Downloads & Free Reading Options - Results

We consider a system of coupled nonlinear Schr{\"o}dinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary wave. Then, we investigate numerically the interaction of two solitary waves supported each on one component. Among the possible outcomes, we find elastic and inelastic interactions, collision with mass extraction and reflexion.

The contact mechanics of individual, very small particles with other particles and walls is studied using a nanoindenter setup that allows normal and lateral displacement control and measurement of the respective forces. The sliding, rolling and torsional forces and torques are tested with borosilicate microspheres, featuring radii of about 10$\mu$m. The contacts are with flat silicon substrates of different roughness for pure sliding and rolling and with silicon based, ion-beam crafted rail systems for combined rolling and torsion. The experimental results are discussed and compared to various analytical predictions and contact models, allowing for two concurrent interpretations of the effects of surface roughness, plasticity and adhesion. This enables us to determine both rolling and torsion friction coefficients together with their associated length scales. Interestingly, even though normal contacts behave elastically (Hertzian), all other modes of motion display effects due to surface roughness and consequent plastic deformation. The influence of adhesion is interpreted in the framework of different models and is very different for different degrees of freedom, being largest for rolling.

Reflection and refraction of electromagnetic waves by artificial periodic composites (metamaterials) can be accurately modeled by an effective medium theory only if the boundary of the medium is explicitly taken into account and the two effective parameters of the medium -- the index of refraction and the impedance -- are correctly determined. Theories that consider infinite periodic composites do not satisfy the above condition. As a result, they cannot model reflection and transmission by finite samples with the desired accuracy and are not useful for design of metamaterial-based devices. As an instructive case in point, we consider the "current-driven" homogenization theory, which has recently gained popularity. We apply this theory to the case of one-dimensional periodic medium wherein both exact and homogenization results can be obtained analytically in closed form. We show that, beyond the well-understood zero-cell limit, the current-driven homogenization result is inconsistent with the exact reflection and transmission characteristics of the slab.

This study is aimed at answering the famous question of how the approximation errors at each iteration of Approximate Dynamic Programming (ADP) affect the quality of the final results considering the fact that errors at each iteration affect the next iteration. To this goal, convergence of Value Iteration scheme of ADP for deterministic nonlinear optimal control problems with undiscounted cost functions is investigated while considering the errors existing in approximating respective functions. The boundedness of the results around the optimal solution is obtained based on quantities which are known in a general optimal control problem and assumptions which are verifiable. Moreover, since the presence of the approximation errors leads to the deviation of the results from optimality, sufficient conditions for stability of the system operated by the result obtained after a finite number of value iterations, along with an estimation of its region of attraction, are derived in terms of a calculable upper bound of the control approximation error. Finally, the process of implementation of the method on an orbital maneuver problem is investigated through which the assumptions made in the theoretical developments are verified and the sufficient conditions are applied for guaranteeing stability and near optimality.

This study is aimed at answering the famous question of how the approximation errors at each iteration of Approximate Dynamic Programming (ADP) affect the quality of the final results considering the fact that errors at each iteration affect the next iteration. To this goal, convergence of Value Iteration scheme of ADP for deterministic nonlinear optimal control problems with undiscounted cost functions is investigated while considering the errors existing in approximating respective functions. The boundedness of the results around the optimal solution is obtained based on quantities which are known in a general optimal control problem and assumptions which are verifiable. Moreover, since the presence of the approximation errors leads to the deviation of the results from optimality, sufficient conditions for stability of the system operated by the result obtained after a finite number of value iterations, along with an estimation of its region of attraction, are derived in terms of a calculable upper bound of the control approximation error. Finally, the process of implementation of the method on an orbital maneuver problem is investigated through which the assumptions made in the theoretical developments are verified and the sufficient conditions are applied for guaranteeing stability and near optimality.

Many numerical simulations, in particular that of a two dimensional incompressible free boundary flow, can be done by performing conformal mapping of the flow domain onto a half plane. The detailed behavior of the conformal mapping, which is closely related to the detailed behavior of the solution to a two dimensional Dirichlet problem, is analysed. A uniform asymptotic expansion to the conformal map of a slender domain is constructed. Its salient features are explained and later generalized by theorems valid for arbitrary domains. It is demonstrated that the conformal map onto a disk can not expand distances beyond a certain bound but can be extremely contracting. The logarithm of its derivative is shown to be well behaved. A general perturbation formula from an arbitrary domain to an arbitrary domain which preserves many features of the infinitesmal perturbation formula is derived, and its use is demonstrated on a fractal. These results utilize two estimates, correct up to a constant factor, of the conformal distance and the location of its geodesics.

