Fractional Uncertainty Research Articles

HORIZON RUN 3: TOPOLOGY AS A STANDARD RULER

We study the Physically Self Bound Cold Dark Matter Halo distribution which we associate with the massive galaxies within the Horizon Run 3 to estimate the accuracy in determination of the cosmological distance scale measured by the topology analysis. We apply the routine "Contour 3D" to 108 Mock Survey of $\pi$ steradians out to redshift z = 0.6, which effectively correspond to the SDSS-III BOSS survey, and compare the topology with that of a Gaussian Random Phase Field. We find that given three separate smoothing lengths $\lambda =$ 15, 21, and 34 $h^{-1}{\rm Mpc}$, the least $\chi^2$ fit genus per unit volume g yields a 1.7 % fractional uncertainty in smoothing length and angular diameter distance to $z = 0.6$. This is an improvement upon former calibrations of and presents a competitive error estimate with next BAO scale techniques. We also present three dimensional graphics of the Horizon Run 3 spherical mock survey to show a wealth of large-scale structures of the universe that are predicted in surveys like BOSS.

Robust DistributedH∞Filtering for Nonlinear Systems with Sensor Saturations and Fractional Uncertainties with Digital Simulation

This paper is concerned with the robust distributedH∞filtering problem for nonlinear systems subject to sensor saturations and fractional parameter uncertainties. A sufficient condition is derived for the filtering error system to reach the requiredH∞performance in terms of recursive linear matrix inequality method. An iterative algorithm is then proposed to obtain the filter parameters recursively by solving the corresponding linear matrix inequality. A numerical example is presented to show the effectiveness of the proposed method.

Convergence Analysis for Three Parareal Solvers

We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem $\mathbf{u}'+A\mathbf{u}=g$. The coarse propagator $\mathcal{G}$ is fixed to the backward-Euler method and three time integrators are chosen for the $\mathcal{F}$-propagator: the trapezoidal rule, the third-order diagonal implicit Runge--Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for $\mathcal{F}$ and $\mathcal{G}$ converges rapidly, but less is known when one uses for $\mathcal{F}$ the trapezoidal rule, or the fourth-order Gauss RK method, especially when the mesh ratio $J(=\Delta T/\Delta t)$ is small. We show that for a specified $\lambda_{\max}$(the maximal eigenvalue of $A$ or its upper bound), there exists some critical $J_{\rm cri}$ such that the parareal solvers derived from these three choices of $\mathcal{F}$ converge as fast as Parareal-Euler, provided $J\geq J_{\rm cri}$. Precisely, for $\mathcal{F}$ the trapezoidal rule and the fourth-order Gauss RK method, $J_{\rm cri}$ depends on $\Delta T, \Delta t$, and $\lambda_{\max}$ and we present concise formulas to calculate $J_{cri}$, while for $\mathcal{F}$ the third-order DIRK method, $J_{cri}=4$, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions.

Atomic clock with 1×10(-18) room-temperature blackbody Stark uncertainty.

The Stark shift due to blackbody radiation (BBR) is the key factor limiting the performance of many atomic frequency standards, with the BBR environment inside the clock apparatus being difficult to characterize at a high level of precision. Here we demonstrate an in-vacuum radiation shield that furnishes a uniform, well-characterized BBR environment for the atoms in an ytterbium optical lattice clock. Operated at room temperature, this shield enables specification of the BBR environment to a corresponding fractional clock uncertainty contribution of 5.5×10(-19). Combined with uncertainty in the atomic response, the total uncertainty of the BBR Stark shift is now 1×10(-18). Further operation of the shield at elevated temperatures enables a direct measure of the BBR shift temperature dependence and demonstrates consistency between our evaluated BBR environment and the expected atomic response.

