used to infer a flux representation for the spectral transports. The resulting semiempirical flux laws can be described as a relaxation to the stationary power laws. The stationary spectrum is consistent with energy sources at low frequencies and dissipation at higher frequencies and vertical wavenumber. The analysis is then extended to include vertically asymmetric wave fields. Vertical wavenumber spectra of horizontal velocity that exhibit a difference between rotary (clockwise and counterclockwise phase rotation of the velocity vector with depth) spectra at large wavelengths tend to be symmetric at smaller scales. This pattern is hypothesized to be a result of nonlinearity within the wave field. In particular, vertical symmetry is linked here to the issue of momentum conservation. A backscattering process is invoked to achieve momentum conservation. This representation of nonlinearity is used in a numerical scheme to assess the spatial evolution of a bottom-generated wave field. The predicted patterns of relaxation and vertical symmetry are in reasonable agreement with finescale observations above rough bathymetry in the Brazil Basin.

A heuristic description of AC losses in superconducting slab with smooth voltage–current characteristics represented by power law

This paper describes the theoretical evaluation of AC losses in the oxide superconductors with smooth transport properties. First, conventional expressions of AC losses are reviewed for the n-value model, in which the local electric field is proportional to the power of local current density. The superconductor under consideration is a slab with infinitely wide faces, to which an external AC magnetic field is applied along the surfaces. It is also assumed that the critical current density and n-value are independent of the magnitude of local magnetic field. As a result of theoretical consideration by combining Maxwell's equations and the n-value model for the voltage–current characteristics, it is pointed out that the loss factor can be determined by only two parameters. Next, a new type of expressions is proposed for the AC loss evaluation, and it is concluded that a discrepancy is less than 17% between analytical and numerical results.

Statistical tools for estimating and combining finite rotations and their uncertainties

Plate reconstructions are a key tool in the study of plate tectonics. Although many methods have been proposed to calculate the finite rotations that describe plate motions, only a few address the problem of the uncertainties in plate reconstructions. This information is critical for a meaningful comparison of rates and directions of motion, predicted by a rotation, with geological field observations at plate boundaries. Similarly, uncertainties of a product of uncertain rotations through a plate circuit are crucial for testing plate geometry or for comparing reference frames. Here we present a method that allows the propagation of the uncertainties in the marine data (magnetic-anomaly and fracture-zone crossings), used to derive the finite rotations, to the uncertainties in the rotation parameters (latitude and longitude of pole, and angle of rotation). The rotation parameters are estimated by minimizing a quadratic objective function. A Taylor series approximation, centred at the true (and unknown) rotation parameters, is used to approximate the non-linear least squares by a linear regression problem. From the design matrix estimated from the data, a covariance matrix describing the uncertainties in the rotation can be estimated. The covariance matrix depends upon the geometry of the plate boundary, the number of data, and the uncertainties in the data. Using a heuristic description of non-linear least squares, we show how the methodology is analogous to standard linear regression. We generalize the method for reconstructing a single plate boundary to solve for the closure of a triple junction, and we present a simple way to combine the covariance matrices of individual rotations to estimate the uncertainties in their product, and show how to derive the uncertainties in the reconstructed points. Since any statistical analysis depends upon assumptions about the errors in the data, we discuss the assumptions inherent in the proposed methodology, and the limitations which result from these assumptions. One of them (referred to as the ‘equal-kappas’ assumption) is especially troublesome when two or more rotations, based on different data distributions, are combined. From similar problems arising in linear regression, we propose a method to solve the plate reconstruction problem when the equal-kappas assumption is not tenable. Finally, we briefly review the sources of error in the magnetic anomaly and fracture-zone crossings that are inverted to derive plate reconstructions, and show how their uncertainties can be evaluated. A series of examples illustrates how these tools, implemented in software, can be used to solve or to test various plate geometries involving a single plate boundary, a triple junction and a combination of both.

Autopilot Designed with Fuzzy Set Theory

In this paper, the design of course-keeping autopilot, is based on the cognitive modelling approach and fuzzy set theory. Using this modelling approach, the aim is to design a control system based on a model of the expert, who is able to specify the general properties of the system, rather than on a model of the system to be controlled. After an heuristic description of the desired manoeuvre characteristics, fuzzy set theory is used for the synthesis of the autopilot. Simulations on a non-linear model of a containership are presented for course-keeping manoeuvres in the presence of wave disturbances.

Pulse-train formation in a gain-switched polymer laser resulting from spatial gain inhomogeneity

We present an experimental study of the dynamics of a polymer laser upon excitation with an ultrashort pump pulse. The laser is unusual in the sense that the cavity length is such that the cavity round-trip time is intermediate between the duration of the pump pulse and the spontaneous-emission time of the gain medium. The cavity is only partly filled with gain medium. The unusual time scales, in combination with the spatial gain distribution, result in the emission of a neat and regular train of short pulses. Its generation can be viewed as a special case of gain switching. A one-dimensional rate-equation model that takes the nonuniformity of the gain explicitly into account is presented. The model describes both the evolution of the inversion, which is also measured experimentally, and the dynamics of the output in great detail, and yields excellent agreement with the experimental results. A heuristic description of the arisal of the pulse train in terms of amplified spontaneous emission and gain depletion is given.

Mel’nikov analysis of a symmetry-breaking perturbation of the NLS equation

The effects of loss of symmetry due to noneven initial conditions on the chaotic dynamics of a Hamiltonian perturbation of the nonlinear Schrödinger equation (NLS) were first numerically studied in [Phys. A 228 (1996)], where it was observed that temporally irregular evolutions can occur even in the absence of homoclinic crossings. In this article, we introduce a symmetry-breaking damped-driven perturbation of the NLS equation in order to develop a Mel’nikov analysis of the noneven chaotic regime. We obtain the following results: (1) spatial symmetry breaking within the chaotic regime causes the wave form to exhibit a more complex dynamics than in [Phys. A 228 (1996)]: center-wing jumping (which characterizes the even chaotic dynamics) about a shifted lattice site alternates (at random times) with the occurrence of modulated traveling wave solutions whose velocity changes sign in a temporally random fashion; (2) we give a heuristic description of the geometry of the full phase space (with no evenness imposed) and compute Mel’nikov type measurements in terms of the complex gradient of the Floquet discriminant. The Mel’nikov analysis yields explicit conditions for the onset of chaotic dynamics which are consistent with the numerical observations; in particular, the imaginary part of the Mel’nikov integral appears to be correlated with purely noneven features of the chaotic wave form.

DEUTERONOMY 31-32 AS A TEXTUAL MEMORIAL

AbstractThis essay analyzes the narrative of Deuteronomy 31–32 as the alternation between two narrative strands: (1) the death of Moses and commission of Joshua, and (2) the writing and reciting of the Torah and song. This alternating structure culminates in a passage that harmonizes the two strands: 32:44-47. In light of this structure, with its emphasis on the and song, I suggest the concept of memorial, based on the text's use of the term (witness), as a heuristic description of the text.

Cryogenic Electron Tunneling within Mixed-Metal Hemoglobin Hybrids: Protein Glassing and Electron-Transfer Energetics

We report that when mixed-metal, [M, Fe] hemoglobin (Hb) hybrids, with Fe in one type of subunit and M = Zn or Mg in the other type, are embedded in clear poly(vinyl alcohol) (PVA) films, they exhibit inter-subunit electron transfer (ET) electron−nuclear tunneling down to cryogenic temperatures (5 K), making them the first protein system other than photosynthetic systems to exhibit such behavior. The rate constant for the (Fe2+Porphyrin) → (MPorphyrin)+ inter-subunit ET reaction shows a roughly temperature-invariant, quantum-tunneling regime from cryogenic temperatures (5 K) up to ca. 200 K. Some of the hybrids (depending on M and the Fe ligand) begin to show a strong increase in this ET rate constant at higher temperatures. This behavior is discussed here in terms of a recent heuristic description of ET in a glassy environment that accounts for the fact that slow solvent relaxation at low temperatures, and in particular upon cooling through a glassing transition, causes the reaction pathway to deviate fr...

The Stochastic Behavior of the Eigenvalue Due to Spatial Perturbation in the Fuel Distribution of the One-Speed Slab Reactor

The behavior of the average eigenvalue keff of the critical one-speed slab reactor is analyzed as a function of local density fluctuation, while keeping the total material loading of the reactor constant, using a combination of analytical and numerical methods. A perturbation of the reference reactor as a binary material medium is used as developed by Pomraning. Two parallel probability distributions are utilized, and the averages as obtained in the corresponding ensembles are compared. These two distributions provide a heuristic description of the physical effects of the spatial perturbation and a methodology that can be extended to practical problems. The sources of increase and decrease in the eigenvalues of the perturbed configurations are identified, and their relative strengths are identified as functions of the reemission factor c. The average eigenvalue is found to always increase for the perturbations and distributions considered and is plotted as function of c from c = 1 to ν. As the number of regions N (even integer) in the binomial distribution is increased, the number of possible perturbed configurations increases rapidly such that the new members of the ensemble are closer to the reference reactor in fuel distribution, and thus each has an eigenvalue keff increasingly closer to 1. Since these new members predominate the ensemble at large N, keff tends to 1 strictly from above as N increases. A similar behavior is observed with the exponential distribution but is tied to the average binary material thickness or the exponential distribution correlation length Λc. The analysis also shows that (using either distribution) for the same c, keff is larger for systems with less scattering in the corresponding reference reactor. In other words, for a fixed c, the maximum keff is when σs = 0, and the minimum is when σa = 0 in the corresponding reference reactors. Some of the conditions on the stochastic perturbation distribution and the cross-section components that are necessary (but may not be sufficient) to produce keff below 1 are identified.

