Some examples of special Lagrangian tori

A short paper giving some examples of smooth hypersurfaces M of degree n+1 in complex projective n-space that are defined by real polynomial equations and whose real slice contains a component diffeomorphic to an n-1 torus, which is then special Lagrangian with respect to the Calabi-Yau metric on M.

Calibrated embeddings in the special Lagrangian and coassociative cases

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.

Sequential anomaly detection in the presence of noise and limited feedback

This paper describes a methodology for detecting anomalies from sequentially observed and potentially noisy data. The proposed approach consists of two main elements: 1) filtering, or assigning a belief or likelihood to each successive measurement based upon our ability to predict it from previous noisy observations and 2) hedging, or flagging potential anomalies by comparing the current belief against a time-varying and data-adaptive threshold. The threshold is adjusted based on the available feedback from an end user. Our algorithms, which combine universal prediction with recent work on online convex programming, do not require computing posterior distributions given all current observations and involve simple primal-dual parameter updates. At the heart of the proposed approach lie exponential-family models which can be used in a wide variety of contexts and applications, and which yield methods that achieve sublinear per-round regret against both static and slowly varying product distributions with marginals drawn from the same exponential family. Moreover, the regret against static distributions coincides with the minimax value of the corresponding online strongly convex game. We also prove bounds on the number of mistakes made during the hedging step relative to the best offline choice of the threshold with access to all estimated beliefs and feedback signals. We validate the theory on synthetic data drawn from a time-varying distribution over binary vectors of high dimensionality, as well as on the Enron email dataset. © 1963-2012 IEEE.

Confidence intervals for half-life deviations from purchasing power parity

Existing point estimates of half-life deviations from purchasing power parity (PPP), around 3-5 years, suggest that the speed of convergence is extremely slow. This article assesses the degree of uncertainty around these point estimates by using local-to-unity asymptotic theory to construct confidence intervals that are robust to high persistence in small samples. The empirical evidence suggests that the lower bound of the confidence interval is between four and eight quarters for most currencies, which is not inconsistent with traditional price-stickiness explanations. However, the upper bounds are infinity for all currencies, so we cannot provide conclusive evidence in favor of PPP either. © 2005 American Statistical Association.

No-Holdback allocation rules for continuous-time assemble-to-order systems

This paper analyzes a class of common-component allocation rules, termed no-holdback (NHB) rules, in continuous-review assemble-to-order (ATO) systems with positive lead times. The inventory of each component is replenished following an independent base-stock policy. In contrast to the usually assumed first-come-first-served (FCFS) component allocation rule in the literature, an NHB rule allocates a component to a product demand only if it will yield immediate fulfillment of that demand. We identify metrics as well as cost and product structures under which NHB rules outperform all other component allocation rules. For systems with certain product structures, we obtain key performance expressions and compare them to those under FCFS. For general product structures, we present performance bounds and approximations. Finally, we discuss the applicability of these results to more general ATO systems. © 2010 INFORMS.

Information relaxations and duality in stochastic dynamic programs

We describe a general technique for determining upper bounds on maximal values (or lower bounds on minimal costs) in stochastic dynamic programs. In this approach, we relax the nonanticipativity constraints that require decisions to depend only on the information available at the time a decision is made and impose a "penalty" that punishes violations of nonanticipativity. In applications, the hope is that this relaxed version of the problem will be simpler to solve than the original dynamic program. The upper bounds provided by this dual approach complement lower bounds on values that may be found by simulating with heuristic policies. We describe the theory underlying this dual approach and establish weak duality, strong duality, and complementary slackness results that are analogous to the duality results of linear programming. We also study properties of good penalties. Finally, we demonstrate the use of this dual approach in an adaptive inventory control problem with an unknown and changing demand distribution and in valuing options with stochastic volatilities and interest rates. These are complex problems of significant practical interest that are quite difficult to solve to optimality. In these examples, our dual approach requires relatively little additional computation and leads to tight bounds on the optimal values. © 2010 INFORMS.

