The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

Algorithms for Colouring Random k-colourable Graphs

The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k greater than or equal to 3. The best known polynomial time approximation algorithms require n(delta) (for a positive constant delta depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be ii-coloured almost surely in polynomial time. In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into ii colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) greater than or equal to n(-1+is an element of) where is an element of is a constant greater than 1/4.

Probability Distribution of the Eigenvalues of the Random String Equation

The probability distribution of the eigenvalues of a second-order stochastic boundary value problem is considered. The solution is characterized in terms of the zeros of an associated initial value problem. It is further shown that the probability distribution is related to the solution of a first-order nonlinear stochastic differential equation. Solutions of this equation based on the theory of Markov processes and also on the closure approximation are presented. A string with stochastic mass distribution is considered as an example for numerical work. The theoretical probability distribution functions are compared with digital simulation results. The comparison is found to be reasonably good.

Search on a hypercubic lattice using a quantum random walk. I. d > 2

Random walks describe diffusion processes, where movement at every time step is restricted to only the neighboring locations. We construct a quantum random walk algorithm, based on discretization of the Dirac evolution operator inspired by staggered lattice fermions. We use it to investigate the spatial search problem, that is, to find a marked vertex on a d-dimensional hypercubic lattice. The restriction on movement hardly matters for d > 2, and scaling behavior close to Grover's optimal algorithm (which has no restriction on movement) can be achieved. Using numerical simulations, we optimize the proportionality constants of the scaling behavior, and demonstrate the approach to that for Grover's algorithm (equivalent to the mean-field theory or the d -> infinity limit). In particular, the scaling behavior for d = 3 is only about 25% higher than the optimal d -> infinity value.

Search on a hypercubic lattice using a quantum random walk. II. d = 2

We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretized according to the staggered lattice fermion formalism. d = 2 is the critical dimension for the spatial search problem, where infrared divergence of the evolution operator leads to logarithmic factors in the scaling behavior. As a result, the construction used in our accompanying article A. Patel and M. A. Rahaman, Phys. Rev. A 82, 032330 (2010)] provides an O(root N ln N) algorithm, which is not optimal. The scaling behavior can be improved to O(root N ln N) by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi Phys. Rev. A 78, 012310 (2008)]. We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimize the proportionality constants of the scaling behavior of the algorithm by numerically tuning the parameters.

Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs

Let n points be placed independently in d-dimensional space according to the density f(x) = A(d)e(-lambda parallel to x parallel to alpha), lambda, alpha > 0, x is an element of R-d, d >= 2. Let d(n) be the longest edge length of the nearest-neighbor graph on these points. We show that (lambda(-1) log n)(1-1/alpha) d(n) - b(n) converges weakly to the Gumbel distribution, where b(n) similar to ((d - 1)/lambda alpha) log log n. We also prove the following strong law for the normalized nearest-neighbor distance (d) over tilde (n) = (lambda(-1) log n)(1-1/alpha) d(n)/log log n: (d - 1)/alpha lambda <= lim inf(n ->infinity) (d) over tilde (n) <= lim sup(n ->infinity) (d) over tilde (n) <= d/alpha lambda almost surely. Thus, the exponential rate of decay alpha = 1 is critical, in the sense that, for alpha > 1, d(n) -> 0, whereas, for alpha <= 1, d(n) -> infinity almost surely as n -> infinity.

Nanomechanical properties of silicon surfaces nanostructured by excimer laser

Excimer laser irradiation at ambient temperature has been employed to produce nanostructured silicon surfaces. Nanoindentation was used to investigate the nanomechanical properties of the deformed surfaces as a function of laser parameters, such as the angle of incidence and number of laser pulses at a fixed laser fluence of 5 J cm(-2). A single-crystal silicon 311] surface was severely damaged by laser irradiation and became nanocrystalline with an enhanced porosity. The resulting laser-treated surface consisted of nanometer-sized particles. The pore size was controlled by adjusting the angle of incidence and the number of laser pulses, and varied from nanometers to microns. The extent of nanocrystallinity was large for the surfaces irradiated at a small angle of incidence and by a high number of pulses, as confirmed by x-ray diffraction and Raman spectroscopy. The angle of incidence had a stronger effect on the structure and nanomechanical properties than the number of laser pulses.

Effect of water vapour stabilization of raw materials on the quality of the inhalation product

Generation of raw materials for dry powder inhalers by different size reduction methods can be expected to influence physical and chemical properties of the powders. This can cause differences in particle size, size distribution, shape, crystalline properties, surface texture and energy. These physical properties of powders influence the behaviour of particles before and after inhalation. Materials with an amorphous surface have different surface energy compared to materials with crystalline surface. This can affect the adhesion and cohesion of particles. Changes in the surface nature of the drug particles results in a change in product performance. By stabilization of the raw materials the amorphous surfaces are converted into crystalline surfaces. The primary aim of the study was to investigate the influence of the surface properties of the inhalation particles on the quality of the product. The quality of the inhalation product is evaluated by measuring the fine particle dose (FPD). FDP is the total dose of particles with aerodynamic diameters smaller than 5,0 μm. The secondary aim of this study was to achieve the target level of the FPD and the stability of the FPD. This study was also used to evaluate the importance of the stabilization of the inhalation powders. The study included manufacturing and analysing drug substance 200 μg/dose inhalation powder batches using non-stabilized or stabilized raw materials. The inhaler formulation consisted of micronized drug substance, lactose <100μm and micronized lactose <10μm. The inhaler device was Easyhaler®. Stabilization of the raw materials was done in different relative humidity, temperature and time. Surface properties of the raw materials were studied by dynamic vapour sorption, scanning electron microscopy and three-point nitrogen adsorption technique. Particle size was studied by laser diffraction particle size analyzer. Aerodynamic particle size distribution from inhalers was measured by new generation impactor. Stabilization of all three raw materials was successful. A clear difference between nonstabilized and stabilized raw materials was achieved for drug substance and lactose <10μm. However for lactose <100μm the difference wasn’t as clear as wanted. The surface of the non-stabilized drug substance was more irregular and the particles had more roughness on the surface compared to the stabilized drug substances particles surface. The surface of the stabilized drug particles was more regular and smoother than non-stabilized. Even though a good difference between stabilized and non-stabilized raw materials was achieved, a clear evidence of the effect of the surface properties of the inhalation particles on the quality of the product was not observed. Stabilization of the raw materials didn’t lead to a higher FPD. Possible explanations for the unexpected result might be too rough conditions in the stabilization of the drug substance or smaller than wanted difference in the degree of stabilization of the main component of the product <100μm. Despite positive effects on the quality of the product were not seen there appears to be some evidence that stabilized drug substance results in smaller particle size of dry powder inhalers.

