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We analyze the coherent dynamics of a fluxonium device (Manucharyan et al 2009 Science 326 113) formed by a superconducting ring of Josephson junctions in which strong quantum phase fluctuations are localized exclusively on a single weak element. In such a system, quantum phase tunnelling by occurring at the weak element couples the states of the ring with supercurrents circulating in opposite directions, while the rest of the ring provides an intrinsic electromagnetic environment of the **qubit**. Taking into account the capacitive coupling between nearest neighbors and the capacitance to the ground, we show that the homogeneous part of the ring can sustain electrodynamic modes which couple to the two levels of the flux **qubit**. In particular, when the number of Josephson junctions is increased, several low-energy modes can have **frequencies** lower than the **qubit** **frequency**. This gives rise to a quasiperiodic dynamics, which manifests itself as a decay of **oscillations** between the two counterpropagating current states at short times, followed by **oscillation**-like revivals at later times. We analyze how the system approaches such a dynamics as the ring's length is increased and discuss possible experimental implications of this non-adiabatic regime.

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- Video

We explain how continuous-variable quantum error-correcting codes can be invoked to protect quantum gates in superconducting circuits against thermal and Hamiltonian noise. The gates are executed by turning on and off a tunable Josephson coupling between an LC **oscillator** and a **qubit** or pair of quits; assuming perfect **qubits**, we show that the gate errors are exponentially small when the **oscillator**'s impedance is large in natural units. The protected gates are not computationally universal by themselves, but a scheme for universal fault-tolerant quantum computation can be constructed by combining them with unprotected noisy operations.

Data Types:

- Video

The **oscillation** amplitude of impingement plane at z=−L in Reactor II at various (a) excitation amplitudes and (b) excitation **frequencies**.
... The relationship between **frequency** and Reynolds number in Reactor II. () self-sustained flapping **oscillation** **frequencies** with excitation, and () the excitation **frequencies** corresponding to excited deflecting **oscillation**.
... Summary of **oscillation** behaviors in T-jets reactors with and without excitation.
... **Oscillation** amplitude of impingement plane at z=−L in Reactor I at various (a) excitation amplitudes and (b) excitation **frequencies**.
... **Oscillation** behavior... Flapping **oscillation** **frequencies** of the impingement plane at z=−L in Reactor I.

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How fast do earthquake waves **oscillate**?
Discretisation of seismic signals, Fourier transformation from time to **frequency** domain, Qualitative analysis of **frequency** spectra

Data Types:

- Video

MESA inlists associated with The Impact of White Dwarf Luminosity Profiles on **Oscillation** **Frequencies**