The Final Proceedings for Analysis and Numerical Solution of Nonlinear Models of Physical Phenomena, 19 May 1997 - 21 May 1997. The Topics covered include: Analysis and numerical computation of solutions of nonlinear systems of partial differential equations modeling physical phenomena Systems arising in the areas of nonlinear optics, mathematical material science, and theoretical fluid mechanics will be addressed.

In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of a small inhomogeneity $\omega_\varepsilon$ (where the coefficients of the equation are altered) located inside a domain $\Omega$ starting from observations of the potential on the boundary $\partial \Omega$. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction, extending the results published in [7] to the case of three-dimensional, parabolic problems. In the second part we implement a reconstruction procedure based on the topological gradient of a suitable cost functional. Numerical results obtained on an idealized three-dimensional left ventricle geometry for different measurement settings assess the feasibility and robustness of the algorithm.

In this work, we focus our studies on the subject of nonlinear discrete self-trapping of S=2 (doubly-charged) vortices in two-dimensional photonic lattices, including theoretical analysis, numerical computation and experimental demonstration. We revisit earlier findings about S=2 vortices with a discrete model, and find that S=2 vortices extended over eight lattice sites can indeed be stable (or only weakly unstable) under certain conditions, not only for the cubic nonlinearity previously used, but also for a saturable nonlinearity more relevant to our experiment with a biased photorefractive nonlinear crystal. We then use the discrete analysis as a guide towards numerically identifying stable (and unstable) vortex solutions in a more realistic continuum model with a periodic potential. Finally, we present our experimental observation of such geometrically extended S=2 vortex solitons in optically induced lattices under both self-focusing and self-defocusing nonlinearities, and show clearly that the S=2 vortex singularities are preserved during nonlinear propagation.

In this work, we focus our studies on the subject of nonlinear discrete self-trapping of S=2 (doubly-charged) vortices in two-dimensional photonic lattices, including theoretical analysis, numerical computation and experimental demonstration. We revisit earlier findings about S=2 vortices with a discrete model, and find that S=2 vortices extended over eight lattice sites can indeed be stable (or only weakly unstable) under certain conditions, not only for the cubic nonlinearity previously used, but also for a saturable nonlinearity more relevant to our experiment with a biased photorefractive nonlinear crystal. We then use the discrete analysis as a guide towards numerically identifying stable (and unstable) vortex solutions in a more realistic continuum model with a periodic potential. Finally, we present our experimental observation of such geometrically extended S=2 vortex solitons in optically induced lattices under both self-focusing and self-defocusing nonlinearities, and show clearly that the S=2 vortex singularities are preserved during nonlinear propagation.

We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precisely blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Berger and Kohn in 1988. Note that our method is applicable to more general equations, in particular those with no scaling invariance.

A real-time thermal field theoretical calculation of shear viscosity has been described in the Kubo formalism for bosonic and fermionic medium. The two point function of viscous stress tensor in the lowest order provides one-loop skeleton diagram of boson or fermion field for bosonic or fermionic matter. According to the traditional diagrammatic technique of transport coefficients, the finite thermal width of boson or fermion is introduced in their internal lines during the evaluation of boson-boson or fermion-fermion loop diagram. These thermal widths of $\phi$ boson and $\psi$ fermion are respectively obtained from the imaginary part of self-energy for $\phi\Phi$ and $\psi\Phi$ loops, where interactions of higher mass $\Phi$ boson with $\phi$ and $\psi$ are governed by the simple $\phi\phi\Phi$ and ${\ov\psi}\psi\Phi$ interaction Lagrangian densities. A two-loop diagram, having same power of coupling constant as in the one-loop diagram, is deduced and its contribution appears much lower than the one-loop values of shear viscosity. Therefore the one-loop results of Kubo-type shear viscosity may be considered as leading order results for this simple $\phi\phi\Phi$ and ${\ov\psi}\psi\Phi$ interactions. This approximation is valid for any values of coupling constant and at the temperatures greater than the mass of constituent particles of the medium.

