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A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.
The relations between the various frequency-moment sum rules of electron liquids, which include even-power moments, are systematically examined by using the Cauchy-Schwarz and Hoelder inequalities. A relation involving the isothermal sound velocity and the kinetic and potential energies is obtained from one of the inequalities in the long-wavelength limit, and is generalized to arbitrary spatial dimensions.
A classical inequality due to Bernstein which estimates the norm of polynomials on any given ellipse in terms of their norm on any smaller ellipse with the same foci is examined. For the uniform and a certain weighted uniform norm, and for the case that the two ellipses are not too close, sharp estimates of this type were derived and the corresponding extremal polynomials were determined. These Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Some new results were also presented for a weighted approximation problem of this type.
The established necessary conditions for optimality in nonlinear control problems that involve state-variable inequality constraints are applied to a class of singularly perturbed systems. The distinguishing feature of this class of two-time-scale systems is a transformation of the state-variable inequality constraint, present in the full order problem, to a constraint involving states and controls in the reduced problem. It is shown that, when a state constraint is active in the reduced problem, the boundary layer problem can be of finite time in the stretched time variable. Thus, the usual requirement for asymptotic stability of the boundary layer system is not applicable, and cannot be used to construct approximate boundary layer solutions. Several alternative solution methods are explored and illustrated with simple examples.
We report a new two-photon polarization correlation experiment for realizing the Einstein-Podolsky-Rosen-Bohm (EPRB) state and for testing Bell-type inequalities. We use the pair of orthogonally-polarized light quanta generated in Type 2 parametric down conversion. Using 1 nm interference filters in front of our detectors, we observe from the output of a 0.5mm beta - BaB2O4 (BBO) crystal the EPRB correlations in coincidence counts, and measure an associated Bell inequality violation of 22 standard deviations. The quantum state of the photon pair is a polarization analog of the spin-1/2 singlet state.
Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This paper develops two analytic methods of incorporating state variable inequality constraints in the Kalman filter. The first method is a general technique of using hard constraints to enforce inequalities on the state variable estimates. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. The second method uses soft constraints to estimate state variables that are known to vary slowly with time. (Soft constraints are constraints that are required to be approximately satisfied rather than exactly satisfied.) The incorporation of state variable constraints increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation results. The use of the algorithm is demonstrated on a linearized simulation of a turbofan engine to estimate health parameters. The turbofan engine model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new algorithms provide improved performance in this example over unconstrained Kalman filtering.
We consider the problem of state estimation for nonlinear time-varying systems whose nonlinearities satisfy an incremental quadratic inequality. These observer results unifies earlier results in the literature; and extend it to some additional classes of nonlinearities. Observers are presented which guarantee that the state estimation error exponentially converges to zero. Observer design involves solving linear matrix inequalities for the observer gain matrices. Results are illustrated by application to a simple model of an underwater.
The μ-conductance measure proposed by Lovász and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a μ fraction of the whole graph. Using μ-conductance enables us to study the network structures in new ways. Here, in this manuscript, we study a modified spectral cut for μ-conductance that is a natural relaxation of the integer program of μ-conductance and show that the optimum of this program has a two-sided Cheeger inequality with μ-conductance.
The day-to-day operation of modern power systems is highly reliant on prompt and adequate situational-awareness. This can be achieved via various system monitoring functions such as anomaly detection, in which static thresholds are commonly utilized to distinguish the normal and the abnormal system states. However, a predetermined static threshold usually lacks the flexibility to adapt to unobserved scenarios. In this paper, we propose two self-adaptive synchrophasor data driven anomaly detection approaches based on Chebyshev’s Inequality. The proposed approaches have been evaluated with Kundur’s 2area system and Mini-WECC system. Experimental results verify that the proposed approaches can dynamically adapt to unprecedented scenarios, and detect anomalous events with lower false alarm rate compared to static threshold based detection.