Two-way optical frequency comparisons at5×10−21relative stability over 100-km telecommunication network fibers

By using two-way frequency transfer, we demonstrate ultra-high resolution comparison of optical frequencies over a telecommunication fiber link of 100 km operating simultaneously digital data transfer. We first propose and experiment a bi-directional scheme using a single fiber. We show that the relative stability at 1 s integration time is 7 10^18 and scales down to 5 10^21. The same level of performance is reached when an optical link is implemented with an active compensation of the fiber noise. We also implement a real-time two-way frequency comparison over a uni-directional telecommunication network using a pair of parallel fibers. The relative frequency stability is 10^15 at 1 s integration time and reaches 2 10^17 at 40 000 s. The fractional uncertainty of the frequency comparisons was evaluated for the best case to 2 10^20. These results open the way to accurate and high resolution frequency comparison of optical clocks over intercontinental fiber networks.

Quantum limit on time measurement in a gravitational field

Good clocks are of importance both to fundamental physics and for applications in astronomy, metrology and global positioning systems. In a recent technological breakthrough, researchers at NIST have been able to achieve a stability of one part in 1018 using an ytterbium clock. This naturally raises the question of whether there are fundamental limits to time keeping. In this article we point out that gravity and quantum mechanics set a fundamental limit on the fractional frequency uncertainty of clocks. This limit comes from a combination of the uncertainty relation, the gravitational redshift and the relativistic time dilation effect. For example, a single ion aluminium clock in a terrestrial gravitational field cannot achieve a fractional frequency uncertainty better than one part in 1022. This fundamental limit explores the interaction between gravity and quantum mechanics on a laboratory scale.

Testing scalar-tensor theories and parametrized post-Newtonian parameters in Earth orbit

We compute the PPN parameters $\gamma$ and $\beta$ for general scalar-tensor theories in the Einstein frame, which we compare to the existing PPN formulation in the Jordan frame for alternative theories of gravity. This computation is important for scalar-tensor theories that are expressed in the Einstein frame, such as chameleon and symmetron theories, which can incorporate hiding mechanisms that predict environment-dependent PPN parameters. We introduce a general formalism for scalar-tensor theories and constrain it using the limit on $\gamma$ given by the Cassini experiment. In particular we discuss massive Brans-Dicke scalar fields for extended sources. Next, using a recently proposed Earth satellite experiment, in which atomic clocks are used for spacecraft tracking, we compute the observable perturbations in the redshift induced by PPN parameters deviating from their general relativistic values. Our estimates suggest that $|\gamma - 1| \sim |\beta -1| \sim 10^{-6}$ may be detectable by a satellite that carries a clock with fractional frequency uncertainty $\Delta f/f \sim 10^{-16}$ in an eccentric orbit around the Earth. Such space experiments are within reach of existing atomic clock technology. We discuss further the requirements necessary for such a mission to detect deviations from Einstein relativity.

Improved limit on a temporal variation of mp/me from comparisons of Yb+ and Cs atomic clocks.

Accurate measurements of different transition frequencies between atomic levels of the electronic and hyperfine structure over time are used to investigate temporal variations of the fine structure constant α and the proton-to-electron mass ratio μ. We measure the frequency of the (2)S1/2→(2)F7/2 electric octupole (E3) transition in (171)Yb(+) against two caesium fountain clocks as f(E3)=642,121,496,772,645.36 Hz with an improved fractional uncertainty of 3.9×10(-16). This transition frequency shows a strong sensitivity to changes of α. Together with a number of previous and recent measurements of the (2)S1/2→(2)D3/2 electric quadrupole transition in (171)Yb(+) and with data from other elements, a least-squares analysis yields (1/α)(dα/dt)=-0.20(20)×10(-16)/yr and (1/μ)(dμ/dt)=-0.5(1.6)×10(-16)/yr, confirming a previous limit on dα/dt and providing the most stringent limit on dμ/dt from laboratory experiments.

Systematic measurement of fast neutron background fluctuations in an urban area using a mobile detection system

Neutron background measurements using a mobile trailer-based system were conducted in Knoxville, Tennessee, USA. The 0.5m2 system, consisting of eight EJ-301 liquid scintillation detectors, was used to collect neutron background measurements in order to better understand the systematic variations in background that depend solely on the street-level measurement position in a downtown area. Data was collected along 5 different streets, and the measurements were found to be repeatable. Using 10-min measurements, the fractional uncertainty in each measured data point was <2%. Compared with fast neutron background count rates measured away from downtown Knoxville, a reduction in background count rates ranging from 10% to 50% was observed in the downtown area, sometimes varying substantially over distances of tens of meters. These reductions are attributed to the net shielding of the cosmic ray neutron flux by adjacent buildings. For reference, the building structure as observed at street level is quantified in part here by a measured angle-of-open-sky metric.

