Non-integer Ratios Research Articles

Auditory rhythm complexity affects cardiac dynamics in perception and synchronization

Accurate perception and production of auditory rhythms are key for human behaviors such as speech and music. Auditory rhythms in music range in their complexity: complex rhythms (based on non-integer ratios between successive tone durations) are more difficult to perceive and produce than simple rhythms (based on integer ratios). The physiological activity supporting this behavioral difference is not well understood. In a within-subjects design, we addressed how rhythm complexity affects cardiac dynamics during auditory perception and production. Musically trained adults listened to and synchronized with simple and complex auditory rhythms while their cardiac activity was recorded. Participants identified missing tones in the rhythms during the Perception condition and tapped on a keyboard to synchronize with the rhythms in the Synchronization condition. Participants were equally accurate at identifying missing tones in simple and complex rhythms during the Perception condition. Tapping synchronization was less accurate and less precise with complex rhythms than with simple rhythms. Linear cardiac analyses showed a slower mean heart rate and greater heart rate variability during perception than synchronization for both simple and complex rhythms; only nonlinear recurrence quantification analyses reflected cardiac differences between simple and complex auditory rhythms. Nonlinear cardiac dynamics were also more deterministic (predictable) during rhythm perception than synchronization. Individual differences during tapping showed that greater heart rate variability was correlated with poorer synchronization. Overall, these findings suggest that linear measures of musicians’ cardiac activity reflect global task variability while nonlinear measures additionally reflect stimulus rhythm complexity.

Sustained Effect of Auditory Entrainment With Coordinated Movement Varies With Temporal Complexity of Sequential Tapping.

While the ability to coordinate movements temporally with rhythmic auditory stimuli is universal, previous investigators showed that accurate rhythm reproduction depends on temporal complexity. To date, the effect of multiple pitches on the timing of rhythmic movements has been assumed. Exploring a possible sustained entrainment effect of auditory stimuli on sequential movement might further elucidate the role of temporal complexity and its interaction with multiple pitch engagement. Thus, we investigated the sustained effect of auditory entrainment and the interaction between temporal complexity and pitch on predefined sequential tapping with tapping sequences predefined before a synchronization-timekeeping task. Temporal complexity was manipulated by increasing the number of non-integer ratios in temporal rhythm. The rhythm sequences were presented with either multiple pitches or a single pitch. We found a reduction in mean asynchronies and ratio error in three rhythms with non-integer ratios, while inter-response interval error was reduced in the integer rhythm and the rhythm with one repetitive integer ratio and one non-integer ratio. Ratio error remanence was less in rhythms with two non-integer ratios. We found no significant difference between the two pitch types. There was a sustained entrainment effect of sequential tapping that varied with differing temporal complexity, and pitch information was not essential for auditory entrainment. These findings provide support for possible interventions aimed at motor learning.

EFFICIENT INCREMENTAL SIMULATION OF “1/f” NOISES

Large-scale simulations and analysis of the original powers of 2 geometric source sum algorithm for simulating [Formula: see text] noise confirm that it provides a reasonably accurate approximation to an exact [Formula: see text] spectral density with a Gaussian amplitude distribution over any arbitrarily large frequency range. This incremental algorithm is computationally efficient with a computation time, after initialization, that varies linearly with the number of samples generated. A new variation allows non-integer and random ratios for the geometric sequence that reduces variations about the exact power-law spectral density and, by varying individual source amplitudes, produces generalized [Formula: see text] noises.

Narrowband Spikes Observed during the 2013 November 7 Flare

Abstract Narrowband spikes have been observed in solar flares for several decades. However, their exact origin is still discussed. To contribute to understanding of these spikes, we analyze the narrowband spikes observed in the 800–2000 MHz range during the impulsive phase of the 2013 November 7 flare. In the radio spectrum, the spikes started with typical broadband clouds of spikes, and then their distribution in frequencies changed into unique, very narrow bands having noninteger frequency ratios. We successfully fitted frequencies of these narrow spike bands by those, calculating dispersion branches and growth rates of the Bernstein modes. For comparison, we also analyzed the model where the narrow bands of spikes are generated at the upper-hybrid frequencies. Using both models, we estimated the plasma density and magnetic field in spike sources. Then, the models are discussed, and arguments in favor of the model with the Bernstein modes are presented. Analyzing frequency profiles of this spike event by the Fourier method, we found the power-law spectra with the power-law indices varying in the −0.8 to −2.75 interval. Because at some times this power-law index was close to the Kolmogorov spectral index (−5/3), we propose that the spikes are generated through the Bernstein modes in turbulent plasma reconnection outflows or directly in the turbulent magnetic reconnection of solar flares.

Review and Analysis of Coaxial Magnetic Gear Pole Pair Count Selection Effects

Magnetic gears perform the same function as mechanical gears using magnetic fields instead of interlocking teeth. A review of the design processes used in the literature demonstrates that a critical design parameter, pole pair count, is often given inadequate consideration. In addition to reviewing the existing prototypes, this article uses a parametric simulation study to analyze the impacts of pole pair counts on gear performance and illustrate how the optimal pole counts vary with gear ratio and various design parameters. This article also introduces new ripple factors, which better correlate with torque ripple than the cogging factor ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C_{T}$ </tex-math></inline-formula> ) used in previous articles, and illustrates why designs with noninteger gear ratios tend to have much smaller torque ripples than designs with integer gear ratios. While selecting the pole counts to minimize symmetry can reduce torque ripple, designs without any symmetry are shown to experience unbalanced magnetic forces on each rotor. Thus, it is recommended to select pole counts that result in an even number of modulators but not an integer gear ratio. This article also reveals that for a fixed gear ratio, a nontrivial optimum pole count minimizes the electromagnetic losses.

Investigation of the effect of speed ratio on workpiece surface topography and grinding power in cylindrical plunge grinding using grooved and non-grooved grinding wheels

Speed ratio is the ratio of the grinding wheel angular speed to the workpiece angular speed. In this paper, it is shown that this parameter has a significant impact on workpiece surface roughness, workpiece texture, and power consumption. Integer speed ratios were investigated between 2 and 7 and selected non-integer speed ratios were studied between 4 and 5. When compared to non-integer speed ratios, integer speed ratios yielded poor surface roughness for non-grooved wheels and visible surface textures for grooved wheels. Furthermore, the non-integer speed ratios exhibited a complex pattern of roughness spikes. The concept of grain synchronization was developed to explain these phenomena. Grain synchronization is the number of grinding wheel revolutions required before grain trajectories overlap—which will happen provided the speed ratio is a rational number. It was shown how grain synchronization can be calculated using the speed ratio. Non-integer speed ratios that resulted in rapid grain synchronization were shown to result in poor surface finish and low grinding power while speed ratio that resulted in longer grain synchronization resulted in better surface finish and higher grinding power suggesting that some speed ratios should be avoided. For a given speed ratio, grooved wheels were shown to produce lower grinding forces and higher surface roughness.

Phase ordering, transformation, and grain growth of two-dimensional binary colloidal crystals: A phase field crystal modeling

The formation and dynamics of a wide variety of binary two-dimensional ordered structures and superlattices are investigated through a phase field crystal model with sublattice ordering. Various types of binary ordered phases, the phase diagrams, and the grain growth dynamics and structural transformation processes, including the emergence of topological defects, are examined. The results are compared to the ordering and assembly of two-component colloidal systems. Two factors governing the binary phase ordering are identified, the coupling and competition between the length scales of two sublattices and the selection of average particle densities of two components. The control and variation of these two factors lead to the prediction of various complex binary ordered patterns, with different types of sublattice ordering for integer vs. noninteger ratios of sublattice length scales. These findings will enable further systematic studies of complex ordering and assembly processes of binary systems particularly binary colloidal crystals.

Fabrication of Hyper-Hemisphere of Cobalt-Chromium Alloys Using Curve Generator Machining

High-precision machining is required for manufacturing hyper-hemispherical artificial joints made of difficult-to-cut metals such as cobalt-chromium (Co-Cr) alloys to provide wear resistance in the human body. The hyper-hemisphere of Co-Cr alloys is finished by curve generator machining, in which the rotation axes of the cutter and workpiece intersect each other at the center of the sphere to be machined. This paper presents a kinematic model to simulate the shape and surface topography on hyper-hemispheres with the cutter loci in curve generator machining. The kinematic model was validated with a cutting test, in which the surface profiles were measured around the pole and equator of the sphere. Simulations were performed to study the cutting process and surface finish. The appropriate cutting parameters were determined to improve the surface finish based on a kinematic simulation. A smooth surface was obtained when small inclinations of the workpiece, large nose radii of the cutter, low workpiece rotation speeds, and non-integer ratios of the tool spindle speed to the workpiece rotation speed were employed. The effects of the setting errors, such as the mounting error of the cutter and alignment error of the spindle and workpiece axes, were estimated via the kinematic simulation. It was found that the surface topography and radius of the sphere depended on the setting errors. The radius and center of the spherical shape were different from those of an ideal sphere by an error in the X-axis in the global coordinate system. The oval shape was caused by an error in the Y-axis. An error in the Z-axis affected the radius of the machined sphere.

Principles of parsimony: fMRI correlates of beat-based versus duration-based sensorimotor synchronization.

(ProQuest: ... denotes formula omitted.)The temporal processing of successive intervals requires two different mechanisms: beat-based and duration-based timing (Essens & Povel, 1985; Grahn & McAuley, 2009; Keele, Nicoletti, Ivry, & Pokorny, 1989; Pashler, 2001; Teki, Grube, & Griffiths, 2012), which seem to engage different cerebral networks (Sakai et al., 1999; Teki, Grube, Kumar, & Griffiths, 2011). Most studies comparing beat-based and duration-based timing focused on perceptual tasks (Grube, Cooper, Chinnery, & Griffiths, 2010; Grube, Lee, Griffiths, Barker, & Woodruff, 2010; Teki et al., 2011) or memory tasks (Sakai et al., 1999). Despite the fact that timing perception and production involve different cerebral networks- though with some overlap (Coull, Cheng, & Meck, 2011; Wiener, Turkeltaub, & Coslett, 2010)-no studies reported human neuroimaging results of direct comparison between beat-based and duration-based sensorimotor synchronization (SMS). The reason might ensue from the difficulty to create SMS tasks in an irregular context that can be compared to corresponding tasks in a regular context.Duration-based timing requires encoding and retaining the absolute duration of discrete intervals. Beat-based timing involves an internal pulsation emerging from the perceived underlying regularity (or beat) of a sequence. The beat is a psychological construct: listeners can extract a pulsation from an auditory signal without it being physically present (Honing, 2013; Sethares, 2007) and without effort (Grahn & Rowe, 2013). Beat-based timing uses this underlying pulsation as a reference (Teki et al., 2011). Rhythmic patterns based on small-integer ratios, such as the 2:1 pattern, induce internal pulsation more strongly than those based on more complex ratios (Parncutt, 1994).SMS refers to synchronizing actions with external events and implies anticipation (see Repp, 2005 for a review). Many studies have shown the benefits of metrical beat subdivisions (i.e., based on integer ratios) on synchronization accuracy (e.g., Patel, Iversen, Chen, & Repp, 2005; Repp, 2003; Wohlschlager & Koch, 2000). In SMS tasks, error correction responses occurred for phase perturbations of the stimuli introduced at the level of the subdivision, that is, not interfering with the expected intertap intervals (ITI), even for subdivisions not directly preceding the target tone (Repp, 2008). This suggests hierarchical metrical processing: all temporal information available contributes to creating an internal model of the stimulus. Thus, regular patterns of noninteger ratios are likely to engage error-correction mechanisms. Such patterns tend to be adjusted and produced as integer ratio patterns (Essens, 1986; Franek , Radil, & Indra, 1988; Sakai et al., 1999).The cortical and subcortical structures most cited for their implication in both timing perception and production are the cerebellum, the supplementary motor area (SMA), and the basal ganglia (e.g., Coull et al., 2011; Grahn & Brett, 2009; Ivry & Spencer, 2004; Matell & Meck, 2000; Thaut, 2003; Wiener et al., 2010; Wing, 2002). In a review, Coull and collaborators (2011) reported differential activations for perception versus production tasks. Both types of tasks, though more rarely perception tasks, resulted in activation of the cerebellum, especially of lobule VI. At the cortical level, perceptual timing tasks activated almost exclusively the pre-SMA. Motor timing tasks also activated the SMA proper. At the level of the basal ganglia, motor tasks involved mostly the putamen, whereas perceptual tasks also activated the caudate nucleus and the globus pallidus.The basal ganglia and the pre-SMA/SMA have been mainly associated with beat processing (Grahn, 2009; Grahn & Brett, 2007, 2009; Schwartze, Keller, Patel, & Kotz, 2011; Teki et al., 2011). These structures show strong anatomical connections (Johansen-Berg et al. …

Stratification of discharge in noble gases from the viewpoint of the discrete dynamics

Based on the analysis of electron phase trajectories in sinusoidal electric fields, a new point of view on discharge stratification is proposed. It is shown that the positive column can be considered as a spatial resonator in which electric fields with a fundamental period length LS or higher mode length qpLS establish, where p and q are integers and p &amp;gt; q. The fundamental mode length LS is equivalent to the distance on which electrons gain energy equal to the lowest excitation threshold. This distance determines a length of the S-striation. Unlike kinetic theory, in the presented model resonance properties of the discharge column are not connected with elastic collision energy losses. A point map is used to obtain the resonance trajectories of electrons in the phase plane. Stable points for the positions of inelastic collisions in the resonance trajectories have been found at the positions of field maxima in the case of integer ratios pq. For non-integer ratios pq, multiple resonance trajectories arise according to a more complex stability criterion. From this point of view, S-, P-, and R-striations in noble gas discharges can be explained. Due to energy losses in elastic collisions, initial electron energy distribution functions converge to the resonance trajectories (the so-called “bunch effect”). The findings of the discrete model agree with results of kinetic theory and experiment. The new approach avoids difficulties of the kinetic theory in the case of exceptionally large relaxation lengths which can even exceed the positive column length.

Serial binary interval ratios improve rhythm reproduction

Musical rhythm perception is a natural human ability that involves complex cognitive processes. Rhythm refers to the organization of events in time, and musical rhythms have an underlying hierarchical metrical structure. The metrical structure induces the feeling of a beat and the extent to which a rhythm induces the feeling of a beat is referred to as its metrical strength. Binary ratios are the most frequent interval ratio in musical rhythms. Rhythms with hierarchical binary ratios are better discriminated and reproduced than rhythms with hierarchical non-binary ratios. However, it remains unclear whether a superiority of serial binary over non-binary ratios in rhythm perception and reproduction exists. In addition, how different types of serial ratios influence the metrical strength of rhythms remains to be elucidated. The present study investigated serial binary vs. non-binary ratios in a reproduction task. Rhythms formed with exclusively binary (1:2:4:8), non-binary integer (1:3:5:6), and non-integer (1:2.3:5.3:6.4) ratios were examined within a constant meter. The results showed that the 1:2:4:8 rhythm type was more accurately reproduced than the 1:3:5:6 and 1:2.3:5.3:6.4 rhythm types, and the 1:2.3:5.3:6.4 rhythm type was more accurately reproduced than the 1:3:5:6 rhythm type. Further analyses showed that reproduction performance was better predicted by the distribution pattern of event occurrences within an inter-beat interval, than by the coincidence of events with beats, or the magnitude and complexity of interval ratios. Whereas rhythm theories and empirical data emphasize the role of the coincidence of events with beats in determining metrical strength and predicting rhythm performance, the present results suggest that rhythm processing may be better understood when the distribution pattern of event occurrences is taken into account. These results provide new insights into the mechanisms underlining musical rhythm perception.

TH-E-BRA-09: Radiation Induced Current Effects on MR Images from an Integrated Linac-MR System

Purpose: During the real-time MR image acquisition of an integrated linac/MRI system, the MRI's RF coils is exposed to pulsed radiation resulting in radiation induced currents (RIC). This work will: (a) visualize the RIC signal as artefacts in k-space and determine its effect on MRI's signal-to-noise ratio (SNR), (b) examine the effects of linac repetition rate (MU/minute) and MRI imaging sequence parameter ‘TR’ on the RIC artefact, (c) use post processing methods to remove the unwanted RIC signal from the MR images. Methods: A small test phantom was imaged with no radiation first. Phantom imaging was then repeated concurrently with linac producing radiation at various repetition rates in two scenarios: (1) the radiation beam was incident on the RF coil unobstructed and (2) a lead block attenuated the radiation beam before reaching the RF coil. Scenario (1) was repeated by obtaining images for several values of TR. Finally, a post-processing algorithm was applied to the corrupted MR k-space data to remove the RIC artefact. Results: The RIC artefact presented as near vertical lines in k-space data for integer ratios of TR to linac pulsing period (rate =180 Hz). For non-integer ratios, the artefact lost its regular pattern and became random in appearance. The RIC artefact disappeared from the k-space data when the linac's radiation was blocked. The image SNR decreased with increasing linac repetition rate. The post-processing method was successful in restoring a significant fraction of the lost image SNR. Conclusions: Signal spikes observed in the k-space data are confirmed to result from RIC. The SNR reduction in MRI images, due to the RIC, is directly related to the linac repetition rate. The artefact's appearance depends on the relationship between linac pulse repetition rate and image sequence timing. Our post-processing algorithm allows the recovery of the lost image SNR. The financial support is provided by the Canadian Institute of Health Research operating grant # MOP 93752.

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ‘slow’ and ‘fast’ motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.

Refinement of taking into account molecule sizes in the lattice gas model

An approach to consideration of non-integer ratios between linear molecule sizes having shapes close to linear chains, plates, and rectangular parallelepipeds was suggested for the lattice gas model. A statistical description of molecule orientation states retains spatial discretization as for molecules with multiple ratios between linear molecule sides, but the general balance of filled nodes and, which is most important, the influence of interparticle potentials reflects real molecule sizes through the nonideality function.

Testing Multiple Coordination Constraints with a Novel Bimanual Visuomotor Task

The acquisition of a new bimanual skill depends on several motor coordination constraints. To date, coordination constraints have often been tested relatively independently of one another, particularly with respect to isofrequency and multifrequency rhythms. Here, we used a new paradigm to test the interaction of multiple coordination constraints. Coordination constraints that were tested included temporal complexity, directionality, muscle grouping, and hand dominance. Twenty-two healthy young adults performed a bimanual dial rotation task that required left and right hand coordination to track a moving target on a computer monitor. Two groups were compared, either with or without four days of practice with augmented visual feedback. Four directional patterns were tested such that both hands moved either rightward (clockwise), leftward (counterclockwise), inward or outward relative to each other. Seven frequency ratios (3∶1, 2∶1, 3∶2, 1∶1, 2∶3. 1∶2, 1∶3) between the left and right hand were introduced. As expected, isofrequency patterns (1∶1) were performed more successfully than multifrequency patterns (non 1∶1). In addition, performance was more accurate when participants were required to move faster with the dominant right hand (1∶3, 1∶2 and 2∶3) than with the non-dominant left hand (3∶1, 2∶1, 3∶2). Interestingly, performance deteriorated as the relative angular velocity between the two hands increased, regardless of whether the required frequency ratio was an integer or non-integer. This contrasted with previous finger tapping research where the integer ratios generally led to less error than the non-integer ratios. We suggest that this is due to the different movement topologies that are required of each paradigm. Overall, we found that this visuomotor task was useful for testing the interaction of multiple coordination constraints as well as the release from these constraints with practice in the presence of augmented visual feedback.

DSP-Based Control of Multi-Rail DC-DC Converter Systems with Non-Integer Switching Frequency Ratios

This paper examines the use of non-integer switching frequency ratios in digitally controlled DC-DC converters. In particular the execution of multiple control algorithms using a Digital Signal Processor (DSP) for this application is analyzed. The variation in delay from when the Analog to Digital Converter (ADC) samples the output voltage to when the duty cycle is updated is identified as a critical factor to be considered when implementing the digital control system. Fixing the delay to its maximum value is found to produce reasonable performance using a conventional DSP. A modification of the DSP's interrupt control logic is proposed here that minimizes the delay and thereby yields improved performance compared with that given by a standard interrupt controller. Applying this technique to a multi-rail power supply system provides the designer with the flexibility to choose arbitrary switching frequencies for individual converters, thereby allowing optimization of the efficiency and performance of the individual converters.

DSP-Based Control of Multi-Rail DC-DC Converter Systems with Non-Integer Switching Frequency Ratios

This paper examines the use of non-integer switching frequency ratios in digitally controlled DC-DC converters. In particular the execution of multiple control algorithms using a Digital Signal Processor (DSP) for this application is analyzed. The variation in delay from when the Analog to Digital Converter (ADC) samples the output voltage to when the duty cycle is updated is identified as a critical factor to be considered when implementing the digital control system. Fixing the delay to its maximum value is found to produce reasonable performance using a conventional DSP. A modification of the DSP’s interrupt control logic is proposed here that minimizes the delay and thereby yields improved performance compared with that given by a standard interrupt controller. Applying this technique to a multi-rail power supply system provides the designer with the flexibility to choose arbitrary switching frequencies for individual converters, thereby allowing optimization of the efficiency and performance of the individual converters.

DSP-Based Control of Multi-Rail DC-DC Converter Systems with Non-Integer Switching Frequency Ratios

This paper examines the use of non-integer switching frequency ratios in digitally controlled DC-DC converters. In particular the execution of multiple control algorithms using a Digital Signal Processor (DSP) for this application is analyzed. The variation in delay from when the Analog to Digital Converter (ADC) samples the output voltage to when the duty cycle is updated is identified as a critical factor to be considered when implementing the digital control system. Fixing the delay to its maximum value is found to produce reasonable performance using a conventional DSP. A modification of the DSP’s interrupt control logic is proposed here that minimizes the delay and thereby yields improved performance compared with that given by a standard interrupt controller. Applying this technique to a multi-rail power supply system provides the designer with the flexibility to choose arbitrary switching frequencies for individual converters, thereby allowing optimization of the efficiency and performance of the individual converters.

Instability of Bose–Einstein condensates in tilted lattices with time-periodical modulation

We study the dynamical stability of Bose–Einstein condensates in an optical lattice with a time-periodic modulation potential and a constant acceleration force simultaneously. We derive the explicit expressions of quasienergies and obtain the stability diagrams of the ground state in parameter spaces. For integer and non-integer (rational) ratios of the acceleration force to the modulation frequency, different dependence relationships of the critical interaction strength on the modulation amplitude are observed. Our results are expected to help experimentalists to determine parameter regions as required.

Students' overuse of proportionality on missing-value problems: how numbers may change solutions

Previous research has shown that when confronted with missingvalue word problems, primary school students strongly tend to use proportional solution approaches, even if these approaches are inappropriate. We investigated whether (besides the missingvalue formulation of word problems) the numbers appearing in word problems are part of the superficial cues that lead students to (over)use proportionality. A test containing proportional and nonproportional word problems was given to 508 4th, 5th, and 6th graders. Numbers in these problems were experimentally manipulated so that the internal and external ratios were either integer or not. For proportional problems, performances decreased as a result of noninteger ratios. For nonproportional problems, noninteger ratios led to a decrease in the overuse of proportionality; only in the case of additive nonproportional problems was this decrease accompanied by an increase in correct answers. This effect decreased from 4th grade to 6th grade. Implications of this study for research and practice are discussed.