Spatial Period Research Articles

Tracking control for a position-dependent periodic signal in a variable-speed rotational system

This paper concerns the problem of tracking a position-related periodic signal for an uncertain nonlinear rotational system. Applying the additive-state-decomposition (ASD) technique decomposes the original uncertain nonlinear system into two subsystems: A primary linear time-invariant system that performs spatial repetitive control, and a secondary nonlinear system that ensures robust stability of the whole system with respect to unknown nonlinear dynamics and exogenous disturbances. This method has four features: (i) The ASD technique provides an effective way to deal with the coupling between the spatial periodicity and other factors (temporal and state-dependent uncertainties and disturbances) in a rotational system. This technique ensures satisfactory tracking and disturbance rejection performance; (ii) A real-time digital implementation of a spatial repetitive controller enables repetitive control with an exact period in the position. This ensures the synchronous execution of repetitive control in the position domain and robust stability in the time domain; (iii) The method of accurate phase compensation in a spatial repetitive controller leads to a significant improvement in steady-state tracking performance; and (iv) An extended-state observer precisely estimates the overall effect of disturbances. Integrating a disturbance estimate into a control input effectively compensates for disturbances. Stability criteria and design algorithms are presented in detail. Finally, experiments of a rotational speed-control system demonstrate the effectiveness of this method.

Cryptic trans-lithospheric fault systems at the western margin of South America: implications for the formation and localization of gold-rich deposit superclusters

We present a review of frontier research advances in the investigation of cryptic structures that transect the South American Andes at oblique strike directions. The intersections between these cryptic structures and the superimposed Andean belt correlate with the spatial distribution of gold-rich mineral deposit clusters. The deposit clusters can be described as superclusters, as they comprise various gold deposit types that formed at multiple times throughout the Phanerozoic, impinging repeatedly on the structural intersections. However, the cryptic inherited fault structures are difficult to detect, because their deeper-seated roots are often overlain by younger supracrustal successions, and/or their exposed surface manifestations are structurally obscured by subsequent tectonic-magmatic activity. Thus, it also remains a challenge to constrain the nature and timing of formation, and the respective subsequent evolutionary path, of these proposed pre-Andean structures. Based on various case studies, we demonstrate that the localization of identified Phanerozoic gold deposit superclusters along the western South American margin is fundamentally controlled by structural inheritance often dating back to at least the Mesoproterozoic. Integration of multi-approach observations and datasets allows insights into a larger-scale tectonic history that showcases the successive inheritance of major structures originating from the Amazonian Craton, over the Paleozoic Gondwana margin, into the Cenozoic magmatic belts of the Andes, and even into recent fractures within the subducting oceanic Nazca plate, recording >1.2-billion-years of progressive structural inheritance and growth at one of the longest-lived tectonic margins in Earth history. In contrast to previous models of the spatial distribution of gold deposits, based on statistical approaches and spatial periodicity in self-organized systems focusing on single subduction and/or accretion episodes and belts, we propose that the structural inheritance and intersections are key to the localization of gold deposits in the Andes. In combination with bulk-geochemical data from magmatic rocks, we suggest that inherited structures maintained a trans-lithospheric connectivity to pre-fertilized gold enriched upper mantle reservoirs, which were tapped during multiple tectono-magmatic reactivation episodes.

Self-assembled gold nanoribbons via surface plasmon polaritons: The role of femtosecond laser

Plasmonic nanostructures have emerged as a critical component in broad fields ranging from optical modulation to biosensing and energy harvesting. Here, an ultrathin titanium (Ti) film (8 nm) is introduced between the gold (Au) film and fused silica (SiO2) substrate to facilitate energy deposition of the surface plasmon polariton (SPP) waves excited at the Ti/SiO2 interface under 1030 nm fs laser irradiation. The femtosecond laser-SPPs patterning results in the formation of Au nanoribbons with a width of about 270 nm and a spatial period of about 700 nm. Notably, centimeter-scale, self-assembled nanoribbons of high uniformity are achieved within several minutes. The localized surface plasmon resonance (LSPR) induced absorption peak is measured at the near-infrared band, confirming the high quality of the fabricated plasmonic nanostructures. Our findings suggest that the femtosecond laser-SPPs patterning technique represents a promising approach for the rapid and cost-effective production of large-scale plasmonic nanostructures.