Robust Ramsey sequences with Raman adiabatic rapid passage

We present a method for robust timekeeping in which alkali-metal atoms are interrogated in a Ramsey sequence based on stimulated Raman transitions with optical photons. To suppress systematic effects introduced by differential ac Stark shifts and optical intensity gradients, we employ atom optics derived from Raman adiabatic rapid passage (ARP). Raman ARP drives coherent transfer between the alkali-metal hyperfine ground states via a sweep of the Raman detuning through the two-photon resonance. Our experimental implementation of Raman ARP reduced the phase sensitivity of Ramsey sequences to Stark shifts in $^{133}\mathrm{Cs}$ atoms by about two orders of magnitude, relative to fixed-frequency Raman transitions. This technique also preserved Ramsey fringe contrast for cloud displacements reaching the $1/{e}^{2}$ intensity radius of the laser beam. In a magnetically unshielded apparatus, second-order Zeeman shifts limited the fractional frequency uncertainty to $\ensuremath{\sim}3.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ after about 2500 s of averaging.

Measurement of Laser Frequencies from CD3OH and CD3OD up to 8.6 THz

Twenty two laser frequencies, whose values range from 1.6 to 8.6 THz, have been measured for the first time using heterodyne techniques. These laser emissions were generated by an optically pumped molecular laser that used either CD3OH or CD3OD as its lasing medium. At least three of the observed laser emissions generated by CD3OH were discovered during this investigation and the first laser frequencies measured for CD3OH above 8 THz are reported. The laser frequencies were measured with fractional uncertainties up to ± 2 × 10−7, of sufficient accuracy to confirm two proposed far-infrared laser assignments. The offset frequency of the CO2 pump laser with respect to its center frequency was also measured for nearly all laser emissions generated by CD3OH.

Delay-dependent robust stability criteria for stochastic neural networks of neutral-type with interval time-varying delay and linear fractional uncertainties

In this paper, we investigate the problem of robust stability for a class of delayed neural networks of neutral-type with linear fractional uncertainties. The activation functions are assumed to be unbounded, non-monotonic and non-differentiable, and the delay is assumed to be time-varying and belonging to a given interval, which means that the lower and upper bounds of the interval time-varying delay are available. By constructing a general form of the Lyapunov–Krasovskii functional, and using the linear matrix inequality (LMI) approach, we derive several delay-dependent stability criteria in terms of LMI. Finally, we give a number of examples to illustrate the effectiveness of the proposed method.

Computer-controlled high-precision Michelson wavemeter

The Michelson wavemeter is a popular instrument in many experiments where the high-precision measurement of a cw laser wavelength is required. In this paper, we describe a simple and inexpensive way to obtain high-precision measurements with this classical physicist’s tool. We exploit the time stamp provided by the high-frequency clock present in modern data acquisition cards to measure the fractional uncertainty of the interference signal. The resulting relative uncertainty value for our current set-up is of the order of 10−8 and can be potentially improved by a factor of 100.

MEASURING TIDES AND BINARY PARAMETERS FROM GRAVITATIONAL WAVE DATA AND ECLIPSING TIMINGS OF DETACHED WHITE DWARF BINARIES