The objective of this article is to study the behavior of electromagnetic field under X-ray diffraction by time-dependent deformed crystals. Derived system of differential equations looks like the Takagi equations in the case of non-stationary crystals. This is a system of multidimensional first-order hyperbolic equations with complex time-dependent coefficients. Efficient difference schemes based on the multicomponent modification of the alternating direction method are proposed. The stability and convergence of devised schemes are proved. Numerical results are shown for the case of an ideal crystal, a crystal heated uniformly according to a linear law and a time-varying bent crystal. Detailed numerical studies indicate the importance of consideration even small crystal changes.

This report summarizes results of stability analyses performed for a slender sharp cone with and without injection of air through a porous strip. The free-stream parameters are related to the experiments in the Caltech s T5 shock tunnel. The analysis is focused on pure aerodynamic effects in the framework of perfect gas model. It was shown that the injection leads to destabilization of acoustic modes in the near-field relaxation region. To reduce this detrimental effect it was suggested to alter the injector surface shape or use suction-blowing of zero net injection. However, stability computations showed that these modifications did not improve the injector performance. The porous wall effect on the acoustic instability was also examined. It was found that porosity stabilizes the boundary layer flow with and without injection. This indicates that the injector performance can be improved, if the porous layer, which is used for the injection, is protruded downstream to cover the near-filed relaxation region.

Based on a point of view that solvency and security are first, this paper considers regular-singular stochastic optimal control problem of a large insurance company facing positive transaction cost asked by reinsurer under solvency constraint. The company controls proportional reinsurance and dividend pay-out policy to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. The paper aims at deriving the optimal retention ratio, dividend payout level, explicit value function of the insurance company via stochastic analysis and PDE methods. The results present the best equilibrium point between maximization of dividend pay-outs and minimization of risks. The paper also gets a risk-based capital standard to ensure the capital requirement of can cover the total given risk. We present numerical results to make analysis how the model parameters, such as, volatility, premium rate, and risk level, impact on risk-based capital standard, optimal retention ratio, optimal dividend payout level and the company's profit.

This paper is a continuation of Ishitani and Kato (2015), in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we show that the function is continuous and has the semigroup property, which is strongly related to the Hamilton-Jacobi-Bellman quasi-variational inequality. Moreover, we show that noise in market impact causes risk-neutral assessment to underestimate the impact cost. We also study typical examples under a log-linear/quadratic market impact function with Gamma-distributed noise.

The overall contractual effort and the theory and numerical solution for the Reacting and Multi-Phase (RAMP2) computer code are described. The code can be used to model the dominant phenomena which affect the prediction of liquid and solid rocket nozzle and orbital plume flow fields. Fundamental equations for steady flow of reacting gas-particle mixtures, method of characteristics, mesh point construction, and numerical integration of the conservation equations are considered herein.

This proposal concerns modeling, analysis, and computational simulations of some non-linear interface problems and applications, that, we believe, are of interests to DOD/ARO missions especially in the fields of computational geometry, computational fluid mechanics, non-linear dynamics, imaging processing, and inverse scattering. Most of our previous work has been focused on linear interface problems with discontinuities. We have been focusing on non-linear interface problems and applications that are of interests of DoD/ARO during the course of the project, particularly for nonlinear interface problems in MR fluids and weighted surface problems. We have successfully developed the immersed interface method for several non-linear interface problems including Poisson-Boltzmann equations on irregular domains, some variational problems, augmented approach for interface problems and problems on irregular domains.

A numerical solution for chemically reacting supersonic gas-particle flows in rocket nozzles and exhaust plumes was described. The gas-particle flow solution is fully coupled in that the effects of particle drag and heat transfer between the gas and particle phases are treated. Gas and particles exchange momentum via the drag exerted on the gas by the particles. Energy is exchanged between the phases via heat transfer (convection and/or radiation). Thermochemistry calculations (chemical equilibrium, frozen or chemical kinetics) were shown to be uncoupled from the flow solution and, as such, can be solved separately. The solution to the set of governing equations is obtained by utilizing the method of characteristics. The equations cast in characteristic form are shown to be formally the same for ideal, frozen, chemical equilibrium and chemical non-equilibrium reacting gas mixtures. The particle distribution is represented in the numerical solution by a finite distribution of particle sizes.

