Optimal Scaling of a Gradient Method for Distributed Resource Allocation

We consider a class of weighted gradient methods for distributed resource allocation over a network. Each node of the network is associated with a local variable and a convex cost function; the sum of the variables (resources) across the network is fixed. Starting with a feasible allocation, each node updates its local variable in proportion to the differences between the marginal costs of itself and its neighbors. We focus on how to choose the proportional weights on the edges (scaling factors for the gradient method) to make this distributed algorithm converge and on how to make the convergence as fast as possible. We give sufficient conditions on the edge weights for the algorithm to converge monotonically to the optimal solution; these conditions have the form of a linear matrix inequality. We give some simple, explicit methods to choose the weights that satisfy these conditions. We derive a guaranteed convergence rate for the algorithm and find the weights that minimize this rate by solving a semidefinite program. Finally, we extend the main results to problems with general equality constraints and problems with block separable objective function

Multi-mesh gear dynamics program evaluation and enhancements

A multiple mesh gear dynamics computer program was continually developed and modified during the last four years. The program can handle epicyclic gear systems as well as single mesh systems with internal, buttress, or helical tooth forms. The following modifications were added under the current funding: variable contact friction, planet cage and ring gear rim flexibility options, user friendly options, dynamic side bands, a speed survey option and the combining of the single and multiple mesh options into one general program. The modified program was evaluated by comparing calculated values to published test data and to test data taken on a Hamilton Standard turboprop reduction gear-box. In general, the correlation between the test data and the analytical data is good

A synoptic view of ionic constitution above the F-layer maximum

Ionic composition above F layer maximum from Ariel I satellite ion mass spectromete

Self-stabilizing Numerical Iterative Computation

Many challenging tasks in sensor networks, including sensor calibration, ranking of nodes, monitoring, event region detection, collaborative filtering, collaborative signal processing, {\em etc.}, can be formulated as a problem of solving a linear system of equations. Several recent works propose different distributed algorithms for solving these problems, usually by using linear iterative numerical methods. In this work, we extend the settings of the above approaches, by adding another dimension to the problem. Specifically, we are interested in {\em self-stabilizing} algorithms, that continuously run and converge to a solution from any initial state. This aspect of the problem is highly important due to the dynamic nature of the network and the frequent changes in the measured environment. In this paper, we link together algorithms from two different domains. On the one hand, we use the rich linear algebra literature of linear iterative methods for solving systems of linear equations, which are naturally distributed with rapid convergence properties. On the other hand, we are interested in self-stabilizing algorithms, where the input to the computation is constantly changing, and we would like the algorithms to converge from any initial state. We propose a simple novel method called \syncAlg as a self-stabilizing variant of the linear iterative methods. We prove that under mild conditions the self-stabilizing algorithm converges to a desired result. We further extend these results to handle the asynchronous case. As a case study, we discuss the sensor calibration problem and provide simulation results to support the applicability of our approach

Robust Quantum Error Correction via Convex Optimization

We present a semidefinite program optimization approach to quantum error correction that yields codes and recovery procedures that are robust against significant variations in the noise channel. Our approach allows us to optimize the encoding, recovery, or both, and is amenable to approximations that significantly improve computational cost while retaining fidelity. We illustrate our theory numerically for optimized 5-qubit codes, using the standard [5,1,3] code as a benchmark. Our optimized encoding and recovery yields fidelities that are uniformly higher by 1-2 orders of magnitude against random unitary weight-2 errors compared to the [5,1,3] code with standard recovery. We observe similar improvement for a 4-qubit decoherence-free subspace code.Comment: 4 pages, including 3 figures. v2: new example

Efficient Optimal Minimum Error Discrimination of Symmetric Quantum States

This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator $\rho_0$. It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator $\Pi_0$, and maximizing $\textrm{Tr}(\rho_0 \Pi_0)$. In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a semidefinite positive operator $X$ commuting with the symmetry operator and such that $X >= \rho_0$. The new formulation gives a deeper insight into the optimization problem and allows to obtain closed-form analytical solutions, as shown by a simple but not trivial explanatory example. Besides the theoretical interest, the result leads to semidefinite programming solutions of reduced complexity, allowing to extend the numerical performance evaluation to quantum communication systems modeled in Hilbert spaces of large dimension.Comment: 5 pages, 1 Table, no figure

Quantum Entanglement Initiated Super Raman Scattering

It has now been possible to prepare chain of ions in an entangled state and thus question arises --- how the optical properties of a chain of entangled ions differ from say a chain of independent particles. We investigate nonlinear optical processes in such chains. We explicitly demonstrate the possibility of entanglement produced super Raman scattering. Our results in contrast to Dicke's work on superradiance are applicable to stimulated processes and are thus free from the standard complications of multimode quantum electrodynamics. Our results suggest the possibility of similar enhancement factors in other nonlinear processes like four wave mixing.Comment: 4 pages, 1 figur

Optically mediated nonlinear quantum optomechanics

We consider theoretically the optomechanical interaction of several mechanical modes with a single quantized cavity field mode for linear and quadratic coupling. We focus specifically on situations where the optical dissipation is the dominant source of damping, in which case the optical field can be adiabatically eliminated, resulting in effective multimode interactions between the mechanical modes. In the case of linear coupling, the coherent contribution to the interaction can be exploited e.g. in quantum state swapping protocols, while the incoherent part leads to significant modifications of cold damping or amplification from the single-mode situation. Quadratic coupling can result in a wealth of possible effective interactions including the analogs of second-harmonic generation and four-wave mixing in nonlinear optics, with specific forms depending sensitively on the sign of the coupling. The cavity-mediated mechanical interaction of two modes is investigated in two limiting cases, the resolved sideband and the Doppler regime. As an illustrative application of the formal analysis we discuss in some detail a two-mode system where a Bose-Einstein condensate is optomechanically linearly coupled to the moving end mirror of a Fabry-P\'erot cavity.Comment: 11 pages, 8 figure

Supersensitive measurement of angular displacements using entangled photons

We show that the use of entangled photons having non-zero orbital angular momentum (OAM) increases the resolution and sensitivity of angular-displacement measurements performed using an interferometer. By employing a 4$\times$4 matrix formulation to study the propagation of entangled OAM modes, we analyze measurement schemes for two and four entangled photons and obtain explicit expressions for the resolution and sensitivity in these schemes. We find that the resolution of angular-displacement measurements scales as $Nl$ while the angular sensitivity increases as $1/(2Nl)$, where $N$ is the number of entangled photons and $l$ the magnitude of the orbital-angular-momentum mode index. These results are an improvement over what could be obtained with $N$ non-entangled photons carrying an orbital angular momentum of $l\hbar$ per photonComment: 6 pages, 3 figure