Morphology of Neutrophils during Their Activation and NETosis: Atomic Force Microscopy Study.

Confocal microscopy and fluorescence staining of cellular structures are commonly used to study neutrophil activation and NETosis. However, they do not reveal the specific characteristics of the neutrophil membrane surface, its nanostructure, and morphology. The aim of this study was to reveal the topography and nanosurface characteristics of neutrophils during activation and NETosis using atomic force microscopy (AFM). We showed the main stages of neutrophil activation and NETosis, which include control cell spreading, cell fragment formation, fusion of nuclear segments, membrane disruption, release of neutrophil extracellular traps (NETs), and final cell disintegration. Changes in neutrophil membrane nanosurface parameters during activation and NETosis were quantified. It was shown that with increasing activation time there was a decrease in the spectral intensity of the spatial periods. Exposure to the activator A23187 resulted in an increase in the number and average size of cell fragments over time. Exposure to the activators A23187 and PMA (phorbol 12-myristate 13-acetate) caused the same pattern of cell transformation from spherical cells with segmented nuclei to disrupted cells with NET release. A23187 induced NETosis earlier than PMA, but PMA resulted in more cells with NETosis at the end of the specified time interval (180 min). In our study, we used AFM as the main research tool. Confocal laser-scanning microscopy (CLSM) images are provided for identification and detailed analysis of the phenomena studied. In this way, we exploited the advantages of both techniques.

Analysis of bending waves and parametric influence on band gaps in periodic track structure

Vibration of railway tracks generated due to rail-wheel interaction is a key research concern. A rail with equally spaced supports can be regarded as a periodic structure. Such a structure possesses unique property of band gap because of that they act as vibro-acoustic filters. Here, the propagation behavior of vertical and lateral bending waves is studied in a periodic ballastless track structure. Bloch-Floquet theorem is used to formulate the dispersion relations for the propagation of two types of waves which is validated by corresponding finite element model. The results show coexistence of both resonance and Bragg type band gaps they are owing to locally resonant units and spatial periodicity in the track. Furthermore, the study also explores the impact of track parameters, such as pad stiffness and sleeper spacing, on the properties of these band gaps.

Atomically precise vacancy-assembled quantum antidots.

Patterning antidots, which are regions of potential hills that repel electrons, into well-defined antidot lattices creates fascinating artificial periodic structures, leading to anomalous transport properties and exotic quantum phenomena in two-dimensional systems. Although nanolithography has brought conventional antidots from the semiclassical regime to the quantum regime, achieving precise control over the size of each antidot and its spatial period at the atomic scale has remained challenging. However, attaining such control opens the door to a new paradigm, enabling the creation of quantum antidots with discrete quantum hole states, which, in turn, offer a fertile platform to explore novel quantum phenomena and hot electron dynamics in previously inaccessible regimes. Here we report an atomically precise bottom-up fabrication of a series of atomic-scale quantum antidots through a thermal-induced assembly of a chalcogenide single vacancy in PtTe2. Such quantum antidots consist of highly ordered single-vacancy lattices, spaced by a single Te atom, reaching the ultimate downscaling limit of antidot lattices. Increasing the number of single vacancies in quantum antidots strengthens the cumulative repulsive potential and consequently enhances the collective interference of multiple-pocket scattered quasiparticles inside quantum antidots, creating multilevel quantum hole states with a tunable gap from the telecom to far-infrared regime. Moreover, precisely engineered quantum hole states of quantum antidots are geometry protected and thus survive on oxygen substitutional doping. Therefore, single-vacancy-assembled quantum antidots exhibit unprecedented robustness and property tunability, positioning them as highly promising candidates for advancing quantum information and photocatalysis technologies.

Daily stream temperature predictions for free-flowing streams in the Pacific Northwest, USA

Supporting sustainable lotic ecosystems and thermal habitats requires estimates of stream temperature that are high in scope and resolution across space and time. We combined and enhanced elements of existing stream temperature models to produce a new statistical model to address this need. Contrasting with previous models that estimated coarser metrics such as monthly or seasonal stream temperature or focused on individual watersheds, we modeled daily stream temperature across the entire calendar year for a broad geographic region. This model reflects mechanistic processes using publicly available climate and landscape covariates in a Generalized Additive Model framework. We allowed covariates to interact while accounting for nonlinear relationships between temporal and spatial covariates to better capture seasonal patterns. To represent variation in sensitivity to climate, we used a moving average of antecedent air temperatures over a variable duration linked to area-standardized streamflow. The moving average window size was longer for reaches having snow-dominated hydrology, especially at higher flows, whereas window size was relatively constant and low for reaches having rain-dominated hydrology. Our model’s ability to capture the temporally-variable impact of snowmelt improved its capacity to predict stream temperature across diverse geography for multiple years. We fit the model to stream temperatures from 1993–2013 and predicted daily stream temperatures for ~261,200 free-flowing stream reaches across the Pacific Northwest USA from 1990–2021. Our daily model fit well (RMSE = 1.76; MAE = 1.32°C). Cross-validation suggested that the model produced useful predictions at unsampled locations across diverse landscapes and climate conditions. These stream temperature predictions will be useful to natural resource practitioners for effective conservation planning in lotic ecosystems and for managing species such as Pacific salmon. Our approach is straightforward and can be adapted to new spatial regions, time periods, or scenarios such as the anticipated decline in snowmelt with climate change.

Deuterium–deuterium fusion in nanowire plasma driven with a nanosecond high-energy laser

Investigating the enhancement of the interaction between laser and plasma is crucial for fundamental and applied physics research studies based on laser-induced acceleration and nuclear reactions. The improvement of energy conversion efficiency resulting in increasing reaction yields has been extensively studied by the interaction of femtosecond (fs) or picosecond (ps) lasers with nanowire targets. However, the effects of nanosecond (ns) lasers interacting with nanowire targets on energy absorption and production yield remain unknown. To address this issue, we conducted a deuterium–deuterium fusion experiment based on the collision of two plasmas induced by the interaction of the kilo-Joule-level nanosecond laser with nanowire targets. The experimental results of neutron detection indicate that the yields of nanowire targets remain at the same level as those of planar targets. We have used the counter-streaming collisionless plasma model to perform a numerical analysis of the output of nuclear reaction products at the center-of-mass energy (Ec.m.) values between 10 and 30 keV, and the calculation results are in good agreement with the experimental results. In addition, a magneto-hydrodynamic numerical simulation was also performed. It shows that the critical density of the target’s surface, which forms on the picosecond time scale, blocks the absorption of laser energy with nanosecond pulse length. Consequently, both our experimental and simulation results indicate that the enhancement factor is limited when a target with a spatial period less than µm is used in conjunction with a ns laser. Therefore, additional research is highly desirable to develop a target structure that can improve the efficiency of energy conversion between the laser and the target.

Tuning the spin of Dirac fermions on the surface of topological insulators in proximity to a helical spin density wave

We investigate the spin tunability of Dirac fermions on the surface of a 3D topological insulator in proximity to a helical spin density wave, acting as an applied one-dimensional periodic potential for spins produced by spiral multiferroic oxide. It is observed that the spin mean values of Dirac fermion undergo oscillations under the influence of such a periodic potential created by the exchange field of magnetization. The tunability of spin is strongly affected by the strength, orientation and period of the exchange field. In particular, the mean values of spin are anisotropic around the Dirac point, depending strongly on the amplitude and spatial period of the periodic potential. We also find that the spin expectation values change significantly by changing the plane of magnetization. Interestingly, the in-plane components of spin mean values perform pronounced oscillations, whereas the out of plane component does not oscillate at all. The oscillations of planar components of spin are originated from the spin-momentum locking on the surface of topological insulator.

Fabrication of functional zirconia surfaces using a two‐beam interference setup employing a picosecond laser system with 532‐nm wavelength: Morphology, microstructure, and wettability

AbstractThe aim of this work was to evaluate the feasibility of the fabrication of microtextures in zirconia using the direct laser interference patterning (DLIP) technique. A green ultra‐short pulsed laser (532 nm, 10 ps) with a two‐beam interference setup was used to produce line‐like structures with a spatial period of 3 µm. For a fixed set of fluence and pulse‐to‐pulse overlap values (6.2 J/cm2, 81%), periodic structures were successfully created for different hatch distances. The average depth of the features ranged from 0.37 µm for a hatch distance of 14.4 µm up to 0.84 µm for a hatch distance of 12.4 µm. However, a further decrease in hatch distance did not result in an increase in depth since the region of the ridges is also ablated. Scanning electron microscopy analysis showed pores formation on the laser grooves, but no sign of micro‐cracking could be observed. Wettability tests showed an increase in hydrophobicity after DLIP. These results bring exciting perspectives on the fabrication of micro‐textures with DLIP on zirconia surface.

