An Optimal Control Derivation of Nonlinear Smoothing Equations

The purpose of this paper is to review and highlight some connections between the problem of nonlinear smoothing and optimal control of the Liouville equation. The latter has been an active area of recent research interest owing to work in mean-field games and optimal transportation theory. The nonlinear smoothing problem is considered here for continuous-time Markov processes. The observation process is modeled as a nonlinear function of a hidden state with an additive Gaussian measurement noise. A variational formulation is described based upon the relative entropy formula introduced by Newton and Mitter. The resulting optimal control problem is formulated on the space of probability distributions. The Hamilton's equation of the optimal control are related to the Zakai equation of nonlinear smoothing via the log transformation. The overall procedure is shown to generalize the classical Mortensen's minimum energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin

Relationship between spin squeezing and single-particle coherence in two-component Bose-Einstein condensates with Josephson coupling

We investigate spin squeezing of a two-mode boson system with a Josephson coupling. An exact relation between the squeezing and the single-particle coherence at the maximal-squeezing time is discovered, which provides a more direct way to measure the squeezing by readout the coherence in atomic interference experiments. We prove explicitly that the strongest squeezing is along the $J_z$ axis, indicating the appearance of atom number-squeezed state. Power laws of the strongest squeezing and the optimal coupling with particle number $N$ are obtained based upon a wide range of numerical simulations.Comment: 4 figures, revtex4, new refs. are adde

An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).Comment: 17 pages, 1 figur

Higher Spin Fronsdal Equations from the Exact Renormalization Group

We show that truncating the exact renormalization group equations of free $U(N)$ vector models in the single-trace sector to the linearized level reproduces the Fronsdal equations on $AdS_{d+1}$ for all higher spin fields, with the correct boundary conditions. More precisely, we establish canonical equivalence between the linearized RG equations and the familiar local, second order differential equations on $AdS_{d+1}$, namely the higher spin Fronsdal equations. This result is natural because the second-order bulk equations of motion on $AdS$ simply report the value of the quadratic Casimir of the corresponding conformal modules in the CFT. We thus see that the bulk Hamiltonian dynamics given by the boundary exact RG is in a different but equivalent canonical frame than that which is most natural from the bulk point of view.Comment: 34 pages, 4 figures; v2: typos fixed, better abstrac

Temperature control of thermal radiation from heterogeneous bodies

We demonstrate that recent advances in nanoscale thermal transport and temperature manipulation can be brought to bear on the problem of tailoring thermal radiation from compact emitters. We show that wavelength-scale composite bodies involving complicated arrangements of phase-change chalcogenide (GST) glasses and metals or semiconductors can exhibit large emissivities and partial directivities at mid-infrared wavelengths, a consequence of temperature localization within the GST. We consider multiple object topologies, including spherical, cylindrical, and mushroom-like composites, and show that partial directivity follows from a complicated interplay between particle shape, material dispersion, and temperature localization. Our calculations exploit a recently developed fluctuating-volume current formulation of electromagnetic fluctuations that rigorously captures radiation phenomena in structures with both temperature and dielectric inhomogeneities.Comment: 17 pages, 7 figuer