We consider a class of hierarchical variational inequality (VI) problems that subsumes VI-constrained optimization and several other problem classes, including the optimal solution selection problem and the optimal Nash equilibrium (NE) seeking problem. Our main contribution is threefold. (i) We consider bilevel VIs with monotone and Lipschitz continuous mappings and devise a single-timescale iteratively regularized extragradient method, named IR-EG 𝚖,𝚖 . We improve the existing iteration complexity results for addressing both bilevel VI and VI-constrained convex optimization problems. (ii) Under the strong monotonicity of the outer-level mapping, we develop a method named IR-EG 𝚜,𝚖 and derive faster guarantees than those in (i). We also study the iteration complexity of this method under a constant regularization parameter. These results appear to be new for both bilevel VIs and VI-constrained optimization. (iii) To our knowledge, complexity guarantees for computing the optimal NE in nonconvex settings do not exist. Motivated by this lacuna, we consider VI-constrained nonconvex optimization problems and devise an inexactly projected gradient method, named IPR-EG, where the projection onto the unknown set of equilibria is performed using IR-EG 𝚜,𝚖 with a prescribed termination criterion and an adaptive regularization parameter. We obtain new complexity guarantees in terms of a residual map and an infeasibility metric for computing a stationary point. Here, we validate the theoretical findings using preliminary numerical experiments for computing the best and the worst NEs.
Optimization of nonlinear systems with inequality constraints
Computation of arcs for optimal flight paths with state variable inequality constraints
Optimum design of curve generating four-bar linkages with inequality constraint
We consider the problem of state estimation from nonlinear time-varying system whose nonlinearities satisfy an incremental quadratic inequality. Observers are presented which guarantee that the state estimation error exponentially converges to zero.
A periodically poled MgO doped LiNbO3 (MgO:LN) non-degenerate photon pair source is utilized for spontaneous parametric down-conversion of 532 nm photons into time-energy entangled pairs of 800 and 1600 nm photons. The entangled photons are separated using previously detailed sorting optics, such that each wavelength is independently directed through one of two modified Mach-Zehnder interferometers also known as a Franson interferometer after which they are fiber-optically guided to high-efficiency photon detectors. Output from the detectors is sent to a high resolution time tagger, where coincidences between the entangled photons are recorded. By varying the length of the long path in one Mach-Zehnder interferometer, it is possible to observe high visibility sinusoidal fringes in the measured coincidence rates (while no variation is seen in single photon detection rates). These fringes due to interference between the photon probability amplitudes are indicative of a violation of the Bell inequality, and confirm inconsistencies with local hidden variable theory for the correlations of the time-energy entangled photon pairs.
A periodically poled MgO - doped LiNbO3 (MgO:LN) non-degenerate photon pair source is utilized for spontaneous parametric down-conversion of 532-nanometer photons into time-energy entangled pairs of 800- and 1600-nanometer photons. The entangled photons are separated using previously detailed sorting optics, such that each wavelength is independently directed through one of two modified Mach-Zehnder interferometers - also known as a Franson interferometer - after which they are fiber-optically guided to high-efficiency photon detectors. Output from the detectors is sent to a high resolution time tagger, where coincidences between the entangled photons are recorded. By varying the length of the long path in one Mach-Zehnder interferometer, it is possible to observe high visibility sinusoidal fringes in the measured coincidence rates (while no variation is seen in single photon detection rates). These fringes - due to interference between the photon probability amplitudes - are indicative of a violation of the Bell inequality, and confirm inconsistencies with local hidden variable theory for the correlations of the time-energy entangled photon pairs.
We generalize the constructions for the multipartite reflected entropy in order to construct spacetimes capable of representing multipartite entanglement wedge cross sections of differing party number as Ryu-Takayanagi surfaces on a single replicated geometry. We devise a general algorithm for such constructions for arbitrary party number and demonstrate how such methods can be used to derive novel inequalities constraining mulipartite entanglement wedge cross sections.
Climate change is already increasing the severity of extreme weather events such as with rainfall during hurricanes. But little research to date investigates if, and to what extent, there are social inequalities in climate change-attributed extreme weather event impacts. Here, we use climate change attribution science paired with hydrological flood models to estimate climate change-attributed flood depths and damages during Hurricane Harvey in Harris County, Texas. Using detailed land-parcel and census tract socio-economic data, we then describe the socio-spatial characteristics associated with these climate change-induced impacts. We show that 30 to 50% of the flooded properties would not have flooded without climate change. Climate change-attributed impacts were particularly felt in Latina/x/o neighborhoods, and especially so in Latina/x/o neighborhoods that were low-income and among those located outside of FEMA’s 100-year floodplain. Our focus is thus on climate justice challenges that not only concern future climate change-induced risks, but are already affecting vulnerable populations disproportionately now.