Emergence of tunable periodic density correlations in a Floquet–Bloch system

In quantum gases, two-body interactions are responsible for a variety of instabilities that depend on the characteristics of both trapping and interactions. These instabilities can lead to the appearance of new structures or patterns. We report on the Floquet engineering of such a parametric instability, on a Bose-Einstein condensate held in a time-modulated optical lattice. The modulation triggers a destabilization of the condensate into a state exhibiting a density modulation with a new spatial periodicity. This new crystal-like order, which shares characteristic correlation properties with a supersolid, directly depends on the modulation parameters: The interplay between the Floquet spectrum and interactions generates narrow and adjustable instability regions, leading to the growth, from quantum or thermal fluctuations, of modes with a density modulation noncommensurate with the lattice spacing. This study demonstrates the production of metastable exotic states of matter through Floquet engineering and paves the way for further studies of dissipation in the resulting phase and of similar phenomena in other geometries.

Plasma Distribution in a Column of a Low-Pressure Microwave Discharge Sustained by a Standing Surface Wave

The structure of a low-pressure microwave discharge sustained by a standing surface electromagnetic wave (SEW) in a quartz tube filled with argon was studied. The standing wave was formed using a set of two flat metal mirrors, which formed an open SEW resonator. The plasma density profile and structure of the electromagnetic field of the SEW were studied in the pressure range from 0.25 to 10 Torr. The excitation of the standing wave allowed us to independently study the longitudinal Ez and transverse Er components of the SEW electric field vector. It was confirmed experimentally that the oscillation phases of the components of the SEW are shifted by π. The excitation of the standing wave in the plasma column leads to the formation of local minimums and maximums of plasma density, whose period equals half the wavelength of the surface wave. At the same time, the spatial period of density modulation is close to the distribution of the Ez component of the standing SEW. It was shown that the formation time of the modulated structure of plasma density is close to the characteristic time of diffusion, while the degree of modulation increases with increasing pressure. It was shown experimentally that it is possible to produce a plasma column with plasma density modulation nemax/nemin ≈ 5 and a length of about 10 wavelengths.

Ekadasa Rudra Ceremony in 1963 and 1979: Power Relation Issue in Bali

This article discusses the enactment of Ekadasa Rudra ceremony in 1963 and 1979 in Pura Agung Besakih, Karangasem, Bali. As the greatest Balinese Hindu ceremony held once in a century, the events were substantial and nationally widespread. The noted events in Indonesian history were in 1963 and 1979, it means that the spatial period was less than twenty years. That problem is an interesting topic to discuss. The discussion uses theory of power relation by Foucault and principle in historical research by Kuntowijoyo, these are determining topic, heuristics, source criticism, interpretation, and historiography. The analysis conducts from explaining each moment of Ekadasa Rudra in both years, then the analysis goes on into the discussion of applying theory of power relation to answer the research question.

Rapid Vegetation Growth due to Shifts in Climate from Slow to Sustained Warming over Terrestrial Ecosystems in China from 1980 to 2018

The fraction of absorbed photosynthetically active radiation (FPAR) is a key biophysiological parameter of terrestrial ecosystems. However, due to a lack of data with adequate spatial resolution and in long enough time series, there have been limitations in exploring the spatiotemporal changes of vegetation and response to climate change. In this study, a 1 km spatial resolution and 8-day period length dataset (FPARANN) was developed covering the years 1980 to 2018 and evaluated on spatiotemporal change consistency by validating with Gross Primary Production (GPP) observations from the Chinese Flux Observation and Research Network (ChinaFLUX), and comparison with other FPAR products. FPARANN provided a comparable performance in capturing seasonal change observed through GPP, according to the coefficient of determination (R2): 0.50, 0.51, 0.70 and 0.74 averaged for all sites, forest sites, grassland sites and cropland flux sites, respectively. The new data had more spatial similarity to the MODIS FPAR product (FPARMCD15A2) with a greater R2 (0.77) and a lower RMSE (0.12) than other products. With a newly developed dataset, combined with FPARANN (1980–2003) and FPARMCD15A2 (2004–2018), an overall increasing trend in FPAR was found for over 81% of the vegetated area of China from 1980 to 2018. FPAR increased more rapidly for over 83.7% of the area in the 2010s, and at a slower pace for over 62.1% of the area in the early 2000s, which was attributed to a decadal shifting of climate change. This study implies the new dataset is useful in quantifying vegetation changes and would be an important data source for future study of the carbon cycle, soil erosion, or evapotranspiration, with great application potential.

