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The fate of natural populations is mediated by complex interactions among vital rates, which can vary within and among years. Although the effects of random, among-year variation in vital rates have been studied extensively, relatively little is known about how periodic, nonrandom variation in vital rates affects populations. This knowledge gap is potentially alarming as global environmental change is projected to alter common periodic variations, such as seasonality. We investigated the effects of changes in vital-rate periodicity on populations of three species representing different forms of adaptation to periodic environments: the yellow-bellied marmot (Marmota flaviventer), adapted to strong seasonality in snowfall; the meerkat (Suricata suricatta), adapted to inter-annual stochasticity as well as seasonal patterns in rainfall; and the dewy pine (Drosophyllum lusitanicum), adapted to fire regimes and periodic post-fire habitat succession. To assess how changes in periodicity affect population growth, we parameterized periodic matrix population models and projected population dynamics under different scenarios of perturbations in the strength of vital-rate periodicity. We assessed the effects of such perturbations on various metrics describing population dynamics, including the stochastic growth rate, log λS. Overall, perturbing the strength of periodicity had strong effects on population dynamics in all three study species. For the marmots, log λS decreased with increased seasonal differences in adult survival. For the meerkats, density dependence buffered the effects of perturbations of periodicity on log λS. Finally, dewy pines were negatively affected by changes in natural post-fire succession under stochastic or periodic fire regimes with fires occurring every 30 years, but were buffered by density dependence from such changes under presumed more frequent fires or large-scale disturbances. We show that changes in the strength of vital-rate periodicity can have diverse but strong effects on population dynamics across different life histories. Populations buffered from inter-annual vital-rate variation can be affected substantially by changes in environmentally driven vital-rate periodic patterns; however, the effects of such changes can be masked in analyses focusing on inter-annual variation. As most ecosystems are affected by periodic variations in the environment such as seasonality, assessing their contributions to population viability for future global-change research is crucial.

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.

Imposing incommensurable periodicity on the periodic atomic lattice can lead to complex structural phases consisting of locally periodic structure bounded by topological defects 1 , 2 , 3 , 4 , 5 , 6 , 7 – 8 . Twisted trilayer graphene (TTG) is an ideal material platform to study the interplay between different atomic periodicities, which can be tuned by twist angles between the layers, leading to moiré-of-moiré lattices 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 – 26 . Interlayer and intralayer interactions between two interfaces in TTG transform this moiré-of-moiré lattice into an intricate network of domain structures at small twist angles, which can harbour exotic electronic behaviours 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 – 26 . Here we report a complete structural phase diagram of TTG with atomic-scale lattice reconstruction. Using transmission electron microscopy (TEM) combined with a new interatomic potential simulation 27 , 28 , we show several large-scale moiré lattices, including triangular, kagome and a corner-shared hexagram-shaped domain pattern. Each domain is bounded by a 2D network of domain-wall lattices. In the limit of small twist angles, two competing structural orders—rhombohedral and Bernal stackings—with a slight energy difference cause unconventional lattice reconstruction with spontaneous symmetry breaking (SSB) and nematic instability, highlighting the importance of long-range interlayer interactions across entire van der Waals layers. The diverse tessellation of distinct domains, whose topological network can be tuned by the adjustment of the twist angles, establishes TTG as a platform for exploring the interplay between emerging quantum properties and controllable nontrivial lattices. Examination of a complete structural phase diagram of twisted trilayer graphene shows that several large-scale moiré domain lattices can be formed, the physical properties of which can be tuned by the twist angles between layers.