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HSOs in cardiomyocytes with blockade of the SR functions. (A) Top, left: epi-illumination image of a myocyte expressed with AcGFP in Z-disks. Fluo-8 was loaded into the myocyte, and 200μM ryanodine and 4μM thapsigargin were present. Laser center is indicated (closed red circle). The yellow arrow indicates the sarcomere used for the analysis. Top, right: time-dependent changes in temperature induced by IR laser irradiation for 10s (2nd irradiation; see Fig. S1B for data of 1st, 2nd and 3rd irradiations). Given the distance from the laser center, the temperature in the sarcomeres indicated by the yellow arrow in Top, left was estimated to be 40.8°C. Bottom, left: enlarged view of the graph (2–6s) showing the occurrence of HSOs. Note that HSOs were induced in a delayed fashion (∼1.5s after the onset of heating). Bottom, right: enlarged view of the graph showing HSOs at the steady-state. Note no periodic F.I. (i.e., [Ca2+]i) changes in any of the graphs, due to the presence of ryanodine and thapsigargin. (B) Effect of heating on peak SLs during HSOs. IR laser irradiation (for 10s each) was applied three times consecutively. Minimal and maximal SLs, obtained at the peak of shortening and lengthening, respectively. (C) Effect of heating on the amplitude of HSOs. Amplitude was defined as maximal SL (at the peak of lengthening) minus minimal SL (at the peak of shortening) [cf. (B)]. (D) Effect of heating on the **frequency** of HSOs. (E) Effect of heating on the sarcomere shortening velocity during HSOs. Velocity was calculated based on our previous study employing SL nanometry [13]. In (B), (C), (D) and (E), data obtained 1.5, 5 and 8.5s after the onset of heating (i.e., beginning, middle and end of heating) were analyzed for 2nd and 3rd heating. Due to time delay of the appearance of HSOs [see (A) and Fig. S1B], data obtained 4, 6.5 and 9s after the onset of heating (i.e., beginning, middle and end of heating) were analyzed for 1st heating. ∗P<0.05 compared with the corresponding value obtained in the beginning of heating. #P<0.05 compared with the corresponding value obtained in the middle of heating. †P<0.05 compared with the corresponding value during the 1st heating. ‡P<0.05 compared with the corresponding value during the 2nd heating.
... Temperature dependence of the occurrence of HSOs. (A) Top: relationship between the distance from the laser center and ΔT in spontaneously beating myocytes, showing whether or not HSOs were induced by heat pulses. Myocytes were set on the microscopic system at the initial temperature of 25°C, and the heat pulse was given for 10s. The values of ΔT that induced HSOs are shown in orange, and those at which HSOs were not observed are shown in gray. The lowest and highest end of the temperature range for HSOs are shown in blue and red triangles, respectively. Bottom: temperature dependence of the state of sarcomeres in neonatal myocytes, i.e., spontaneous beating without HSOs (green) or spontaneous beating with HSOs (yellow), or contraction [with no **oscillations** (red)]. Cell numbers are indicated on left. Of the total 16 myocytes tested, four myocytes (#13–16) did not exhibit HSOs (one even at 41.5°C). Note that the lowest temperatures at which HSOs occur in various myocytes are likely lower than those shown in this graph (Table 2). (B) Top: same as in (A) top in non-beating myocytes, showing whether or not HSOs were induced by heat pulses. As in (A), myocytes were set on the microscopic system at the initial temperature of 25°C, and the heat pulse was given for 10s. The values of ΔT that induced HSOs are shown in orange, and those at which HSOs were not observed are shown in gray. The lowest and highest end of the temperature range for HSOs are shown in blue and red triangles, respectively. Bottom: temperature dependence of the state of sarcomeres in neonatal myocytes, either relaxation (green), HSOs (yellow) or contraction [with no **oscillations** (red)]. Cell numbers are indicated on left. Of the total 8 myocytes tested, one myocyte (#5) did not exhibit relaxation, presumably because the lowest temperature at which observation started was relatively high (i.e., 40.6°C).
... Summary of maximal and minimal temperatures for HSOs in beating and non-beating myocytes. Tmax (Tmin), maximal (minimal) temperature at which HSOs were observed in the experiments of Fig. 3. Note that the real Tmin value may be even lower than the value (∼38.6°C) in this table and hence closer to the body temperature of rat neonates [22] in beating myocytes, because in all myocytes showing HSOs, the **oscillations** were already induced at the lowest temperatures (37–40°C; see Fig. 3A) given. ΔT, Tmax minus Tmin. n, 12 and 8 for beating and non-beating myocytes, respectively.
... HSOs in spontaneously beating cardiomyocytes. (A) Epi-illumination image of a myocyte expressed with AcGFP in Z-disks. Fluo-8 loaded into the myocyte. Laser center is indicated (closed red circle). A yellow arrow indicates the sarcomere for the analysis in (C). (B) Top: changes in temperature induced by IR laser irradiation for 10s. Bottom: relationship of the distance from laser center and ΔT. Given the distance from the laser center, the temperature in the sarcomeres, indicated by the yellow arrow in (A), was estimated to be 39.7°C. (C) Left, top: time-dependent changes in SL and F.I. in the myocyte in (A) upon IR laser irradiation (2nd irradiation; see Fig. S1A for data of 1st and 2nd irradiations). F.I. was obtained from the whole myocyte in the view window. Heat pulse was given for 10s, i.e., from 3 to 13s (in gray). Due to the temperature sensitivity of fluorescence [23], F.I. was decreased (or increased) as heat pulse was given (or ceased). Left, bottom: enlarged view of the graph showing SL on top from 6 to 10s. HSOs are clearly seen coexisting with [Ca2+]i-dependent spontaneous beating. Right, top: FFT analysis for the changes in F.I. from 4.7 to 12.5s. Right, bottom: same as in Right, top for the changes in SL. Note that while only one peak is present for F.I., two different components, i.e., slow and fast components, are seen for SL (with the former corresponding to that for F.I.; see arrows). (D) Left: effects of IR laser irradiations on the **frequency** of Ca2+ transient. Closed red circles with error bars, average values; open circles in various colors without error bars, individual data. ∗Pfrequency of HSOs. Closed black circles with error bars, average values; open circles in various colors without error bars, individual data. The HSO **frequency** was not significantly changed during the course of heating (both 1st and 2nd heating). Data obtained 1.5, 5 and 8.5s after the onset of heating were analyzed. n=24 (12 cells).
... Summary of the effects of heating on the cross-bridge attachment rate and the properties of HSOs. Our mathematical model predicts that heating increases amplitude and decreases **frequency** via an increase in the attachment rate constant of cross-bridges, α. Arrows, directions of change. See [19,20] for details of our model.

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- Video
- Tabular Data
- Document

What **oscillates** in a seismometer?
Construction and operation principle of geophones and seismometers, effects of different **oscillation** **frequencies** on the measurement of ground motion, damping of the **oscillating** system.