Scale invariance and critical balance in electrostatic drift-kinetic turbulence

The equations of electrostatic drift kinetics are observed to possess a symmetry associated with their intrinsic scale invariance. Under the assumptions of spatial periodicity, stationarity, and locality, this symmetry implies a particular scaling of the turbulent heat flux with the system's parallel size, from which its scaling with the equilibrium temperature gradient can be deduced under some additional assumptions. This macroscopic transport prediction is then confirmed numerically for a reduced model of electron-temperature-gradient-driven turbulence in slab geometry. The system realises this scaling through a turbulent cascade from large to small perpendicular spatial scales. The route of this cascade through wavenumber space (i.e. the relationship between parallel and perpendicular scales in the inertial range) is shown to be determined by a balance between nonlinear-decorrelation and parallel-dissipation timescales. This type of ‘critically balanced’ cascade, which maintains a constant energy flux despite the presence of parallel dissipation throughout the inertial range (as well as order-unity dissipative losses at the outer scale) is expected to be a generic feature of plasma turbulence. The outer scale of the turbulence, on which the turbulent heat flux depends, is determined by the breaking of drift-kinetic scale invariance due to the existence of large-scale parallel inhomogeneity (the parallel system size).

Self-Similarity and Spatial Periodicity in Cerebral Cortical Patterning: Structural Design Notes for Neural Tissue Architects

Tissue engineering is a powerful tool with which to systematically identify the determinants of biological functions. Applied to the design and fabrication of biomimetic brains, tissue engineering serves to disentangle the complex anatomy of neural circuits and pathways by recapitulating structure-function relationships in simplified model systems. The complex neuroanatomy of the cerebral cortex, with its enigmatic columnar and stratified cytoarchitectonic organization, represents a major challenge toward isolating the minimal set of elements that are required to assemble neural tissues with cognitive functions. Whereas considerable efforts have highlighted important genetic and physical correlates of early cortical tissue patterning, no substantive attempt to identify the determinants of how the cortices acquire their relatively conserved, narrow range of numbered layers is evident in the literature. Similarly, it is not yet clear whether cortical columns and laminae are functionally relevant or epiphenomena of embryonic neurodevelopment. Here, we demonstrate that spatial frequencies (m−1) derived from the width-to-height ratios of cerebral cortical columns predict sinusoids with a narrow range of spatial cycles over the average cortical thickness. The resulting periodicities, denoted by theoretical wavenumbers, reflect the number of observed cortical layers among humans and across several other species as revealed by a comparative anatomy approach. We present a hypothesis that cortical columns and their periodic layers are emergent of the intrinsic spatial dimensions of neurons and their nested, self-similar aggregate structures including minicolumns. Finally, we discuss the implications of periodic tissue patterns in the context of neural tissue engineering.

An Optical-Fiber-Based Key for Remote Authentication of Users and Optical Fiber Lines.

We have shown the opportunity to use the unique inhomogeneities of the internal structure of an optical fiber waveguide for remote authentication of users or an optic fiber line. Optical time domain reflectometry (OTDR) is demonstrated to be applicable to observing unclonable backscattered signal patterns at distances of tens of kilometers. The physical nature of the detected patterns was explained, and their characteristic spatial periods were investigated. The patterns are due to the refractive index fluctuations of a standard telecommunication fiber. We have experimentally verified that the patterns are an example of a physically unclonable function (PUF). The uniqueness and reproducibility of the patterns have been demonstrated and an outline of authentication protocol has been proposed.