Probing the ultimate limits of plasmonic enhancement.

Metals support surface plasmons at optical wavelengths and have the ability to localize light to subwavelength regions. The field enhancements that occur in these regions set the ultimate limitations on a wide range of nonlinear and quantum optical phenomena. We found that the dominant limiting factor is not the resistive loss of the metal, but rather the intrinsic nonlocality of its dielectric response. A semiclassical model of the electronic response of a metal places strict bounds on the ultimate field enhancement. To demonstrate the accuracy of this model, we studied optical scattering from gold nanoparticles spaced a few angstroms from a gold film. The bounds derived from the models and experiments impose limitations on all nanophotonic systems.

The empirical analysis of cigarette tax avoidance and illicit trade in Vietnam, 1998-2010.

Illicit trade carries the potential to magnify existing tobacco-related health care costs through increased availability of untaxed and inexpensive cigarettes. What is known with respect to the magnitude of illicit trade for Vietnam is produced primarily by the industry, and methodologies are typically opaque. Independent assessment of the illicit cigarette trade in Vietnam is vital to tobacco control policy. This paper measures the magnitude of illicit cigarette trade for Vietnam between 1998 and 2010 using two methods, discrepancies between legitimate domestic cigarette sales and domestic tobacco consumption estimated from surveys, and trade discrepancies as recorded by Vietnam and trade partners. The results indicate that Vietnam likely experienced net smuggling in during the period studied. With the inclusion of adjustments for survey respondent under-reporting, inward illicit trade likely occurred in three of the four years for which surveys were available. Discrepancies in trade records indicate that the value of smuggled cigarettes into Vietnam ranges from $100 million to $300 million between 2000 and 2010 and that these cigarettes primarily originate in Singapore, Hong Kong, Macao, Malaysia, and Australia. Notable differences in trends over time exist between the two methods, but by comparison, the industry estimates consistently place the magnitude of illicit trade at the upper bounds of what this study shows. The unavailability of annual, survey-based estimates of consumption may obscure the true, annual trend over time. Second, as surveys changed over time, estimates relying on them may be inconsistent with one another. Finally, these two methods measure different components of illicit trade, specifically consumption of illicit cigarettes regardless of origin and smuggling of cigarettes into a particular market. However, absent a gold standard, comparisons of different approaches to illicit trade measurement serve efforts to refine and improve measurement approaches and estimates.

Cross-Domain Multitask Learning with Latent Probit Models

Learning multiple tasks across heterogeneous domains is a challenging problem since the feature space may not be the same for different tasks. We assume the data in multiple tasks are generated from a latent common domain via sparse domain transforms and propose a latent probit model (LPM) to jointly learn the domain transforms, and the shared probit classifier in the common domain. To learn meaningful task relatedness and avoid over-fitting in classification, we introduce sparsity in the domain transforms matrices, as well as in the common classifier. We derive theoretical bounds for the estimation error of the classifier in terms of the sparsity of domain transforms. An expectation-maximization algorithm is derived for learning the LPM. The effectiveness of the approach is demonstrated on several real datasets.

A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

Wave transformation and wave-driven flow across a steep coral reef

Observations of waves, setup, and wave-driven mean flows were made on a steep coral forereef and its associated lagoonal system on the north shore of Moorea, French Polynesia. Despite the steep and complex geometry of the forereef, and wave amplitudes that are nearly equal to the mean water depth, linear wave theory showed very good agreement with data. Measurements across the reef illustrate the importance of including both wave transport (owing to Stokes drift), as well as the Eulerian mean transport when computing the fluxes over the reef. Finally, the observed setup closely follows the theoretical relationship derived from classic radiation stress theory, although the two parameters that appear in the model-one reflecting wave breaking, the other the effective depth over the reef crest-must be chosen to match theory to data. © 2013 American Meteorological Society.