Abbe's Sine Condition

Ernst Abbe (1840–1905), professor of physics and mathematics at the University of Jena, Germany, and major partner in the Carl Zeiss company, made important contributions to the theory and practice of optical microscopy. His compound microscope was a superb optical design based on a theoretical understanding of diffraction and minimization of the effects of aberrations. Abbe enunciated his famous sine condition regarding the axial point in the object plane of a centered image-forming system such as a microscope or a telescope. When this condition is satisfied, “aberration-free” imaging of the object points located in the vicinity of the optical axis is assured. This chapter provides an heuristic description of the sine condition, which, in the words of Conrady, is “one of the most remarkable and labor-saving theorems in the whole realm of applied optics”. As the chapter follows a rather unconventional approach towards explaining the sine condition, it is worthwhile to highlight its main features at the outset. An introduction of the necessary geometric-optical concepts provides the basis for defining the sine condition. This is followed by establishing, for an axial object point, a one-to-one mapping between the principal planes of the imaging system. The wavefront entering the system at the first principal plane (p.p.) is thus related to that emerging from the second p.p. To describe the imaging of near-axis regions, we switch to a wave-optical viewpoint.

Self-replicating spots in reaction-diffusion systems

In this article we discuss the self-replicating spot, a particlelike phenomenon that occurs in reaction-diffusion ~RD! systems @1#. The spots consist of localized regions in which the concentrations of the reactants differ from the surrounding concentration fields. They grow, reaching a critical size at which time they divide in two. The two resulting spots again grow and divide. This process, which is visually similar to cell division, continues indefinitely. The long-time behavior depends on the precise values of the external control parameters, but typically consists of a chaotic ‘‘soup’’ in which many spots compete for resources as illustrated in Fig. 1. Those spots that find adequate resources continue to grow and divide. Those that are unable to find adequate resources decay into the background. The spots observed in @1# were found during an attempt to model labyrinthine patterns in the ferrocyanide-iodate-sulfate ~FIS! reaction @2#. Since then, replicating spot patterns have been observed both numerically and experimentally in the FIS reaction @3,4#. There are obvious differences in the Gaspar-Showalter model of the FIS reaction @5# and of the models discussed by us and others. This fact suggests that replication is a generic feature characterizing a broad class of reaction-diffusion systems. In @6#, we presented some arguments in support of this proposition. These arguments included both a heuristic description of the process of replication and demonstrations of analytic features common to several related model RD systems. It turns out that replication is more general than our analysis accounts for. Nevertheless, we think it worthwhile to spell out the details of the theory presented in @6#. We remark here that various aspects of the replication phenomenon have been discussed by other authors @7,8#. Kerner and Osipov have a large body of work on largeamplitude dissipative structures including an analysis of the static division of one-dimensional pulses as the system size is changed. Gurevich and Mints have a body of work on replication of thermal hot spots in composite superconductors. In the article by Petrov, Scott, and Showalter replication

Heuristics for a One-Warehouse Multiretailer Distribution Problem with Performance Bounds

We investigate the one warehouse multiretailer distribution problem with traveling salesman tour vehicle routing costs. We model the system in the framework of the more general production/distribution system with arbitrary non-negative monotone joint order costs. We develop polynomial time heuristics whose policy costs are provably close to the cost of an optimal policy. In particular, we show that given a submodular function which is close to the true order cost then we can find a power-of-two policy whose cost is only moderately greater than the cost of an optimal policy. Since such submodular approximations exist for traveling salesman tour vehicle routing costs we present a detailed description of heuristics for the one warehouse multiretailer distribution problem. We formulate a nonpolynomial dynamic program that computes optimal power-of-two policies for the one warehouse multiretailer system assuming only that the order costs are non-negative monotone. Finally, we perform computational tests which compare our heuristics to optimal power of two policies for problems of up to sixteen retailers. We also perform computational tests on larger problems; these tests give us insight into what policies one should employ.

An empirical model for early resistance changes due to electromigration

A new heuristic description for electromigration-induced early resistance changes is given. The basis is formed by two coupled partial differential equations, one for vacancies, and one for imperfections. These equations are solved numerically for a grain boundary bamboo structure. It is shown that this model is capable of simulating the typical effects as observed in early resistance change measurements. These early resistance changes are due to the redistributions of the vacancies and the generation of imperfections. Simulations are performed that closely match the measured resistance change curves.