Simple Second-order Equation Research Articles

Phase-integral solution of the radial Dirac equation

A phase-integral (WKB) solution of the radial Dirac equation is constructed, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the classical transition points. The potential is allowed to be the time component of a four-vector, a Lorentz scalar, a pseudoscalar, or any combination of these. The key point in the construction is the transformation from two coupled first-order equations constituting the radial Dirac equation to a single second-order Schrödinger-type equation. This transformation can be carried out in infinitely many ways, giving rise to different second-order equations but with the same spectrum. A unique transformation is found that produces a particularly simple second-order equation and correspondingly simple and well-behaved phase-integral solutions. The resulting phase-integral formulas are applied to unbound and bound states of the Coulomb potential. For bound states, the exact energy levels are reproduced.

Bracing Demand in Axially Loaded Cold-Formed Steel Stud Walls

The problem of determining adequate bracing for multiple studs in a wall with similar imperfections is addressed. Previous studies have determined the necessary brace strength and stiffness required to adequately brace a single axially loaded cold-formed steel stud and this information has been incorporated into design standards. Using elastic nonlinear frame analysis software, the required bracing strength and stiffness demand for a single compression member that was derived by Winter and recommended by previous studies was replicated. The model was then expanded to include walls with up to 30 studs, braced at both the midpoint and third point. For these models the required brace stiffness and the resulting brace forces were recorded and compared to the requirements for a single stud. Analysis of the data indicates that the required brace strength accumulates directly as to the number of braced studs. The required brace stiffness for bracing multiple studs may be determined through the use of a simple second-order equation.

Investigation of the Impedance Characteristic of Human Arm for Development of Robots to Cooperate with Humans.

In the near future many aspects of our lives will be encompassed by tasks performed in cooperation with robots. The application of robots in home automation, agricultural production and medical operations etc. will be indispensable. As a result robots need to be made human-friendly and to execute tasks in cooperation with humans. Control systems for such robots should be designed to work imitating human characteristics. In this study, we have tried to achieve these goals by means of controlling a simple one degree-of-freedom cooperative robot. Firstly, the impedance characteristic of the human arm in a cooperative task is investigated. Then, this characteristic is implemented to control a robot in order to perform cooperative task with humans. A human followed the motion of an object, which is moved through desired trajectories. The motion is actuated by the linear motor of the one degree-of-freedom robot system. Trajectories used in the experiments of this method were minimum jerk (the rate of change of acceleration) trajectory, which was found during human and human cooperative task and optimum for muscle movement. As the muscle is mechanically analogous to a spring-damper system, a simple second-order equation is used as models for the arm dynamics. In the model, we considered mass, stiffness and damping factor. Impedance parameter is calculated from the position and force data obtained from the experiments and based on the “Estimation of Parametric Model”. Investigated impedance characteristic of human arm is then implemented to control a robot, which performed cooperative task with human. It is observed that the proposed control methodology has given human like movements to the robot for cooperating with human.

Novel SIR-estimation-based power control in a CDMA mobile radio system under multipath environment

A novel decisive algorithm based on signal-to-interference power ratio estimation has been proposed to control the power transmission of the mobiles in a code-division multiple-access mobile radio system under multipath environment. A simple second-order equation, together with three different sets of measurement and power-control periods, is first used to predict the power gains of the channels. Based on the predicted power gains, five decision rules have been proposed to determine the power-control commands for the mobiles. The novel algorithm with the best prediction method and decision rule has been observed to accomplish a lower outage probability when compared with the conventional power-control algorithm with optimum power-control threshold.

Alternative equations of motion for the radiating electron

We have previously written down a simple second-order equation for the radiating electron and pointed out that its solutions are well behaved. A key feature of this equation is the presence of a term involving the time derivative of the external fieldf(t). Here, we show that a completely equivalent, but less elegant, equation may be written down which contains no derivatives off(t) but, instead, an infinite number of derivatives of the coordinate. However, it has the merit of displaying explicitly how our result differs from that of Abraham-Lorentz.

Radiation reaction in electrodynamics and the elimination of runaway solutions

The familiar Abraham-Lorentz theory of radiation reaction in classical non-relativistic electrodynamics exhibits many problems such as “runaway solutions” and violation of causality. As shown by many authors, such problems can be alleviated by dropping the assumption of a point electron. We also drop this assumption (by introducing a form-factor with a large cutoff frequency Ω) but we present a new approach based on the use of the generalized quantum Langevin equation. For an electric dipole interaction, an exact treatment is possible and we obtain a new equation of motion which, in spite of being third order, does not lead to runaway solutions or solutions which violate causality (the sole proviso being that Ω cannot exceed an upper limit of 3Mc 3 2e 2 =1.60×10 23 s -1 ). Furthermore, Ω appears in the third-derivative term but we show that, to a very good approximation, this term may be dropped so that we end up with a simple second-order equation which does not contain Ω and whose solutions are well-behaved.

Properties of powders deposited by silane/hydrogen and silane/methane plasmas

Particulate matter is often produced by the glow discharge decomposition of silane/hydrogen and silane/methane mixtures during the production of amorphous thin films. This material, produced in two forms of rf-excited reactors, has been studied by electron microscopy and diffraction and thermomanometric analysis. The latter material is produced under conditions in which physical vapour deposition processes dominate, i.e. high plasma current density and high reactive gas pressure. The material produced in a silane/hydrogen plasma has an average size of 20 nm while that from the silane/methane plasma has an average size of 100 nm. Material deposited on the walls of the chamber is amorphous with a high hydrogen content while material deposited on the central electrode is partly crystalline. The former material produces columnar structures which reflect, on a large scale, the structures formed in thin film deposits. The latter material has a lower hydrogen content due to densification and bond reconstruction caused by ion bombardment. The two distinct processes by which hydrogen evolves from the amorphous powder were modelled by simple second-order equations with activation energies of 1.3 and 2.5 eV.

Group Interactions in Polyelectrolytes. V. The Kinetics of the Amination of Chloromethylated Polystyrene with 2-Aminobutanol

Abstract The amination of chloromethylated polystyrene with 2-aminobutanol in dioxane was studied. The apparent rate constatns, kapp, as computed by the simple second-order equation, were found to increase linearly with the degree of amination; this was in contrast with the amination with n-butylamine, in which the kapp decreased as the amination proceeded. By assuming that the reactivity of the chloromethyl group is enhanced when it is situated next to the already-aminated group as a result of the “hydrophilic effect” of the hydroxyl group, rate equations which give the elementary rate constants were derived. In the presence of a large excess of amine, the over-all kinetics can be expressed by: dx⁄dt=[ab⁄(3k1−k2)][3k1(k1−k2)e−3k1at+2k1k2e−k2at] where x is the concentration of chloride ions at time t; where a is the initial concentration of amine, and b, that of the chloromethyl group, and where k1 and k2 are the rate constants of the elementary reactions which are independent of the neighboring group and under the influence of it respectively. The values of kapp calculated on the basis of k1 and k2 were also found to agree with the observed values.