Vortex-induced vibration of an elastically mounted cylinder subject to planar inflow shear

We have numerically investigated the effect of planar inflow shear on the vortex-induced vibration of a three-dimensional circular cylinder mounted on elastic supports in streamwise and cross-stream directions. At a Reynolds number Re=300, the reduced velocity (U∗) is varied in the range U∗=2−12 for a fixed mass ratio m∗=2. The effect of inflow shear is brought about by varying the shear parameter βy=0,0.05,0.1. We have observed four distinct branches in the structural response curve: Initial Branch (IB), High Amplitude Branch-1 (HAB1), High Amplitude Branch-2 (HAB2), and Desynchronization Branch (DS). The response of the cylinder is locked in with the near wake during IB, HAB1, and HAB2, where the inline oscillation amplitude of the cylinder triggers for increasing βy. The cylinder oscillates with a tilt against the high-velocity side at larger βy for HAB1 and HAB2 responses. During IB response at βy=0, the near wake shows the formation of vortex filaments clustered through the spanwise extent. However, in the HAB1 and HAB2 regimes, the propagation of secondary instability yields random occurrences of vortex splits in the wake. Similar consequences of vortex splits are found at βy=0.1 for IB, HAB1, and HAB2 regimes, where we observe fragmented vortex filaments wrapping around the vortex rings. For βy=0, we have found mode B-type wake instability in the IB response at a plane x=2, where the vortex structures form a staggered array with a spatial periodicity λ≈D. At z=0 plane, the vortex shedding mode is “2S” for the IB response at βy=0.05, while the “P+S” mode is at βy=0.1. The vortex strands experience stretching in the DS branch, where no specific shedding mode is discerned. The cross-stream evolution of time-averaged streamwise velocity shows a data collapse with an asymmetric localization above the cylinder for increasing βy. The streamwise and spanwise velocity fluctuations at βy=0.1 are insignificant compared to the cross-stream. However, for the IB and DS regimes, the growth of the rms velocities reveals spanwise invariance. In contrast, we found small-scale fluctuations of cross-stream velocities about the mean in the HAB1 regime. The temporal growth of the sectional drag coefficients (CDs) along the span shows a negative value in the IB regime for βy=0,0.5. However, we observe the quasiperiodic behavior of CDs along the span for the HAB1 regime at βy=0. The spanwise growth of CDs for the DS regimes shows intermittent transitions via the positive, zero, and negative values for different βy. For both IB and DS regimes at βy=0, the time evolution of sectional lift coefficients (CLs) is asymmetric along the span and represents a helical pattern. However, for increasing βy, the growth of CLs for HAB1 and HAB2 regimes is intermittent. An increase in βy triggers the magnitudes of CLs in the DS branch, where the spanwise evolution shows symmetric periodic bands of sectional lift coefficients.

Multi-Scale Structuring of CoCrMo and AZ91D Magnesium Alloys Using Direct Laser Interference Patterning

In this work, the technique of Direct Laser Interference Patterning (DLIP) was used to fabricate micrometric structures at the surface of Cobalt-Chromium-Molybdenum and AZ91D magnesium alloys. Line-like patterns with spatial periods of 5 μm were textured using an ultra-short pulsed laser (10 ps pulse duration and 1064 nm wavelength) with a two-beam interference setup. The surface topography, morphology, and chemical modifications were analysed using Confocal Microscopy, Scanning Electron Microscopy, and Energy Dispersive Spectroscopy (EDS), respectively. Laser fluence and pulse overlap were varied to evaluate their influence on the final structure. Homogeneous structures were achieved for the CoCrMo alloy for every condition tested, with deeper structures (up to 0.85 μm) being achieved for higher energy levels (higher overlap and/or fluence). For high energy, sub-micrometric secondary structures, so-called LIPSS, could also be observed on the CoCrMo. The EDS analysis showed some oxidation after the laser texturing. Regarding the AZ91D alloy, deeper structures could be achieved (up to 2.5 μm), but more melting and oxidation was observed, forming spherical oxide particles. Nonetheless, these results bring new perspectives on the fabrication of microtextures on the surface of CoCrMo and AZ91D using DLIP.

Condensation phases in a non-Hermitian optical lattices system

We explore the condensate phase of single-species bosons in an optical lattice through the use of dissipative Bose–Hubbard models. The ground state of system possesses a real-energy, which corresponds to non-Hermitian Hamiltonians with non-reciprocal hopping phase factors. Through the application of inhomogeneous dynamical Gutzwiller mean-field theory (DGMFT), we predict a fascinating phenomenon: the ground state displays a periodic arrangement of superfluid islands amidst a Mott insulator sea. Additionally, the number of superfluid islands is proportional to the spatial period squared.