Data Types:

- Video

Loss of correlation exhibited by mitochondrial **oscillators** in the high-**frequency**, low-amplitude domain of the power spectrum after treatment affecting ROS production, scavenging, or spreading. Isolated cardiomyocytes loaded with TMRM as described for the other experiments were treated for 30min with 64μM 4′Cl-DZP (n=4, two experiments) or 15μM rotenone (n=4, two experiments) or for 2h with 135μM TMPyP (n=4, two experiments). Treated cells were imaged at 110-ms time resolution and the TMRM fluorescence time series analyzed by PSA. (A) We determined the Pearson correlation coefficient, r, of the high-**frequency**, low-amplitude region of the spectrum as that represented by **frequencies** >0.3Hz. Represented are the absolute values of r. We determined 0.3Hz as delimiting the high-**frequency** region on the basis of analysis of white noise spectra since it contains 93% of the data points (panel E; the dashed line points out the 0.3Hz **frequency**, which to the right corresponds to the high-**frequency**, low-amplitude domain of the spectrum, as in panels B–E). This analysis was also applied to randomized time series (see Fig. 4 B) or mitochondria **oscillating** outside the mitochondrial cluster (panel D). The region of the spectrum >0.3Hz corresponds to random behavior characterized by r=0.051±0.001 (n=4) as opposed to r=0.70±0.05 (n=4) exhibited by the mitochondrial network under control conditions (panel B) or an **oscillating** mitochondrial cluster (panel C). In panels B–E, the dashed lines represent the linear fitting for the two separate regions of the spectrum to emphasize that the change in slope mainly happens in the high **frequency** domain (see also Fig. 5). In panel A, WN is white noise and MOC stands for mitochondria outside the cluster.
... Physiological and pathophysiological behaviors of the mitochondrial network in heart cells. Freshly isolated ventricular cardiomyocytes were loaded with 100nM TMRM and imaged by two photon microscopy (150-ms time resolution) as described in Materials and Methods. The results obtained from a stack of 3720 images are shown (see the video of this experiment in Supplementary Material). Before the mitochondrial network reaches criticality (9,15,30), the ΔΨm (as measured by TMRM) **oscillates** at high **frequencies** and small amplitudes. After criticality, the network behavior evolves into “pathophysiological” behavior characterized by low-**frequency**, high-amplitude **oscillations** (8,10). The return plot of the time series shown in the inset was calculated by representing the fluorescent signal, Xn, with a lag of 150ms with respect to itself, Xn+1. This graph allows a quick visualization of the richness of high-**frequency**, low-amplitude **oscillations** present in the physiological as opposed to the low-**frequency**, high-amplitude **oscillations** present in the pathophysiological regime.
... Inverse power law behavior and the amplitude versus **frequency** relationship exhibited by the mitochondrial **oscillator**. (A) **Oscillations** were simulated with our computational model of the mitochondrial **oscillator** ((10); see also Fig. S4 in Supplementary Material). The double log graph of the amplitude versus **frequency** (1/period) was plotted from ΔΨm **oscillations** with amplitudes in the range of 2–124mV and periods ranging from 70 to 430ms, respectively. The simulations for shunt=0.1 and SOD concentrations from 0.9×10−4 to 1.3×10−3 were performed with the set of parameters described in the legend of Fig. 2 (see Cortassa et al. (10) and Cortassa et al. (14)) for detailed parameter descriptions). B) From the simulations, we selected five oscillatory periods in the high **frequency** domain (between 70 and 300ms) and one from the low-**frequency** (1-min period) domain and attributed each one of them proportionally to a network composed by 500 mitochondria as described in Supplementary Material (see also Fig. S5). A matrix containing a total of 500 columns (mitochondria) and 6,000 rows was constructed. The time steps represented by the rows correspond to a fixed integration step of 20ms for the numerical integration of the system of ordinary differential equations (see Supplementary Material). We applied RDA and PSA to the average value of each row of the matrix at, e.g., time 1, T1, that represents the experimental average value of fluorescent intensity of the ΔΨm probe (corresponding to mV) obtained every 110ms from 500 mitochondria (on average) from each image of our stack.
... **Frequency** and amplitude modulation of the mitochondrial **oscillator** model through changes in the balance between ROS production and ROS scavenging. (A) **Oscillation** periods of 25ms and 143ms are shown for SOD concentrations of 0.75μM and 1.07μM, respectively. The model parameters used to run the simulations for shunt=0.0744 (defined as the fraction of the electron flow in the respiratory chain diverted to the generation of superoxide anion, O2−) were concentration of respiratory chain carriers, ρREN=2.5×10−6mM; concentration of F1F0 ATPase, ρF1=2.03×10−3mM; [Ca2+]i=0.1μM; Kcc=0.01mM; kSOD1=2.4×106mM−1s−1; kCAT1=1.7×104mM−1s−1; GT=0.5mM; maximal rate of the adenine nucleotide translocase, VmaxANT=5mMs−1; maximal rate of the mitochondrial Na-Ca exchanger, VmaxNaCa= 0.015mMs−1. The O2−concentrations correspond to the mitochondrial matrix space and were calculated as described in Cortassa et al. (14). Remaining parameters were set as described in Cortassa et al. (10) and Cortassa et al. (14). (B) Under similar parametric conditions, the **frequency** and amplitude of the **oscillations** in O2− delivered to the cytoplasm as a function of the fractional O2− production in the high-**frequency** domain (ms). Within the oscillatory region (shaded), the oscillatory period constantly decreased, whereas the amplitude reached a peak and then decreased as a function of the increase in ROS production. A similar analysis was performed in the low-**frequency** domain (seconds to minutes) for SOD concentrations of 1.87μM (not shown). In the latter case, the amplitude of O2− (0.86mM) did not change as a function of the shunt (from 0.05 to 0.25), whereas the period decreased from 276s to 62s, respectively.
... PSA of TMRM fluorescence time series from the mitochondrial network of cardiomyocytes. Experiments were carried out as described in the legend of Fig. 1 and Materials and Methods. The time series of TMRM fluorescence was subjected to FFT as described in Materials and Methods. (A) PSA: The power spectrum was obtained from the FFT of the TMRM signal as the double log plot of the amplitude (power) versus the **frequency**. This relationship obeys a homogeneous power law (1/fβ; with f, **frequency**, and β, the spectral exponent) and is statistically self-similar, which means that there is no dominant **frequency**. The PSA reveals a broad spectrum of **oscillation** in normally polarized mitochondria with a spectral exponent of β=1.79, whereas a random process (white noise) gives a β ∼ 0, meaning that there is no relationship between the amplitude and the **frequency** in a random signal. A β=1.0 (Supplementary Material, Fig. S3) or 2.0 (Fig. 4 B, bottom panels) corresponds to pink or brown noise, respectively. The inverse power law spectrum arises from the coupling of **frequency** and amplitude in an orderly statistical sequence. The periods, in seconds or milliseconds, at the bottom of panel A are intended to facilitate the interpretation of the high- and low-**frequency** domains of the spectrum. (B) When the time series of the TMRM fluorescent signal is randomized (mid, left), we obtain a value of β close to zero (mid, right) as opposed to a β=1.79 in the nonrandomized signal (right, top). The spectral exponent β=1.79 (right, top) is consistent with long-range correlations that after signal randomization becomes white noise, with loss of correlation properties β=0.25 (≅ 0) (mid, right) (12).

