Exact controllability of generalized Hammerstein type equation

In this article, we study the exact controllability of an abstract model described by the controlled generalized Hammerstein type integral equation $$ x(t) = int_0^t h(t,s)u(s)ds+ int_0^t k(t,s,x)f(s,x(s))ds, quad 0 leq t leq T less than infty, $$ where, the state $x(t)$ lies in a Hilbert space $H$ and control $u(t)$ lies another Hilbert space $V$ for each time $t in I=[0,T]$, $T$ greater than 0. We establish the controllability result under suitable assumptions on $h, k$ and $f$ using the monotone operator theory.

Noise reduction with perforated three-duct muffler components

This paper describes the authors’ distributed parameter approach for derivation of closed-form expressions for the four-pole parameters of the perforated three-duct muffler components. In this method, three simultaneous second-order partial differential equations are first reduced to a set of six first-order ordinary differential equations. These equations are then uncoupled by means of a modal matrix. The resulting 6 × 6 matrix is reduced to the 2 × 2 transfer matrix using the relevant boundary conditions. This is combined with transfer matrices of other elements (upstream and downstream of this perforated element) to predict muffler performance like noise reduction, which is also measured. The correlation between experimental and theoretical values of noise reduction is shown to be satisfactory.

Switching Transients in Single-Phase Induction Motors

The subject of transients in polyphase induction motors and synchronous machines has been studied in very great detail by several investigators, but no published literature exists dealing exclusively with the analysis of the problem of transients in single-phase induction motors. This particular problem has been studied in this paper by applying the Laplace transform. The results of actual computation of the currents and developed electrical torque are compared with the data obtained by setting up the integro-differential equations of the machine on an electronic differential analyzer. It is shown that if the motor is switched on to the supply when the potential passes through its zero value, there is a pulsating fundamental frequency torque superimposed on the average steady-state unidirectional torque. If, on the other hand, the switch is closed when the applied potential passes through its maximum value, the developed electrical torque settles down to its final steady-state value during the first cycle of the supply voltage.

The x¿n-x Plane for Analysis of Certain Second-Order Nonlinear Systems

Analysis of certain second-order nonlinear systems, not easily amenable to the phase-plane methods, and described by either of the following differential equations xÿn-2ÿ+ f(x)xÿ2n+g(x)xÿn+h(x)=0 ÿ+f(x)xÿn+h(x)=0 n≫0 can be effected easily by drawing the entire portrait of trajectories on a new plane; that is, on one of the xÿnÿx planes. Simple equations are given to evaluate time from a trajectory on any of these n planes. Poincaré's fundamental phase plane xÿÿx is conceived of as the simplest case of the general xÿnÿx plane.

Techniques for Analysis of Certain Nonlinear Systems

This paper suggests the use of simple transformations like ÿ=kx, kx2 for second-order nonlinear differential equations to effect rapid plotting of the phase-plane trajectories. The method is particularly helpful in determining quickly the trajectory slopes along simple curves in any desired region of the phase plane. New planes such as the tÿ-x, tÿ2-x are considered for the study of some groups of nonlinear time-varying systems. Suggestions for solving certain higher-order nonlinear systems are also made.

Variation of the gas and radiation content in the sub-Keplerian accretion disk around black holes and its impact to the solutions

We investigate the variation of the gas and the radiation pressure in accretion disks during the infall of matter to the black hole and its effect to the flow. While the flow far away from the black hole might be non-relativistic, in the vicinity of the black hole it is expected to be relativistic behaving more like radiation. Therefore, the ratio of gas pressure to total pressure (beta) and the underlying polytropic index (gamma) should not be constant throughout the flow. We obtain that accretion flows exhibit significant variation of beta and then gamma, which affects solutions described in the standard literature based on constant beta. Certain solutions for a particular set of initial parameters with a constant beta do not exist when the variation of beta is incorporated appropriately. We model the viscous sub-Keplerian accretion disk with a nonzero component of advection and pressure gradient around black holes by preserving the conservations of mass, momentum, energy, supplemented by the evolution of beta. By solving the set of five coupled differential equations, we obtain the thermo-hydrodynamical properties of the flow. We show that during infall, beta of the flow could vary up to similar to 300%, while gamma up to similar to 20%. This might have a significant impact to the disk solutions in explaining observed data, e.g. super-luminal jets from disks, luminosity, and then extracting fundamental properties from them. Hence any conclusion based on constant gamma and beta should be taken with caution and corrected. (C) 2011 Elsevier B.V. All rights reserved.

Prediction of Breakout Noise from a Rectangular Duct with Compliant Walls

Breakout noise from HVAC ducts is important at low frequencies, and the coupling between the acoustic waves and the structural waves plays a critical role in the prediction of the transverse transmission loss. This paper describes the analytical calculation of breakout noise by incorporating three-dimensional effects along with the acoustical and structural wave coupling phenomena. The first step in the breakout noise prediction is to calculate the inside duct pressure field and the normal duct wall vibration by using the solution of the governing differential equations in terms of Green's function. The resultant equations are rearranged in terms of impedance and mobility, which results in a compact matrix formulation. The Green's function selected for the current problem is the cavity Green's function with modification of wave number in the longitudinal direction in order to incorporate the terminal impedance. The second step is to calculate the radiated sound power from the compliant duct walls by means of an ``equivalent unfolded plate'' model. The transverse transmission loss from the duct walls is calculated using the ratio of the incident power due to surface source inside the duct to the acoustic power radiated from the compliant duct walls. Analytical results are validated with the FE-BE numerical models.

Five-stage Milstein methods for SDEs

In this paper, we consider the problem of computing numerical solutions for Ito stochastic differential equations (SDEs). The five-stage Milstein (FSM) methods are constructed for solving SDEs driven by an m-dimensional Wiener process. The FSM methods are fully explicit methods. It is proved that the FSM methods are convergent with strong order 1 for SDEs driven by an m-dimensional Wiener process. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the methods proposed in this paper are larger than the Milstein method and three-stage Milstein methods.

Longitudinal Magnetic Field Effect on Nonlocal Ultrasonic Vibration Analysis of Single-Walled Carbon Nanotubes Based on Wave Propagation Approach

This paper studies the effect of longitudinal magnetic field on ultrasonic vibration in single walled carbon nanotubes (CNTs) based on nonlocal continuum medium theory. Governing partial differential equations of CNTs are derived by considering the Lorentz magnetic forces applied on CNTs induced by a longitudinal magnetic field through Maxwell equations. The vibration characteristics of CNTs under a longitudinal magnetic field are obtained by solving the governing equations via wave propagation approach. The effects of longitudinal magnetic field on vibration of CNTs are discussed through numerical experiments. The present analysis show that vibration frequencies of CNTs drops dramatically in the presence of the magnetic field for various circumferential wavenumbers. Such effect is also observed for various boundary conditions of the CNT. New features for the effect of longitudinal magnetic field on ultrasonic vibration of CNTs, presented in this paper are useful in the design of nano-drive device, nano-oscillator and actuators and nano-electron technology, where carbon nanotubes act as basic elements.

Determination of alpha(s)(M-tau(2)) from improved fixed order perturbation theory

We revisit the extraction of alpha(s)(M-tau(2)) from the QCD perturbative corrections to the hadronic tau branching ratio, using an improved fixed-order perturbation theory based on the explicit summation of all renormalization-group accessible logarithms, proposed some time ago in the literature. In this approach, the powers of the coupling in the expansion of the QCD Adler function are multiplied by a set of functions D-n, which depend themselves on the coupling and can be written in a closed form by iteratively solving a sequence of differential equations. We find that the new expansion has an improved behavior in the complex energy plane compared to that of the standard fixed-order perturbation theory (FOPT), and is similar but not identical to the contour-improved perturbation theory (CIPT). With five terms in the perturbative expansion we obtain in the (MS) over bar scheme alpha(s)(M-tau(2)) = 0.338 +/- 0.010, using as input a precise value for the perturbative contribution to the hadronic width of the tau lepton reported recently in the literature.

