Magnetic field gradient effects on the magnetorotational instability

The magnetorotational instability (MRI), also known as the Balbus -- Hawley instability, is thought to have an important role on the initiation of turbulence and angular momentum transport in accretion discs. In this work, we investigate the effect of the magnetic field gradient in the azimuthal direction on MRI. We solve the magnetohydrodynamic equations by including the azimuthal component of the field gradient. We find the dispersion relation and calculate the growth rates of the instability numerically. The inclusion of the azimuthal magnetic field gradient produces a new unstable region on wavenumber space. It also modifies the growth rate and the wavelength range of the unstable mode: the higher the magnitude of the field gradient, the greater the growth rate and the wider the unstable wavenumber range. Such a gradient in the magnetic field may be important in T Tauri discs where the stellar magnetic field has an axis which is misaligned with respect to the rotation axis of the disc.Comment: 8 pages, 3 figures, accepted for publication in A

Development of an LED display system for cross-track distance and velocity for Loran-C flight

The methodology for estimating cross-track velocity by combining rate-gyro and Loran-C data is illustrated in block diagrams. At present, preliminary analysis has established values for K sub 1, K sub 2, the parameters of the digital control loops. A computer program was written to implement a digital simulation of the system as illustrated. Given a model for the noise in the rate-gyro and Loran-C receiver, and their dynamic response, the simulation provides a working model to establish good control loop parameters. The layout of the LED display for flight testing of Loran-C approach flying, which was constructed during a visit to Langley Research Center, is shown. Four bar-graph LED displays are paired to provide cross-track distance and velocity from a Loran-C defined runway centerline. Two seven-segment LED displays are used to provide alphanumeric readout of range to touchdown and desired height. A metal case was built, a circuit board designed, and manufactured with the assistance of NASA Langley personnel

Smart Traction Control Systems for Electric Vehicles Using Acoustic Road-type Estimation

The application of traction control systems (TCS) for electric vehicles (EV) has great potential due to easy implementation of torque control with direct-drive motors. However, the control system usually requires road-tire friction and slip-ratio values, which must be estimated. While it is not possible to obtain the first one directly, the estimation of latter value requires accurate measurements of chassis and wheel velocity. In addition, existing TCS structures are often designed without considering the robustness and energy efficiency of torque control. In this work, both problems are addressed with a smart TCS design having an integrated acoustic road-type estimation (ARTE) unit. This unit enables the road-type recognition and this information is used to retrieve the correct look-up table between friction coefficient and slip-ratio. The estimation of the friction coefficient helps the system to update the necessary input torque. The ARTE unit utilizes machine learning, mapping the acoustic feature inputs to road-type as output. In this study, three existing TCS for EVs are examined with and without the integrated ARTE unit. The results show significant performance improvement with ARTE, reducing the slip ratio by 75% while saving energy via reduction of applied torque and increasing the robustness of the TCS.Comment: Accepted to be published by IEEE Trans. on Intelligent Vehicles, 22 Jan 201

Symmetry Analysis of Initial and Boundary Value Problems for Fractional Differential Equations in Caputo sense

In this work we study Lie symmetry analysis of initial and boundary value problems for partial differential equations (PDE) with Caputo fractional derivative. We give generalized definition and theorem for the symmetry method for PDE with Caputo fractional derivative, according to Bluman's definition and theorem for the symmetry analysis of PDE system. We investigate the symmetry analysis of initial and boundary value problem for fractional diffusion and the third order fractional partial differential equation (FPDE). Also we give some solutions

Canonical sequences of optimal quantization for condensation measures

Let $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel probability measure on $\mathbb R$ with compact support. Such a measure $P$ is called a condensation measure, or an an inhomogeneous self-similar measure, associated with the condensation system $(\{S_1, S_2\}, (\frac 13, \frac 13, \frac 13), \nu)$. Let $D(\mu)$ denote the quantization dimension of a measure $\mu$ if it exists. Let $\kappa$ be the unique number such that $(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}+(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}=1$. In this paper, we have considered four different self-similar measures $\nu:=\nu_1, \nu_2, \nu_3, \nu_4$ satisfying $D(\nu_1)>\kappa$, $D(\nu_2)\kappa$, and $D(\nu_4)=\kappa$. For each measure $\nu$ we show that there exist two sequences $a(n)$ and $F(n)$, which we call as canonical sequences. With the help of the canonical sequences, we obtain a closed formula to determine the optimal sets of $F(n)$-means and $F(n)$th quantization errors for the condensation measure $P$ for each $\nu$. Then, we show that for each measure $\nu$ the quantization dimension $D(P)$ of the condensation measure $P$ exists, and satisfies: $D(P)=\max\{\kappa, D(\nu)\}$. Moreover, we show that for $D(\nu_1)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; on the other hand, for $\nu=\nu_2, \nu_3, \nu_4$, the $D(P)$-dimensional lower quantization coefficient is infinity. This shows that for $D(\nu)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients can be either finite, positive and unequal, or it can be infinity