Decision-making in a fuzzy environment

Decision making where goals or constraints are not sharply defined boundaries and fuzzy using dynamic programmin

Estimation of internal source distributions using external field measurements in radiative transfer

Intensity of emergent radiation for finite homogeneous slab which absorbs radiation and scatters it isotropicall

Numerical estimation of derivatives with an application to radiative transfer in spherical shells

Numerical estimation of derivatives with application to radiative transfer in spherical shell

Invariant imbedding and perturbation techniques applied to diffuse reflection from spherical shells

Invariant imbedding and perturbation techniques applied to diffuse reflection from spherical shell

The invariant imbedding equation for the dissipation function of a homogeneous finite slab

Differential-integral equation for dissipation function and derivation of conservation relationship connecting reflection, transmission and dissipation functions of finite sla

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schr\"{o}dinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.Comment: 8 pages, 12 figures, 1 tabl

Gigantic transmission band edge resonance in periodic stacks of anisotropic layers

We consider Fabry-Perot cavity resonance in periodic stacks of anisotropic layers with misaligned in-plane anisotropy at the frequency close to a photonic band edge. We show that in-plane dielectric anisotropy can result in a dramatic increase in field intensity and group delay associated with the transmission resonance. The field enhancement appears to be proportional to forth degree of the number N of layers in the stack. By contrast, in common periodic stacks of isotropic layers, those effects are much weaker and proportional to N^2. Thus, the anisotropy allows to drastically reduce the size of the resonance cavity with similar performance. The key characteristic of the periodic arrays with the gigantic transmission resonance is that the dispersion curve omega(k)at the photonic band edge has the degenerate form Delta(omega) ~ Delta(k)^4, rather than the regular form Delta(omega) ~ Delta(k)^2. This can be realized in specially arranged stacks of misaligned anisotropic layers. The degenerate band edge cavity resonance with similar outstanding properties can also be realized in a waveguide environment, as well as in a linear array of coupled multimode resonators, provided that certain symmetry conditions are in place.Comment: To be submitted to Phys. Re

Path integrals and symmetry breaking for optimal control theory

This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of stochastic control problems the non-linear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton-Jacobi equation to the Schr\"odinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process, that can be computed by stochastic integration or by the evaluation of a path integral. It is shown, how in the deterministic limit the PMP formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as MC sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in number of examples. Examples are given that show the qualitative difference between stochastic and deterministic control and the occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA

Negativity as a distance from a separable state

The computable measure of the mixed-state entanglement, the negativity, is shown to admit a clear geometrical interpretation, when applied to Schmidt-correlated (SC) states: the negativity of a SC state equals a distance of the state from a pertinent separable state. As a consequence, a SC state is separable if and only if its negativity vanishes. Another remarkable consequence is that the negativity of a SC can be estimated "at a glance" on the density matrix. These results are generalized to mixtures of SC states, which emerge in certain quantum-dynamical settings.Comment: 9 pages, 1 figur

Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragus. A recent paper by Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragus-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragus control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR