Relative Dispersion Research Articles

Numerical and experimental analysis of Lagrangian dispersion in two-dimensional chaotic flows.

We present a review and a new assessment of the Lagrangian dispersion properties of a 2D model of chaotic advection and diffusion in a regular lattice of non stationary kinematic eddies. This model represents an ideal case for which it is possible to analyze the same system from three different perspectives: theory, modelling and experiments. At this regard, we examine absolute and relative Lagrangian dispersion for a kinematic flow, a hydrodynamic model (Delft3D), and a laboratory experiment, in terms of established dynamical system techniques, such as the measure of (Lagrangian) finite-scale Lyapunov exponents (FSLE). The new main results concern: (i) an experimental verification of the scale-dependent dispersion properties of the chaotic advection and diffusion model here considered; (ii) a qualitative and quantitative assessment of the hydro-dynamical Lagrangian simulations. The latter, even though obtained for an idealized open flow configuration, contributes to the overall validation of the computational features of the Delft3D model.

Key factors affecting the decision-making process for buildings projects in Egypt

The success of optimal system selection depends on many factors, such as the nature of the project itself, design team collaboration, team leader's personality, and client input. Unfortunately, despite numerous existing studies on key impacting factors in various fields of construction design, the majority of them failed to focus on the key factors affecting the structure design decision-making process. This study aims to identify, investigate, and rank factors perceived to affect decision-making for the optimal system in the construction industry, particularly that of buildings, during the early stages of the entire life cycle, in terms of their relative importance. Practitioners and experts from Egypt comprising a statistically representative sample, were invited to participate in a structured questionnaire survey to achieve this objective. The questionnaire was composed of 17 factors classified under three primary stage categories: feasibility study, planning, and design and engineering. The relative importance index and dispersion analysis were used to obtain statistics and calculate the collected data. The results show that “The impact of financial difficulties on the decision-making” is the first key factor, “The impact of errors and omissions due to the lack of communication and coordination on the decision-making” is the second key factor, and “The Impact of inaccurate evaluation of projects duration on the decision-making” and “The impact of design changes upon contractor request on the decision-making?” constitute the third key factors. Therefore, this work can guide decision-makers for effective construction management and help achieve a competitive quality and cost-effective project.

Assessing the Old-Growth Dependency of Two Saproxylic Beetle Species in the Southern Appalachian Mountains

AbstractThe southern Appalachian Mountains were intensively logged during the early 1900s, leaving little remaining old-growth forest. Much of the region is now second-growth forest, which may not be suitable to specialist saproxylic species. Moreover, if suitable habitat exists, poorly dispersing species may not be able to colonize it. To investigate this, we assessed the distribution and old-growth dependency of two low-mobility saproxylic beetles in Great Smoky Mountains National Park. Using both field surveys and community science data, we found Megalodacne heros (Say 1823) (Coleoptera: Erotylidae) to be limited to lower elevations regardless of disturbance history, while Phellopsis obcordata (Kirby 1837) (Coleoptera: Zopheridae) was restricted to inside or near old-growth forests. Although trees were generally smaller in second-growth, we detected no habitat limitation for P. obcordata: fungal hosts were present in second-growth areas and the beetle was present on trees as small as 11 cm in diameter. This suggests its distribution is shaped by its low dispersal capability and need for temporal continuity of deadwood habitat, therefore qualifying it as an indicator species. For P. obcordata, old-growth acted as refugia during landscape-wide, anthropogenic disturbances in the early 1900s, though we can draw no conclusions about M. heros from our dataset. The difference in sensitivity to human disturbance displayed between species may be linked to their relative dispersal abilities: P. obcordata is entirely flightless while M. heros is capable of some flight. This study highlights the value of using saproxylic invertebrates with limited dispersal ability for assessing impacts from anthropogenic forest disturbances.

