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Rhythmic structure enables precise temporal expectations that are essential to human communication, including speech and music. Computational models have been developed to account for how humans perceive, produce, and learn rhythmic sequences. However, it is unclear how different types of models relate to each other and how they can be evaluated. In this review and perspective, we discuss how three major classes of models-entrainment, probabilistic, and timekeeper models-have been used to study rhythmic expectations. We critically assess each model class in terms of its level of explanation, the rhythmic behaviors it captures, its ability to account for learning and enculturation, and its ability to integrate other features, such as pitch. We show that entrainment, probabilistic, and timekeeper models differ substantially in the aspects of rhythmic expectations they can capture. To move the field forward, we propose that model comparison and integration are crucial. We identify key challenges to this effort, such as the varying nature of the input and output signals and divergent modeling goals. To address these challenges, we arrive at several practical recommendations: to equate input and output signals when comparing models, to consider several model outcomes beyond goodness-of-fit measures in model evaluation, to use model-integration efforts to inform theory building, and to make code and data openly accessible. Ultimately, understanding how models of rhythmic expectations relate, and how features in these models account for behavioral, neural, and cognitive aspects of rhythmic expectations, will deepen our understanding of a core aspect of human behavior.

As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c -almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every c -almost periodic function in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) is also c m -almost periodic in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) for each non-zero integer number m . This study is performed for both representative cases of functions defined on the real axis and with values in a Banach space and the complex functions defined on vertical strips in the complex plane.

The event rate, energy distribution and time-domain behaviour of repeating fast radio bursts (FRBs) contain essential information regarding their physical nature and central engine, which are as yet unknown 1 , 2 . As the first precisely localized source, FRB 121102 (refs. 3 – 5 ) has been extensively observed and shows non-Poisson clustering of bursts over time and a power-law energy distribution 6 – 8 . However, the extent of the energy distribution towards the fainter end was not known. Here we report the detection of 1,652 independent bursts with a peak burst rate of 122 h −1 , in 59.5 hours spanning 47 days. A peak in the isotropic equivalent energy distribution is found to be approximately 4.8 × 10 37 erg at 1.25 GHz, below which the detection of bursts is suppressed. The burst energy distribution is bimodal, and well characterized by a combination of a log-normal function and a generalized Cauchy function. The large number of bursts in hour-long spans allows sensitive periodicity searches between 1 ms and 1,000 s. The non-detection of any periodicity or quasi-periodicity poses challenges for models involving a single rotating compact object. The high burst rate also implies that FRBs must be generated with a high radiative efficiency, disfavouring emission mechanisms with large energy requirements or contrived triggering conditions. For FRB 121102, 1,652 burst events are detected over 47 days, with a peak burst rate of 122 per hour, a bimodal burst rate energy distribution, and no periodicity or quasi-periodicity.

High magnetic fields suppress cuprate superconductivity to reveal an unusual density wave (DW) state coexisting with unexplained quantum oscillations. Although routinely labeled a charge density wave (CDW), this DW state could actually be an electron-pair density wave (PDW). To search for evidence of a field-induced PDW, we visualized modulations in the density of electronic states N(r) within the halo surrounding Bi₂Sr₂CaCu₂O₈ vortex cores. We detected numerous phenomena predicted for a field-induced PDW, including two sets of particle-hole symmetric N(r) modulations with wave vectors QP and 2QP , with the latter decaying twice as rapidly from the core as the former. These data imply that the primary field-induced state in underdoped superconducting cuprates is a PDW, with approximately eight CuO₂ unit-cell periodicity and coexisting with its secondary CDWs.

Landslide movements typically show a series of progressively shorter quiescent phases, punctuated by sudden bursts during an acceleration crisis. We propose that such intermittent rupture phenomena can be described by a log‐periodic power law singularity model. Amounting mathematically to a generalization of the power law exponent from real to complex numbers, this model captures the partial break of continuous scale invariance to discrete scale invariance that is inherent to the intermittent dynamics of damage and rupture processes in heterogeneous geomaterials. By performing parametric and nonparametric tests on a large data set of 52 landslides, we present empirical evidence and theoretical arguments demonstrating the statistical significance of log‐periodic oscillations decorating power law finite‐time singularities during landslide crises. Log‐periodic landslide motions may stem from the interaction between frictional stress drop along geological structures and stress corrosion damage in rock bridges, as well as the interplay of inertia, damage, and healing.

Supersolids are exotic materials combining the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid. The supersolid phase of matter was predicted 50 years ago 1 – 3 for solid helium 4 – 8 . Ultracold quantum gases have recently been made to exhibit periodic order typical of a crystal, owing to various types of controllable interaction 9 – 13 . A crucial feature of a D -dimensional supersolid is the occurrence of D  + 1 gapless excitations, reflecting the Goldstone modes associated with the spontaneous breaking of two continuous symmetries: the breaking of phase invariance, corresponding to the locking of the phase of the atomic wave functions at the origin of superfluid phenomena, and the breaking of translational invariance due to the lattice structure of the system. Such modes have been the object of intense theoretical investigations 1 , 14 – 18 , but they have not yet been observed experimentally. Here we demonstrate supersolid symmetry breaking through the appearance of two distinct compressional oscillation modes in a harmonically trapped dipolar Bose–Einstein condensate, reflecting the gapless Goldstone excitations of the homogeneous system. We observe that the higher-frequency mode is associated with an oscillation of the periodicity of the emergent lattice and the lower-frequency mode characterizes the superfluid oscillations. This work also suggests the presence of two separate quantum phase transitions between the superfluid, supersolid and solid-like configurations. Two distinct compressional oscillation modes are characterized in a dipolar Bose–Einstein condensate; this demonstrates the breaking of two symmetries owing to the coexisting superfluid and crystal properties of a supersolid.

NRC publication: Yes

Atomically thin layers of two-dimensional materials can be assembled in vertical stacks that are held together by relatively weak van der Waals forces, enabling coupling between monolayer crystals with incommensurate lattices and arbitrary mutual rotation 1 , 2 . Consequently, an overarching periodicity emerges in the local atomic registry of the constituent crystal structures, which is known as a moiré superlattice 3 . In graphene/hexagonal boron nitride structures 4 , the presence of a moiré superlattice can lead to the observation of electronic minibands 5 – 7 , whereas in twisted graphene bilayers its effects are enhanced by interlayer resonant conditions, resulting in a superconductor–insulator transition at magic twist angles 8 . Here, using semiconducting heterostructures assembled from incommensurate molybdenum diselenide (MoSe 2 ) and tungsten disulfide (WS 2 ) monolayers, we demonstrate that excitonic bands can hybridize, resulting in a resonant enhancement of moiré superlattice effects. MoSe 2 and WS 2 were chosen for the near-degeneracy of their conduction-band edges, in order to promote the hybridization of intra- and interlayer excitons. Hybridization manifests through a pronounced exciton energy shift as a periodic function of the interlayer rotation angle, which occurs as hybridized excitons are formed by holes that reside in MoSe 2 binding to a twist-dependent superposition of electron states in the adjacent monolayers. For heterostructures in which the monolayer pairs are nearly aligned, resonant mixing of the electron states leads to pronounced effects of the geometrical moiré pattern of the heterostructure on the dispersion and optical spectra of the hybridized excitons. Our findings underpin strategies for band-structure engineering in semiconductor devices based on van der Waals heterostructures 9 . Excitonic bands in MoSe 2 /WS 2 heterostructures can hybridize, resulting in a resonant enhancement of moiré superlattice effects.

Electrophysiological signals exhibit both periodic and aperiodic properties. Periodic oscillations have been linked to numerous physiological, cognitive, behavioral and disease states. Emerging evidence demonstrates that the aperiodic component has putative physiological interpretations and that it dynamically changes with age, task demands and cognitive states. Electrophysiological neural activity is typically analyzed using canonically defined frequency bands, without consideration of the aperiodic (1/f-like) component. We show that standard analytic approaches can conflate periodic parameters (center frequency, power, bandwidth) with aperiodic ones (offset, exponent), compromising physiological interpretations. To overcome these limitations, we introduce an algorithm to parameterize neural power spectra as a combination of an aperiodic component and putative periodic oscillatory peaks. This algorithm requires no a priori specification of frequency bands. We validate this algorithm on simulated data, and demonstrate how it can be used in applications ranging from analyzing age-related changes in working memory to large-scale data exploration and analysis.A method for parameterizing electrophysiological neural power spectra into periodic and aperiodic components is introduced, addressing limitations of common approaches. The method is validated in simulation and demonstrated on real data applications.

Fast radio bursts (FRBs) are extragalactic astrophysical transients 1 whose brightness requires emitters that are highly energetic yet compact enough to produce the short, millisecond-duration bursts. FRBs have thus far been detected at frequencies from 8 gigahertz (ref. 2 ) down to 300 megahertz (ref. 3 ), but lower-frequency emission has remained elusive. Some FRBs repeat 4 – 6 , and one of the most frequently detected, FRB 20180916B 7 , has a periodicity cycle of 16.35 days (ref. 8 ). Using simultaneous radio data spanning a wide range of wavelengths (a factor of more than 10), here we show that FRB 20180916B emits down to 120 megahertz, and that its activity window is frequency dependent (that is, chromatic). The window is both narrower and earlier at higher frequencies. Binary wind interaction models predict a wider window at higher frequencies, the opposite of our observations. Our full-cycle coverage shows that the 16.3-day periodicity is not aliased. We establish that low-frequency FRB emission can escape the local medium. For bursts of the same fluence, FRB 20180916B is more active below 200 megahertz than at 1.4 gigahertz. Combining our results with previous upper limits on the all-sky FRB rate at 150 megahertz, we find there are 3–450 FRBs in the sky per day above 50 Jy ms. Our chromatic results strongly disfavour scenarios in which absorption from strong stellar winds causes FRB periodicity. We demonstrate that some FRBs are found in ‘clean’ environments that do not absorb or scatter low-frequency radiation. The fast radio burst FRB 20180916B repeats with a periodicity of 16 days, and is now found to emit down to a frequency of 120 MHz, much lower than previously observed.