A numerical solution for chemically reacting supersonic gas-particle flows in rocket nozzles and exhaust plumes was described. The gas-particle flow solution is fully coupled in that the effects of particle drag and heat transfer between the gas and particle phases are treated. Gas and particles exchange momentum via the drag exerted on the gas by the particles. Energy is exchanged between the phases via heat transfer (convection and/or radiation). Thermochemistry calculations (chemical equilibrium, frozen or chemical kinetics) were shown to be uncoupled from the flow solution and, as such, can be solved separately. The solution to the set of governing equations is obtained by utilizing the method of characteristics. The equations cast in characteristic form are shown to be formally the same for ideal, frozen, chemical equilibrium and chemical non-equilibrium reacting gas mixtures. The particle distribution is represented in the numerical solution by a finite distribution of particle sizes.

A numerical solution for chemically reacting supersonic gas-particle flows in rocket nozzles and exhaust plumes was described. The gas-particle flow solution is fully coupled in that the effects of particle drag and heat transfer between the gas and particle phases are treated. Gas and particles exchange momentum via the drag exerted on the gas by the particles. Energy is exchanged between the phases via heat transfer (convection and/or radiation). Thermochemistry calculations (chemical equilibrium, frozen or chemical kinetics) were shown to be uncoupled from the flow solution and, as such, can be solved separately. The solution to the set of governing equations is obtained by utilizing the method of characteristics. The equations cast in characteristic form are shown to be formally the same for ideal, frozen, chemical equilibrium and chemical non-equilibrium reacting gas mixtures. The particle distribution is represented in the numerical solution by a finite distribution of particle sizes.

This article is from BMJ Open , volume 3 . Abstract Background: The reliability of binary exposure classification methods is routinely reported in occupational health literature because it is viewed as an important component of evaluating the trustworthiness of the exposure assessment by experts. The Kappa statistics (κ) are typically employed to assess how well raters or classification systems agree in a variety of contexts, such as identifying exposed participants in a population-based epidemiological study of risks due to occupational exposures. However, the question we are really interested in is not so much the reliability of an exposure assessment method, although this holds value in itself, but the validity of the exposure estimates. The validity of binary classifiers can be expressed as a method's sensitivity (SN) and specificity (SP), estimated from its agreement with the error-free classifier. Methods and results: We describe a simulation-based method for deriving information on SN and SP that can be derived from κ and the prevalence of exposure, since an analytic solution is not possible without restrictive assumptions. This work is illustrated in the context of comparison of job-exposure matrices assessing occupational exposures to polycyclic aromatic hydrocarbons. Discussion: Our approach allows the investigators to evaluate how good their exposure-assessment methods truly are, not just how well they agree with each other, and should lead to incorporation of information of validity of expert assessment methods into formal uncertainty analyses in epidemiology.

This article is from BMJ Open , volume 3 . Abstract Background: The reliability of binary exposure classification methods is routinely reported in occupational health literature because it is viewed as an important component of evaluating the trustworthiness of the exposure assessment by experts. The Kappa statistics (κ) are typically employed to assess how well raters or classification systems agree in a variety of contexts, such as identifying exposed participants in a population-based epidemiological study of risks due to occupational exposures. However, the question we are really interested in is not so much the reliability of an exposure assessment method, although this holds value in itself, but the validity of the exposure estimates. The validity of binary classifiers can be expressed as a method's sensitivity (SN) and specificity (SP), estimated from its agreement with the error-free classifier. Methods and results: We describe a simulation-based method for deriving information on SN and SP that can be derived from κ and the prevalence of exposure, since an analytic solution is not possible without restrictive assumptions. This work is illustrated in the context of comparison of job-exposure matrices assessing occupational exposures to polycyclic aromatic hydrocarbons. Discussion: Our approach allows the investigators to evaluate how good their exposure-assessment methods truly are, not just how well they agree with each other, and should lead to incorporation of information of validity of expert assessment methods into formal uncertainty analyses in epidemiology.