Highlights from 25 Years of the GeoPT Programme: What Can be Learnt for the Advancement of Geoanalysis

Data submitted over the past 25 years to GeoPT, the highly successful proficiency‐testing programme for the geochemical analysis of geological materials, organised by the International Association of Geoanalysts, provide a valuable resource that permits detailed investigation of contrasting results associated with different sample preparation and measurement principles. Highlighted issues include the following: recurring problems with the dissolution of the refractory minerals zircon and chromite, which produce a large dispersion in data obtained when acid digestion is involved; issues related to different XRF sample preparation methods, whereby a significant divergence of pressed powder pellet results compared with those from fused glass discs is observed; high relative dispersion of data both at low mass fractions, and those higher than normally found in silicate rocks, due to incomplete method validation, in particular due to an overconfidence in estimating reporting limits and to the employment of limited working ranges. In addition, an example for Sr in an ancient pegmatite with extremely high Rb abundance is presented, where ICP‐MS results amount to only to a third of the XRF results, severely underestimated due to the disregard of the radiogenic ingrowth from 87Rb decay. Recommendations are made both for improving data quality and the selection of test materials for future GeoPT rounds.

Resource Scheduling Method for Optimizing the Distribution Path of Fresh Agricultural Products under Low-Carbon Environmental Constraints

Aiming at the development concept of relative dispersion of urban multiregional business centers, saving energy, and emission reduction, this article proposes an algorithm for solving the problem of green logistics distribution to fresh agricultural products considering low-carbon and environmental protection, to realize the economic and environmental protection of cold chain logistics. Firstly, we comprehensively consider the fixed costs of delivery vehicles, transportation costs, damage costs of fresh agricultural products, cooling costs, carbon emissions costs, and penalty costs due to service time windows that do not meet customer requirements as the objective function to construct green logistics distribution location-routing optimization model. Then, Tent chaotic perturbation method is introduced to optimize the genetic algorithm and an improved genetic algorithm is used to solve the distribution location-routing optimization model to obtain the best scheme. Finally, the proposed algorithm is verified experimentally based on the MATLAB simulation platform. Experimental results show that the total distance and total cost of the distribution plan obtained by the proposed algorithm are 128.96 km and 12,593 yuan, respectively, which are superior to other comparison algorithms and can be used as a reference for logistics enterprises’ distribution decision-making.

Hybrid-Domain Evaluation PTS with Adaptive Selection Methods for PAPR Reduction

The partial transmit sequence (PTS) technique is a fairly suitable scheme to mitigate the high peak-to-average power ratio (PAPR) problem inherent in 5G multicarrier systems, especially considering a high-order QAM modulation design. However, the high computational complexity level and the speed of the convergence for optimizing the phases of the transmitting signal restrict this technique in practical applications. In this paper, a low-complexity frequency-domain-evaluated PTS (F-PTS) based on a spacing multiobjective (SMO) processing algorithm is proposed to reduce the PAPR values. The PAPR performance are accurately predicted in terms of modifying relative dispersion in the frequency domain. As a result, the complexity of searching the optimal phase factors and IFFT computing is simplified. Moreover, a frequency-domain- and time-domain-evaluating PTS (FTD-PTS) is employed to search the optimal solution with a reasonable complexity. Simulation results verify that the operation rate of F-PTS is significantly improved after transferring the exhaustive search strategy of PTS into the SMO algorithm, and the F-PTS PAPR reduction performance is just 0.3 dB away from theoretical optimal performance. The FTD-PTS spends an acceptable operation rate to obtain optimal PAPR reduction performance, which subtracts 0.5 and 0.6 dB more than PSO-PTS and conventional PTS at CCDF=10−3, respectively.

Landau parameters and entrainment matrix of cold stellar matter: effect of the symmetry energy and strong magnetic fields

Nuclear matter properties based on a relativistic approach suitable for the description of multi-component systems are calculated. We use a set of nuclear relativistic mean-field models that satisfy acceptable nuclear matter properties and neutron star observations. The effects of the density dependence of the symmetry energy and of the Landau quantization due to the presence of a strong external magnetic field are discussed. Properties such as the proton fraction, the Landau mass, Landau parameters and entrainment matrix, the adiabatic index and speed of sound are calculated for cold β-equilibrium matter. A large dispersion on the calculated properties is obtained at two to three times saturation density ρ0 . The proton Landau mass can be as low as one third of the vacuum nucleon mass at 2-3 ρ0 . Similar effects are obtained for the Landau parameters, in particular, the ones involving protons, where the relative dispersion of F0 pp and F1 pp is as high as 30% to 50% at 2-3 ρ0 . These parameters are particularly sensitive to the symmetry energy. The effect of the magnetic field on the nuclear properties is small for fields as high as 1018G except for a small range of densities just above the crust-core transition. Tables with the EoS, and the parameters, are provided in the supplementary data section.

Статистическая модель адаптивной системы компенсации шумовых помех в радиолокаторе при обнаружении гидрометеорных образований в атмосфере

A simulation model of an adaptive combined spatial-polarization system for detecting hydrometeorolog-ical formations in the atmosphere are created. The creation of this model is based on statistical modeling of both the electromagnetic field created by a useful signal and interference vibrations and the configu-rable parameters of the system under study. The laws of distribution of input random processes are de-scribed by Gaussian statistics. The adaptation of the weighting coefficients to the interference situation is carried out using a gradient algorithm. The dependences of the relative dispersion of the process at the output of the system under study on the number of the adaptation step are constructed, which indicate high efficiency of detecting the reflected signal from hydrometeors under the influence of noise interfer-ence arbitrarily oriented in space.

The role of metallic and acid sites of Ru-Nb-Si catalysts in the transformation of levulinic acid to γ-valerolactone

The transformation of γ-valerolactone (GVL) into levulinic acid (LA) requires the use of bifunctional catalysts allowing the hydrogenation and lactonization steps to occur. The aim of this study is to clarify the role played by both redox (Ru) and acid (Nb) sites of the ruthenium-niobium-silicon mixed oxides catalysts, synthesized by sol-gel route, when this reaction is performed in mild conditions. The catalytic performances of the materials are related to an in-depth morphologic and structural characterization, demonstrating that the interaction between the two metals and their relative amount and dispersion are crucial in determining their activity. The results confirm the hydrogenation-dehydration consecutive reaction network and evidence the possibility to selectively produce either GVL or the intermediate γ-hydroxypentanoic acid by an accurate choice of the catalyst composition.

Plasma oscillations and spectral index in non-extensive statistics

In this paper, we have discussed the dependency of the plasma oscillations on the numbers of degrees of freedom (dimensionality) involved in the spectral indices (q or κ) of non-extensive statistical mechanics. Generally, the spectral indices have an inherent dimensional dependency, which may be ignored when a reduced velocity distribution function (VDF) has been derived from the higher dimensional ones, e.g., when studying the longitudinal oscillations or the related topics. In such cases, if we ignore the equipartition of energy, the higher dimensional dependency of the original VDF may be still included in the spectral index of our solutions. On the other hand, in many studies of the plasma oscillations in non-extensive statistics, the ordinary formalism of canonical probability distribution (the old version) had been used instead of the escort probability distribution (the modern version). The escort formalism has some advantages, as opposed to the ordinary version, which may provide more appropriate solutions for the plasma oscillations. In this study, we have re-examined this problem by utilizing the modern version of kappa distribution formalism labeled with an invariant kappa index as of zero dimensionality spectral index, κ0, which is independent of the dimensionality, the degrees of freedom, or the numbers of particles. The related dispersion relation and Landau damping rate of plasma oscillations (Langmuir waves) have been analytically derived and studied by using the Vlasov–Poisson equations for the weakly damped modes. The physical and thermodynamics features of the problem have been considered in terms of the variation of the invariant spectral index κ0.

Mechanisms of dispersal and colonisation in a wind-borne cereal pest, the haplodiploid wheat curl mite

Dispersal and colonisation determine the survival and success of organisms, and influence the structure and dynamics of communities and ecosystems in space and time. Both affect the gene flow between populations, ensuring sufficient level of genetic variation and improving adaptation abilities. In haplodiploids, such as Aceria tosichella (wheat curl mite, WCM), a population may be founded even by a single unfertilised female, so there is a risk of heterozygosity loss (i.e. founder effect). It may lead to adverse outcomes, such as inbreeding depression. Yet, the strength of the founder effect partly depends on the genetic variation of the parental population. WCM is an economically important pest with a great invasive potential, but its dispersal and colonisation mechanisms were poorly studied before. Therefore, here we assessed WCM dispersal and colonisation potential in relation to the genetic variation of the parental population. We checked whether this potential may be linked to specific pre-dispersal actions (e.g. mating before dispersal and collective behaviour). Our study confirms that dispersal strategies of WCM are not dependent on heterozygosity in the parental population, and the efficient dispersal of this species depends on collective movement of fertilised females.

Characterization of volcanoclastic succession in South West Sardinia: In situ vs. laboratory assessment of compressional wave velocity

Non-destructive ultrasonic analyses for the measure of compressional wave velocity (VP) can represent a non-invasive technique to characterize different lithologies and to set data from geophysical logs. This study shows a comparison between in situ and laboratory measurement of VP and its correlation with density, porosity and water content with reference to three geologic formations (i.e. the Monte Crobu, Nuraxi and Paringianu rhyolites), with the final aim to obtain essential data to support the experimental campaigns on the Sotacarbo Fault Lab, a research facility to study CO2 leakage in southwest Sardinia. The considered formations are characterized by a significant heterogeneity, highlighted by the relative wide dispersion of the experimental data (characterized by a high relative standard deviation). In any case, as expected, a direct and slightly positive correlation between VP and bulk density has been found, whereas an opposite trend has been found for the correlations between VP and both porosity and water content. A significant impact on VP value can be ascribed to the pore structure of the considered lithotypes: if pores are non-interconnected, sample saturation in water bath does not involve any significant increasing of VP. And water content can even involve a reduction of P-wave velocity in specific conditions that can reduce the capacity to absorb water. The whole study indicates a mild correlation between in situ and laboratory measures of compressional wave velocity.

First Kyushu record of Remora australis (Echeneidae) from Miyazaki Prefecture, eastern Kyushu, Japan, with comments on its dispersal in relation to the ecology of the host marine mammals

First Kyushu record of <i>Remora australis</i> (Echeneidae) from Miyazaki Prefecture, eastern Kyushu, Japan, with comments on its dispersal in relation to the ecology of the host marine mammals

The effect of different energy portions on the 2D/3D stability swapping for 13-atom metal clusters.

The complexity of Cu13, Ag13, and Au13 coinage-metal clusters was investigated through their energy contributions via a density functional theory study, considering improvements in the PBE functional, such as van der Waals (vdW) corrections, spin-orbit coupling (SOC), Hubbard term (+U), and their combinations. Investigating two-dimensional (planar 2D) and three-dimensional (distorted 3D, CUB - cuboctahedral, and ICO - icosahedral) configurations, we found that vdW corrections are dominant in modulating the stability swapping between 2D and ICO (3D) for Ag13 (Au13), whereas for Cu13 its role is increasing the relative stability between 2D (least stable) and 3D (most stable), setting ICO as the reference. Among the energy portions that constitute the relative total energy, the dimensionality difference correlates with the magnitude of the relative dispersion energy (large for 2D/ICO and small for 3D/ICO) as the causal factor responsible for an eventual stability swapping. For instance, empirical vdW corrections may favor Ag13 as ICO, while semi empirical ones tend to swap the stability by favoring 2D. The same tendency is observed for Au13, except when SOC is included, which enlarges the stability of 3D over 2D. Energy decomposition analysis combined with the natural orbitals for the chemical valence approach confirmed the correlations between the dimensionality difference and the magnitude of the relative dispersion energies. Our structural analysis protocol was able to capture the local distortion effects (or even their absence) through the quantification of the Hausdorff chirality measure. Here, ICO, CUB, and 2D are achiral configurations for all coinage-metal clusters, whereas Cu13 as 3D presents a slight chirality when vdW correction based on many body dispersion is used, at the same time Ag13 as 3D turned out to be chiral for all calculation protocols as evidence of the role of the chemical composition.

Parameterizations of different hydrometeor spectral relative dispersion in the convective clouds