The discovery of the most compact detached white dwarf (WD) binary SDSS J065133.33+284423.3 has been discussed in terms of probing the tidal effects in white dwarfs. This system is also a verification source for the space-based gravitational wave (GW) detector, evolved Laser Interferometer Space Antenna (eLISA) which will observe short-period compact Galactic binaries with $P_{orb}\lesssim 5$ hrs. We address the prospects of doing tidal studies using eLISA binaries by showing the fractional uncertainties in the orbital decay rate and the rate of that decay, $\dot{f}, \ddot{f}$ expected from both the GW and EM data for some of the high-$f$ binaries. We find that $\dot{f}$ and $\ddot{f}$ can be measured using GW data only for the most massive WD binaries observed at high-frequencies. Form timing the eclipses for $\sim 10$ years, we find that $\dot{f}$ can be known to $\sim 0.1% $ for J0651. We find that from GW data alone, measuring the effects of tides in binaries is (almost) impossible. We also investigate the improvement in the knowledge of the binary parameters by combining GW amplitude and inclination with EM data with and without $\dot{f}$. In our previous work we found that EM data on distance constrained 2-$\sigma$ uncertainty in chirp mass to $15-25%$ whereas adding $\dot{f}$ reduces it to $0.11%$. EM data on $\dot{f}$ also constrains 2-$\sigma$ uncertainty in distance to $35%-19%$. EM data on primary mass constrains the secondary mass $m_2$ to factors of 2 to $\sim40%$ whereas adding $\dot{f}$ reduces this to $25%$. And finally using single-line spectroscopic constrains 2-$\sigma$ uncertainties in both the $m_2, d$ to factors of 2 to $\sim 40%$. Adding EM data on $\dot{f}$ reduces these 2-$\sigma$ uncertainties to $\leq 25%$ and $6%-19%$ respectively. Thus we find that EM measurements of $\dot{f}$ and radial velocity will be valuable in constraining binary parameters.

CONSTRAINING PARAMETERS OF WHITE-DWARF BINARIES USING GRAVITATIONAL-WAVE AND ELECTROMAGNETIC OBSERVATIONS

The space-based gravitational wave (GW) detector, \emph{evolved Laser Interferometer Space Antenna} (eLISA) is expected to observe millions of compact Galactic binaries that populate our Milky Way. GW measurements obtained from the eLISA detector are in many cases complimentary to possible electro-magnetic (EM) data. In our previous papers, we have shown that the EM data can significantly enhance our knowledge of the astrophysically relevant GW parameters of the Galactic binaries, such as the amplitude and inclination. This is possible due to the presence of some strong correlations between GW parameters that are measurable by both EM and GW observations, for example the inclination and sky position. In this paper, we quantify the constraints in the physical parameters of the white-dwarf binaries, i.e. the individual masses, chirp mass and the distance to the source that can be obtained by combining the full set of EM measurements such as the inclination, radial velocities, distances and/or individual masses with the GW measurements. We find the following $2-\sigma$ fractional uncertainties in the parameters of interest. The EM observations of distance constrains the the chirp mass to $\sim 15-25 %$, whereas EM data of a single-lined spectroscopic binary constrains the secondary mass and the distance with factors of 2 to $\sim 40 %$. The single-line spectroscopic data complemented with distance constrains the secondary mass to $\sim 25-30%$. Finally EM data on double-lined spectroscopic binary constrains the distance to $\sim 30%$. All of these constraints depend on the inclination and the signal strength of the binary systems. We also find that the EM information on distance and/or the radial velocity are the most useful in improving the estimate of the secondary mass,inclination and/or distance.

Simplest molecules as candidates for precise optical clocks.

The precise measurement of transition frequencies in cold, trapped molecules has applications in fundamental physics, and extremely high accuracies are desirable. We determine suitable candidates by considering the simplest molecules with a single electron, for which the external-field shift corrections can be calculated theoretically with high precision. Our calculations show that H(2)(+) exhibits particular transitions whose fractional systematic uncertainties may be reduced to 5×10(-17) at room temperature. We also generalize the method of composite frequencies, introducing tailored linear combinations of individual transition frequencies that are free of the major systematic shifts, independent of the strength of the external perturbing fields. By applying this technique, the uncertainty of the composite frequency is reduced compared to what is achievable with a single transition, e.g., to the 10(-18) range for HD(+). Thus, these molecules are of metrological relevance for future studies.