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- Video
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Relation between the **oscillation** **frequency** and the coupling strength.
... Capacitive coupled RC-**oscillators**.
... RC-**oscillators**... Coupled **oscillators**... (a) Single RC **oscillator** and (b) small-signal equivalent circuit.
... Quadrature **oscillator**... Simulated **frequency**.
... Van der Pol **oscillators**... **Frequency** of **oscillation** with the **oscillators** uncoupled and coupled (CX=20fF).

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Snapshots of **oscillations**. The original movies are presented in the Supplementary material. In the left panel (Supplementary movie-2), white circles evolved from the middle of the original blue circles. These spread over the whole reaction volume and the colors then reversed again. By the end, the whole reaction volume had become blue. In the right panel (Supplementary movie-3), two propellers evolved after fronts had undergone damping. These propellers continued to rotate for several hours. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
... Bulk **oscillation** of hydrogenase. Absorption change of an unbuffered hydrogenase solution containing 2 mM benzyl viologen and 200 nM hydrogenase, with 50 μl (blue line) or 100 μl (red line) of gaseous H2 added at the beginning of the reaction. The measurement was performed in an anaerobic cell with a path-length of 0.6 mm. After addition of H2, the cell was vigorously shaken in order to distribute the H2 evenly in the solution. Stirring the reaction did not affect the **oscillation** behavior of the reaction: stirred (blue line) and non-stirred (red line) reactions both exhibited **oscillations**. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
... **Oscillating** reaction... The experiment was performed in a thin-layer reaction chamber [13], which was flushed with gaseous N2 for 10 min. After the air was replaced by N2, the gas inlet was closed and gaseous H2 was injected into the atmosphere (final H2 concentration 10%). The movie was accelerated 100-fold (1 s in the movie corresponds to 100 s in real time). The total length of the movie is 30 s. The whole movie is repeated and white squares are positioned onto the regions where local **oscillations** are clearly visible.

Data Types:

- Image
- Video