Narrowband random excitation of a limit cycle system

The response of the Van der Pol oscillator to stationary narrowband Gaussian excitation is considered. The central frequency of excitation is taken to be in the neighborhood of the system limit cycle frequency. The solution is obtained using a non-Gaussian closure approximation on the probability density function of the response. The validity of the solution is examined with the help of a stochastic stability analysis. Solution based on Stratonovich''s quasistatic averaging technique is also obtained. The comparison of the theoretical solutions with the digital simulations shows that the theoretical estimates are reasonably good.

Nature of nitrogen adsorbed on transition metal surfaces as revealed by electron spectroscopy and cognate techniques

The nature of the chemisorbed states of nitrogen on various transition metal surfaces is discussed comprehensively on the basis of the results of electron spectroscopic investigations augmented by those from other techniques such as LEED and thermal desorption. A brief discussion of the photoemission spectra of free N2, a comparison of adsorbed N2 and CO as well as of physisorption of N2 on metal surfaces is also presented. We discuss the chemisorption of N2 on the surfaces of certain metals (e.g. Ni, Fe, Ru and W) in some detail, paying considerable attention to the effect of electropositive and electronegative surface modifiers. Features of the various chemisorbed states (one or more weakly chemisorbed gamma-states, strongly chemisorbed alpha-states with bond orders between 1 and 2. and dissociative chemisorbed beta-states) on different surfaces are described and relations between them indicated. While the gamma-state could be a precursor of the alpha-state, the alpha-state could be the precursor of the beta-state and this kind of information is of direct relevance to ammonia synthesis. The nature of adsorption of N2 on the surfaces of some metals (e.g. Cr, Co) deserves further study and such investigations might as well suggest alternative catalysts for ammonia synthesis.

Rocking response of rectangular rigid blocks under random noise base excitations

The response of a rigid rectangular block resting on a rigid foundation and acted upon simultaneously by a horizontal and a vertical random white-noise excitation is considered. In the equation of motion, the energy dissipation is modeled through a viscous damping term. Under the assumption that the body does not topple, the steady-state joint probability density function of the rotation and the rotational velocity is obtained using the Fokker-Planck equation approach. Closed form solution is obtained for a specific combination of system parameters. A more general but approximate solution to the joint probability density function based on the method of equivalent non-linearization is also presented. Further, the problem of overturning of the block is approached in the framework of the diffusion methods for first passage failure studies. The overturning of the block is deemed incipient when the response trajectories in the phase plane cross the separatrix of the conservative unforced system. Expressions for the moments of first passage time are obtained via a series solution to the governing generalized Pontriagin-Vitt equations. Numerical results illustra- tive of the theoretical solutions are presented and their validity is examined through limited amount of digital simulations.

In-Network Computation in Random Wireless Networks: A PAC Approach to Constant Refresh Rates with Lower Energy Costs

We propose a method to compute a probably approximately correct (PAC) normalized histogram of observations with a refresh rate of Theta(1) time units per histogram sample on a random geometric graph with noise-free links. The delay in computation is Theta(root n) time units. We further extend our approach to a network with noisy links. While the refresh rate remains Theta(1) time units per sample, the delay increases to Theta(root n log n). The number of transmissions in both cases is Theta(n) per histogram sample. The achieved Theta(1) refresh rate for PAC histogram computation is a significant improvement over the refresh rate of Theta(1/log n) for histogram computation in noiseless networks. We achieve this by operating in the supercritical thermodynamic regime where large pathways for communication build up, but the network may have more than one component. The largest component however will have an arbitrarily large fraction of nodes in order to enable approximate computation of the histogram to the desired level of accuracy. Operation in the supercritical thermodynamic regime also reduces energy consumption. A key step in the proof of our achievability result is the construction of a connected component having bounded degree and any desired fraction of nodes. This construction may also prove useful in other communication settings on the random geometric graph.

Interior Medial Axis Transform computation of 3D objects bound by free-form surfaces

This paper presents an algorithm for generating the Interior Medial Axis Transform (iMAT) of 3D objects with free-form boundaries. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description of the iMAT. The algorithm generates the iMAT by a tracing technique that marches along the object's boundary. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction points and the search for these junction points is done in an efficient way. The algorithm works for multiply-connected objects as well. Results of the implementation are provided. (C) 2010 Elsevier Ltd. All rights reserved.

Inelastic response of beams under sinusoidal and random loads

A new analytical model has been suggested for the hysteretic behaviour of beams. The model can be directly used in a response analysis without bothering to locate the precise point where the unloading commences. The model can efficiently simulate several types of realistic softening hysteretic loops. This is demonstrated by computing the response of cantilever beams under sinusoidal and random loadings. Results are presented in the form of graphs for maximum deflection, bending moment and shear