Spectral relative dispersion of different hydrometeors is vital to accurately describe sedimentation. Here, the Weather Research and Forecasting model with spectral bin microphysics is used to simulate convective clouds in Shouxian of Anhui province in China to study the spectral relative dispersion of different hydrometeors. Firstly, regardless of clean or polluted conditions, the relative dispersion of ice crystal spectra and its volume-mean diameter are negatively correlated, while the relative dispersion of other hydrometeor spectra is positively related to their respective volume-mean diameter. The correlations for cloud droplets and raindrops are affected by the process of collision–coalescence; the correlations for ice crystals, graupel particles, and snow particles could be affected by the deposition, riming, and aggregation processes, respectively. Secondly, relative dispersion parameterizations are developed based on a comprehensive consideration of the relationships between the relative dispersion and volume-mean diameter under both polluted and clean conditions. Finally, the relative dispersion parameterizations are applied to terminal velocity parameterizations. The results show that for cloud droplets, ice crystals, graupel particles, and snow particles, assuming the shape parameter in the Gamma distribution is equal to 0 underestimates the shape parameter and overestimates the relative dispersion; and for raindrops, assuming the shape parameter is equal to 0 is close to the relative dispersion parameterizations. The most appropriate constant shape parameters are recommended for different hydrometeors. The relative dispersion parameterizations developed here shed new light for further optimizing the terminal velocity parameterizations in models.摘要离散度的诊断对模式中沉降过程的准确描述至关重要. 本文利用WRF模式结合谱分档方案模拟安徽寿县地区的对流云, 研究不同水成物的离散度. 首先, 无论在清洁还是污染条件下, 除冰晶谱的离散度与体积平均直径间呈现负相关关系外, 云滴, 雨滴, 霰粒子与雪粒子谱离散度与体积平均直径呈现正相关关系; 云滴和雨滴受碰并过程影响, 冰晶, 霰粒子和雪粒子分别受凝华过程, 淞附过程和聚并增长影响. 其次, 综合考虑污染与清洁条件下离散度和体积平均直径之间的相关关系, 建立了离散度的参数化方案. 最后, 把该离散度方案应用到下落末速度的参数化方案中, 结果表明, 对于云滴, 冰晶, 霰粒子和雪粒子, 在Gamma分布中假设谱形参数等于0会低估谱形参数而高估离散度. 对于雨滴而言, 假设谱形参数等于0与参数化方案结果接近. 针对不同的水成物, 给出了最合适的谱形参数定值. 本文发展的离散度方案为进一步优化模式中下落末速度参数化方案提供参考.

Tunable Continuous‐Variable Tripartite Entanglement via Cascaded Third‐Order Nonlinear Processes in a Ring Cavity

AbstractIn this study, a tunable continuous‐variable (CV) triple‐mode entanglement scheme generated by cascading a spontaneous parametric four‐wave and a sum‐frequency four‐wave mixing process is proposed. The entanglement feature of the triple‐mode is demonstrated by using a sufficient inseparability criterion. It is confirmed that the triple‐mode CV entanglement property, including the degree of violation of the sufficient inseparability criterion and the entanglement range in the frequency domain, can be modulated dynamically by modifying the nonlinear susceptibility and optical parameters. In particular, the intensity of the pumping field affects the modulation of the degree of entanglement, and the splitting of the entanglement spectrum comes from electromagnetically induced transparency. Although changing the relative detuning between the Stokes and anti‐Stokes fields can increase the entanglement degree of the triple‐mode, it results in the expansion of the entanglement range in frequency as well. Furthermore, although nonlinear susceptibility directly improves the degree of triple‐mode CV entanglement, optical detuning related nonlinear dispersion has an indirect effect as well. Such an optical triple‐mode CV entanglement with tunable properties can find applications in all‐optical quantum communication and pave the way for the generation and manipulation of multichannel quantum networks.

Relative dispersion with finite inertial ranges

Relative dispersion experiments are often analysed using theoretical predictions from two- and three-dimensional turbulence. These apply to infinite inertial ranges, assuming the same dispersive behaviour over all scales. With finite inertial ranges, the metrics are less conclusive. We examine this using pair separation probability density functions (PDFs), obtained by integrating a Fokker–Planck equation with different diffusivity profiles. We consider time-based metrics, such as the relative dispersion, and separation-based metrics, such as the finite scale Lyapunov exponent (FSLE). As the latter cannot be calculated from a PDF, we introduce a new measure, the cumulative inverse separation time (CIST), which can. This behaves like the FSLE, but advantageously has analytical solutions in the inertial ranges. This allows the establishment of consistency between the time- and space-based metrics, something which has been lacking previously. We focus on three dispersion regimes: non-local spreading (as in a two-dimensional enstrophy inertial range), Richardson dispersion (as in an energy inertial range) and diffusion (for uncorrelated pair motion). The time-based metrics are more successful with non-local dispersion, as the corresponding PDF applies from the initial time. Richardson dispersion is barely observed, because the self-similar PDF applies only asymptotically in time. In contrast, the separation-based CIST correctly captures the dependencies, even with a short (one decade) inertial range, and is superior to the traditional FSLE at large scales. Nevertheless, it is advantageous to use all measures together, to seek consistent indications of the dispersion.