SCALE EFFECTS ON ULTRASONIC WAVE DISPERSION CHARACTERISTICS OF MONOLAYER GRAPHENE EMBEDDED IN AN ELASTIC MEDIUM

Ultrasonic wave propagation in a graphene sheet, which is embedded in an elastic medium, is studied using nonlocal elasticity theory incorporating small-scale effects. The graphene sheet is modeled as an one-atom thick isotropic plate and the elastic medium/substrate is modeled as distributed springs. For this model, the nonlocal governing differential equations of motion are derived from the minimization of the total potential energy of the entire system. After that, an ultrasonic type of wave propagation model is also derived. The explicit expressions for the cut-off frequencies are also obtained as functions of the nonlocal scaling parameter and the y-directional wavenumber. Local elasticity shows that the wave will propagate even at higher frequencies. But nonlocal elasticity predicts that the waves can propagate only up to certain frequencies (called escape frequencies), after which the wave velocity becomes zero. The results also show that the escape frequencies are purely a function of the nonlocal scaling parameter. The effect of the elastic medium is captured in the wave dispersion analysis and this analysis is explained with respect to both local and nonlocal elasticity. The simulations show that the elastic medium affects only the flexural wave mode in the graphene sheet. The presence of the elastic matrix increases the band gap of the flexural mode. The present results can provide useful guidance for the design of next-generation nanodevices in which graphene-based composites act as a major element.

Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues. Part I: zero and extrapolated boundary conditions

The mathematical model for diffuse fluorescence spectroscopy/imaging is represented by coupled partial differential equations (PDEs), which describe the excitation and emission light propagation in soft biological tissues. The generic closed-form solutions for these coupled PDEs are derived in this work for the case of regular geometries using the Green's function approach using both zero and extrapolated boundary conditions. The specific solutions along with the typical data types, such as integrated intensity and the mean time of flight, for various regular geometries were also derived for both time-and frequency-domain cases. (C) 2013 Optical Society of America

Non-rotating beams isospectral to a given rotating uniform beam

In this paper, we seek to find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The Barcilon-Gottlieb transformation is used to convert the fourth order governing equation of a non-rotating beam, to a canonical fourth order eigenvalue problem. If the coefficients in this canonical equation match with the coefficients of the uniform rotating beam equation, then the non-rotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients leads to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of non-rotating beams that are isospectral to a uniform rotating beam. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. We also verify numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. Finally, the example of beams having a rectangular cross-section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the rotating beam. (c) 2012 Elsevier Ltd. All rights reserved.

A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems

Using a Girsanov change of measures, we propose novel variations within a particle-filtering algorithm, as applied to the inverse problem of state and parameter estimations of nonlinear dynamical systems of engineering interest, toward weakly correcting for the linearization or integration errors that almost invariably occur whilst numerically propagating the process dynamics, typically governed by nonlinear stochastic differential equations (SDEs). Specifically, the correction for linearization, provided by the likelihood or the Radon-Nikodym derivative, is incorporated within the evolving flow in two steps. Once the likelihood, an exponential martingale, is split into a product of two factors, correction owing to the first factor is implemented via rejection sampling in the first step. The second factor, which is directly computable, is accounted for via two different schemes, one employing resampling and the other using a gain-weighted innovation term added to the drift field of the process dynamics thereby overcoming the problem of sample dispersion posed by resampling. The proposed strategies, employed as add-ons to existing particle filters, the bootstrap and auxiliary SIR filters in this work, are found to non-trivially improve the convergence and accuracy of the estimates and also yield reduced mean square errors of such estimates vis-a-vis those obtained through the parent-filtering schemes.

Frustration induced oscillator death on networks

An array of identical maps with Ising symmetry, with both positive and negative couplings, is studied. We divide the maps into two groups, with positive intra-group couplings and negative inter-group couplings. This leads to antisynchronization between the two groups which have the same stability properties as the synchronized state. Introducing a certain degree of randomness in signs of these couplings destabilizes the anti-synchronized state. Further increasing the randomness in signs of these couplings leads to oscillator death. This is essentially a frustration induced phenomenon. We explain the observed results using the theory of random matrices with nonzero mean. We briefly discuss applications to coupled differential equations. (C) 2013 AIP Publishing LLC.