Power-law Function Research Articles

Measurement of Born cross sections of e+e−→Ξ0Ξ¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{+}{e}^{-}\ o {\\Xi}^0{\\overline{\\Xi}}^0 $$\\end{document} and search for charmonium(-like) states at s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s} $$\\end{document} = 3.51–4.95 GeV

Using e+e− collision data collected by the BESIII detector at BEPCII corresponding to an integrated luminosity of 30 fb−1, we measure Born cross sections and effective form factors for the process e+e−→Ξ0Ξ¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{+}{e}^{-}\ o {\\Xi}^0{\\overline{\\Xi}}^0 $$\\end{document} at forty-five center-of-mass energies between 3.51 and 4.95 GeV. The dressed cross section is fitted, assuming a power-law function plus a charmonium(-like) state, i.e., ψ(3770), ψ(4040), ψ(4160), ψ(4230), ψ(4360), ψ(4415) or ψ(4660). No significant charmonium(-like) state decaying into Ξ0Ξ¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Xi}^0{\\overline{\\Xi}}^0 $$\\end{document} is observed. Upper limits at the 90% confidence level on the product of the branching fraction and the electronic partial width are provided for each decay. In addition, ratios of the Born cross sections and the effective form factors for e+e−→Ξ0Ξ¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{+}{e}^{-}\ o {\\Xi}^0{\\overline{\\Xi}}^0 $$\\end{document} and e+e−→Ξ−Ξ¯+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{+}{e}^{-}\ o {\\Xi}^{-}{\\overline{\\Xi}}^{+} $$\\end{document} are also presented to test isospin symmetry and the vector meson dominance model.

Post-irradiation darkening model for EBT-3 films characterized using a single lot calibration approach

Objective.In this study, we present a model to correct the progressive post-irradiation darkening of EBT3 films. The model allows for a clinical use of EBT3 using application and calibration films scanned with different post-irradiation times.Approach.The model is a post-irradiation time- and dose-dependent power-law function, projecting the scanned transmittance of application films to the transmittance matching the same post-irradiation time of calibration films. The model was characterized for two EBT3 production lots within the dose range 0.1-12.8 Gy. A first characterization was performed utilizing calibration films scanned repeatedly for 54 d post-irradiation (lot 1), while a fast re-characterization of a second lot used three post-irradiation times (lot 2). For a long-term validation of the model, 16 film strips were irradiated at 2 Gy on different time points starting from the day of film calibration up to 43 d afterwards (lot 1). For the multiple-dose validation of the model, 8 strips were irradiated with dose levels ranging 0-12 Gy deposited 25 d after the calibration (lot 2). As a proof of principle, the model was applied to four clinical patient-specific quality assurance film measurements with prescribed dose/fraction ranging 2.66 Gy-8 Gy.Main results. The post-irradiation transmittance decreased for higher doses up to -2.5% at 12.8 Gy, and 54 d post-irradiation. With a lot-specific model correction, the mean dose accuracy of validation strips that ranged from initial -3.4% (triple-channel) and -9.90% (blue-channel) reduced to within 3.0% (all colour channels) for doses above 1 Gy. The median dose difference with the planned dose improved from -3.5% to -1.1%, and the 3%/2 mm local gamma ranged from (48.5-92.5)% to (81.2-99.2)%.Significance.A film darkening model corrects the transmittance of EBT3 films and increases the flexibility of existing dosimetry protocols. The correction ensures dose accuracies within 3%.

Size-frequency distribution of submarine mass movements on the Palomares continental slope (W Mediterranean)

In this work, over 3620 km2 from the Palomares continental slope, which is located in the W. Mediterranean Sea, was analysed to quantify the impact of recent mass movements on this margin. A total of 936 landslides were identified, mapped and characterised by defining several morphometric variables that outline the accumulated impact of landslides equivalent to 918 km2 and 10.34 km3 of eroded sediment on the continental slope. The smallest event area was 0.0014 km2, whereas the largest event area was 32.48 km2. Smaller scars with a higher headwall gradient tend to dominate when the environment is steeper, and major mass movements are located on open slopes and structural highs. However, the slight or null correlations between variables indicate that a wide range of sizes may occur on any slope gradient and at any depth.The Palomares continental slope is intensively affected by mass movements. Compared with other passive margins (e.g., the U.S. Atlantic continental margin), landslides mobilised a limited amount of sediment, although it is comparable to other Mediterranean areas where small- to moderate-sized events are characteristic.The cumulative size distribution can be defined by a power-law function that describes events larger than 0.7 km2 with an exponent of α = 1.269. These results are consistent with those of other published inventories, including onshore cases. This result allows us to assume that the scale-invariant properties of the events are mapped. Scale-invariant properties can be explained by different models; self-organised criticality (SOC) is probably the most assumed by the scientific community, although alternative models may be nominated. Each model has important implications in terms of the landslide distribution and long-term landslide history of any slope. Alternative scenarios, such as submarine slopes, with more precise landslide inventories may contribute to new hazard assessment models that consider scaling exponents derived from size–frequency distributions.