Interactions among electrons create novel many-body quantum phases of matter with wavefunctions that reflect electronic correlation effects, broken symmetries and collective excitations. Many quantum phases have been discovered in magic-angle twisted bilayer graphene (MATBG), including correlated insulating 1 , unconventional superconducting 2 – 5 and magnetic topological 6 – 9 phases. The lack of microscopic information 10 , 11 of possible broken symmetries has hampered our understanding of these phases 12 – 17 . Here we use high-resolution scanning tunnelling microscopy to study the wavefunctions of the correlated phases in MATBG. The squares of the wavefunctions of gapped phases, including those of the correlated insulating, pseudogap and superconducting phases, show distinct broken-symmetry patterns with a √3 × √3 super-periodicity on the graphene atomic lattice that has a complex spatial dependence on the moiré scale. We introduce a symmetry-based analysis using a set of complex-valued local order parameters, which show intricate textures that distinguish the various correlated phases. We compare the observed quantum textures of the correlated insulators at fillings of ±2 electrons per moiré unit cell to those expected for proposed theoretical ground states. In typical MATBG devices, these textures closely match those of the proposed incommensurate Kekulé spiral order 15 , whereas in ultralow-strain samples, our data have local symmetries like those of a time-reversal symmetric intervalley coherent phase 12 . Moreover, the superconducting state of MATBG shows strong signatures of intervalley coherence, only distinguishable from those of the insulator with our phase-sensitive measurements. High-resolution scanning tunnelling microscopy is used to observe the quantum textures of the many-body wavefunctions of the correlated insulating, pseudogap and superconducting phases in magic-angle graphene.

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.

NRC publication: Yes

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.

Magic-angle twisted trilayer graphene (MATTG) exhibits a range of strongly correlated electronic phases that spontaneously break its underlying symmetries 1 , 2 . Here we investigate the correlated phases of MATTG using scanning tunnelling microscopy and identify marked signatures of interaction-driven spatial symmetry breaking. In low-strain samples, over a filling range of about two to three electrons or holes per moiré unit cell, we observe atomic-scale reconstruction of the graphene lattice that accompanies a correlated gap in the tunnelling spectrum. This short-scale restructuring appears as a Kekulé supercell—implying spontaneous inter-valley coherence between electrons—and persists in a wide range of magnetic fields and temperatures that coincide with the development of the gap. Large-scale maps covering several moiré unit cells further reveal a slow evolution of the Kekulé pattern, indicating that atomic-scale reconstruction coexists with translation symmetry breaking at a much longer moiré scale. We use auto-correlation and Fourier analyses to extract the intrinsic periodicity of these phases and find that they are consistent with the theoretically proposed incommensurate Kekulé spiral order 3 , 4 . Moreover, we find that the wavelength characterizing moiré-scale modulations monotonically decreases with hole doping away from half-filling of the bands and depends weakly on the magnetic field. Our results provide essential insights into the nature of the correlated phases of MATTG in the presence of strain and indicate that superconductivity can emerge from an inter-valley coherent parent state. Scanning tunnelling microscopy imaging of the correlated phases of magic-angle twisted trilayer graphene shows marked signatures of interaction-driven spatial symmetry breaking.

Aims/hypothesisAgainst a background of a near-universally increasing incidence of childhood type 1 diabetes, recent reports from some countries suggest a slowing in this increase. Occasional reports also describe cyclical variations in incidence, with periodicities of between 4 and 6 years.MethodsAge/sex-standardised incidence rates for the 0- to 14-year-old age group are reported for 26 European centres (representing 22 countries) that have registered newly diagnosed individuals in geographically defined regions for up to 25 years during the period 1989–2013. Poisson regression was used to estimate rates of increase and test for cyclical patterns. Joinpoint regression software was used to fit segmented log-linear relationships to incidence trends.ResultsSignificant increases in incidence were noted in all but two small centres, with a maximum rate of increase of 6.6% per annum in a Polish centre. Several centres in high-incidence countries showed reducing rates of increase in more recent years. Despite this, a pooled analysis across all centres revealed a 3.4% (95% CI 2.8%, 3.9%) per annum increase in incidence rate, although there was some suggestion of a reduced rate of increase in the 2004–2008 period. Rates of increase were similar in boys and girls in the 0- to 4-year-old age group (3.7% and 3.7% per annum, respectively) and in the 5- to 9-year-old age group (3.4% and 3.7% per annum, respectively), but were higher in boys than girls in the 10- to 14-year-old age group (3.3% and 2.6% per annum, respectively). Significant 4 year periodicity was detected in four centres, with three centres showing that the most recent peak in fitted rates occurred in 2012.Conclusions/interpretationDespite reductions in the rate of increase in some high-risk countries, the pooled estimate across centres continues to show a 3.4% increase per annum in incidence rate, suggesting a doubling in incidence rate within approximately 20 years in Europe. Although four centres showed support for a cyclical pattern of incidence with a 4 year periodicity, no plausible explanation for this can be given.