ℓ 2 –ℓ ∞ reliable control for discrete time‐delay systems with fractional uncertainties and saturated package losses

This study deals with the l 2 -l ∞ reliable control problem for a class of discrete time-delay systems. The considered system involves fractional uncertainties, saturated package losses and stochastic non-linearities as well as possible actuator failures. A sensor model is proposed to depict the phenomenon of saturated package losses which better reflect the reality in a networked environment. In addition, stochastic non-linearities with statistical characteristics cover several well-studied non-linear functions as special cases. The focus of this study is placed upon the design of a reliable controller such that the closed-loop system satisfies a prescribed noise attenuation level in an l 2 -l ∞ sense. By utilising stochastic analysis methods, some sufficient conditions are established to guarantee both the exponentially mean-square stability and the l 2 -l ∞ performance. Owing to the obtained conditions with a non-linear equality constraint, the cone complementary linearisation method is exploited to cast them into a convex optimisation problem, which can be readily solved by using standard numerical software. Finally, a simulation example is exploited to demonstrate the applicability of the proposed design approach.

Robust reliable sampled-data control for offshore steel jacket platforms with nonlinear perturbations

In this article, we consider the robust reliable sample-data control problem for an offshore steel jacket platform with input time-varying delay and possible occurrence of actuator faults subject to nonlinear self-exited hydrodynamic forces. The main objective of this work is to design a state feedback reliable sample-data controller such that for all admissible uncertainties as well as actuator failure cases, the resulting closed-loop system is robustly exponentially stable. By constructing an appropriate Lyapunov–Krasovskii functional and using linear matrix inequality (LMI) approach, a new set of sufficient condition is derived in terms of LMIs for the existence of robust reliable sample-data control law. In particular, the uncertainty under consideration in system parameters includes linear fractional norm-bounded uncertainty. Further, Schur complement and Jenson’s integral inequality are used to substantially simplify the derivation in the main results. More precisely, the controller gain matrix for the nonlinear offshore steel jacket platform can be achieved by solving the LMIs, which can be easily facilitated by using some standard numerical packages. Finally, a numerical example with simulation result is provided to illustrate the applicability and effectiveness of the proposed reliable sampled-data control scheme.

Reliable gain‐scheduled control design for networked control systems

In this article, the problem of reliable gain‐scheduled H∞ performance optimization and controller design for a class of discrete‐time networked control system (NCS) is discussed. The main aim of this work is to design a gain‐scheduled controller, which consists of not only the constant parameters but also the time‐varying parameter such that NCS is asymptotically stable. In particular, the proposed gain‐scheduled controller is not only based on fixed gains but also the measured time‐varying parameter. Further, the result is extended to obtain a robust reliable gain‐scheduled H∞ control by considering both unknown disturbances and linear fractional transformation parametric uncertainties in the system model. By constructing a parameter‐dependent Lyapunov–Krasovskii functional, a new set of sufficient conditions are obtained in terms of linear matrix inequalities (LMIs). The existence conditions for controllers are formulated in the form of LMIs, and the controller design is cast into a convex optimization problem subject to LMI constraints. Finally, a numerical example based on a station‐keeping satellite system is given to demonstrate the effectiveness and applicability of the proposed reliable control law. © 2014 Wiley Periodicals, Inc. Complexity 21: 214–228, 2015

First accuracy evaluation of NIST-F2

We report the first accuracy evaluation of NIST-F2, a second-generation laser-cooled caesium fountain primary standard developed at the National Institute of Standards and Technology (NIST) with a cryogenic (liquid nitrogen) microwave cavity and flight region. The 80 K atom interrogation environment reduces the uncertainty due to the blackbody radiation shift by more than a factor of 50. Also, the Ramsey microwave cavity exhibits a high quality factor (>50 000) at this low temperature, resulting in a reduced distributed cavity phase shift. NIST-F2 has undergone many tests and improvements since we first began operation in 2008. In the last few years NIST-F2 has been compared against a NIST maser time scale and NIST-F1 (the US primary frequency standard) as part of in-house accuracy evaluations. We report the results of nine in-house comparisons since 2010 with a focus on the most recent accuracy evaluation. This paper discusses the design of the physics package, the laser and optics systems and the accuracy evaluation methods. The type B fractional uncertainty of NIST-F2 is shown to be 0.11 × 10−15 and is dominated by microwave amplitude dependent effects. The most recent evaluation (August 2013) had a statistical (type A) fractional uncertainty of 0.44 × 10−15.