P063 Prevalence in the Use of Complementary/Alternative Medicine in Patients With Inflammatory Bowel Disease from Centro Médico Nacional 20 de Noviembre.

Frequently, existing therapies for inflammatory bowel disease (IBD) do not achieve complete remission of the disease and are commonly associated with adverse effects. Therefore, patients regularly turn to complementary or alternative medicine (CAM). International CAM use in IBD patients ranges from 21% to 60%. There are no reported statistics of CAM use in the Mexican population who suffer IBD. This study´s objective was to determine the prevalence and characterize the use of CAM in our IBD population. Observational, cross-sectional, retrospective study in patients diagnosed with IBD belonging to our Institution's IBD center. The results were analyzed with measures of relative frequency, central tendency, and dispersion. A total of 52 patients with IBD (78% ulcerative colitis and 22% Crohn's disease) were included, 38.5% were exposed to CAM. A total of 27% used it as an aid in their IBD treatment. The remaining 11.5% used it for other reasons. Patients exposed to CAM for IBD averaged 51 ± 19 years, 64% were female and 71% had at least an undergraduate degree. The most used CAMs were acupuncture (42.8%), herbal (35.7%) and homeopathy (35.7%). Half of the patients used concomitantly two or more modalities, and 91.6% of the patients knew their diagnosis at the time that CAM was being used. The median exposure time to CAM was 4.5 ± 40.7 months. The main reasons for the use of CAM in IBD patients was as a complementary therapy in 58.3%, as the perception of lack of improvement was present in 33.3% of patients. Twenty-five percent of patients used CAM after a healthcare professional recommended it. CAM was used in 83.2% of patients with conventional treatment (Mesalamine: 74.6%, Mesalamine + Azathioprine: 8.3%) and only 16.5% were on biological therapy. A total of 58.3% CAM users perceived improvement in their symptoms. Currently 25% still use some modality of CAM. No statistically significant differences were found in sociodemographic variables and clinical outcomes when comparing the IBD group exposed to CAM vs the unexposed group. Exposure to CAM for IBD treatment had a prevalence of 27%. The main CAM modalities for IBD patients were acupuncture, herbal, and homeopathy. No relationship was demonstrated between the use of CAM in IBD patients and their clinical outcomes.

Culturable bacteria are more common than fungi in floral nectar and are more easily dispersed by thrips, a ubiquitous flower visitor.

Variation in dispersal ability among taxa affects community assembly and biodiversity maintenance within metacommunities. Although fungi and bacteria frequently coexist, their relative dispersal abilities are poorly understood. Nectar-inhabiting microbial communities affect plant reproduction and pollinator behavior, and are excellent models for studying dispersal of bacteria and fungi in a metacommunity framework. Here, we assay dispersal ability of common nectar bacteria and fungi in an insect-based dispersal experiment. We then compare these results with the incidence and abundance of culturable flower-inhabiting bacteria and fungi within naturally occurring flowers across two coflowering communities in California across two flowering seasons. Our microbial dispersal experiment demonstrates that bacteria disperse via thrips among artificial habitat patches more readily than fungi. In the field, incidence and abundance of culturable bacteria and fungi were positively correlated, but bacteria were much more widespread. These patterns suggest shared dispersal routes or habitat requirements among culturable bacteria and fungi, but differences in dispersal or colonization frequency by thrips, common flower visitors. The finding that culturable bacteria are more common among nectar sampled here, in part due to superior thrips-mediated dispersal, may have relevance for microbial life history, community assembly of microbes, and plant-pollinator interactions.

Quality Differentiation and Spatial Clustering among Restaurants

To explore the relationship between spatial location and quality differentiation, we build a dataset of over 30,000 restaurants rated by TripAdvisor, across large UK cities. We document several new stylized facts. Top-rated restaurants tend to be spatially more concentrated around the city center than bottom-rated ones. Whereas top-rated restaurants tend to locate close to other top restaurants, bottom-rated restaurants tend to locate away from each other and closer to top-rated ones. Our theoretical model can explain the main features of observed spatial patterns. Consistently with the predictions of our model, our regression analysis finds that an increase in the population density in the city center is associated with a decrease in the spatial dispersion of both top and bottom restaurants, this reduction being larger in magnitude for top ones. Also, a larger quality difference between top and bottom restaurants increases both the absolute and relative dispersion of top restaurants.