X-Ray Bright Active Galactic Nuclei in Local Dwarf Galaxies: Insights from eROSITA

Although supermassive black holes (SMBHs) reside in the heart of virtually every massive galaxy, it remains debated whether dwarf galaxies commonly host SMBHs. Because low-mass galaxies may retain memory of the assembly history of their black holes (BHs), probing the BH occupation fraction of local dwarf galaxies might offer insights into the growth and seeding mechanisms of the first BHs. In this work, we exploit the Western half of the eROSITA all-sky survey (covering 20,000 deg2) and compile a catalog of accreting SMBHs in local (D < 200 Mpc) dwarf galaxies. Cleaning our sample from X-ray background sources, X-ray binaries, and ultraluminous X-ray sources, we identify 74 active galactic nucleus (AGN)–dwarf galaxy pairs. Using this large and uniform sample, we derive the luminosity function of the dwarf galaxy AGN, fitting it with a power-law function and obtaining dN/dLX=(15.9±2.2)×LX−1.63±0.05 . Measuring the offset between the dwarf galaxies' centroids and the X-ray sources, we find that ≈50% of the AGN are likely off-nuclear, in agreement with theoretical predictions. We compare the BH-to-stellar mass relation of our sample with the local and high-redshift relations, finding that our sources better adhere to the former, which suggests that local AGN across different mass scales undergo similar growth histories. Finally, we compare our sources with semianalytical models: while our sample’s shallowness prevents distinguishing between different seeding models, we find that the data favor models that keep SMBHs in dwarf galaxies active at a moderate rate, motivating model improvement by comparison to AGN in the dwarf galaxy regime.

An empirical tool for predicting the presence or absence of coseismic displacements at GNSS stations

Unmodeled displacements in GNSS times series, induced by instrumental artifacts or geophysical events, create significant biases in station trajectory parameters that can propagate into the reference frame itself. While non-tectonic ‘jumps’, such as equipment changes, affect only a specific GNSS station, seismically-induced displacements can affect large numbers of sites, severely threatening the frame’s stability. Manually reviewing individual GNSS time series for such effects is highly impractical because there can be thousands of GNSS stations in a frame, and the total number of earthquakes Mw ≥ 6.0 since GPS became fully operational is + 5100. To avoid this time-consuming task, automated methods rely on empirical power-law functions to determine which earthquake-station pairs require coseismic displacement parameters. Still, ‘conservative’ power-law functions tend to add coseismic offsets to stations that do not need them, which can also threaten the stability of the frame. In this work, we present an empirical formulation that was obtained using 809 global seismic events to fit power-law parameters that do not overestimate the region of influence of earthquakes. Our method is based on a two-level selection process: level 1 is isotropic and only considers the epicentral distance between the stations and the earthquake, and level 2 uses the geophysical parameters of the earthquake to predict a ‘tighter’ displacement pattern to select which stations require coseismic trajectory parameters. We applied our level 2 method to a database of ~ 4700 event-station pairs and showed that it removed ~ 55% of the total pairs, all of which had been falsely selected by level 1.

New Versus Past Silica Crush Curve Experiments: Application to Dimorphos Benchmarking Impact Simulations

Crush curves are of fundamental importance to numerical modeling of small and porous astrophysical bodies. The empirical literature often measures them for silica grains, and different studies have used various methods, sizes, textures, and pressure conditions. Here, we review past studies and supplement further experiments in order to develop a full and overarching understanding of the silica crush curve behavior. We suggest a new power-law function that can be used in impact simulations of analog materials similar to microgranular silica. We perform a benchmarking study to compare this new crush curve to the parametric quadratic crush curve often used in other studies, based on the study case of the DART impact onto the asteroid Dimorphos. We find that the typical quadratic crush curve parameters do not closely follow the silica crushing experiments, and as a consequence, they under (over) estimate compression close (far) from the impact site. The new crush curve presented here, applicable to pressures between a few hundred Pa and up to 1.1 GPa, might therefore be more precise. Additionally, it is not calibrated by case-specific parameters, and can be used universally for comet- or asteroid-like bodies, given an assumed composition similar to microgranular silica.

Thin-film evaporation in microchannels: A combined analytical and computational fluid dynamics approach to assessing meniscus curvature impact

Computational fluid dynamics (CFD) is employed to investigate the effects of variation in bulk meniscus curvature on evaporative heat and mass transfer in microchannels. Evaporation in microchannels presents a classic multiscale problem, encompassing both microscopic and macroscopic length scales in the thin-film and bulk meniscus regions, respectively. Therefore, the liquid-vapor interface or meniscus shape is first determined through thin-film evaporation analysis using numerical approaches found in the literature. However, a new approach for establishing boundary conditions is also proposed, allowing for the representation of varying curvatures within the bulk meniscus region. Then, the obtained meniscus thickness profiles are exported to a two-dimensional CFD framework. The primary advantage of this developed CFD framework lies in its capability to simultaneously analyze evaporative heat and mass transfer from both microscopic and macroscopic regions, a feature that enhances research endeavors concerning micro and miniature heat pipes. The developed model is applied to various rectangular microchannel sizes, ranging from 10 to 100 μm in width, with wall superheats varying from 0.1 to 3.0 K. The aim is to quantify the effects of variations in bulk meniscus curvature on the evaporative characteristics of the extended meniscus (comprising both thin-film and bulk meniscus regions), using the concept of thermal resistance. Consequently, based on the CFD simulation results, multiple regression analysis is employed to formulate the total thermal resistance in terms of independent parameters, namely the microchannel width, wall superheat, and the radius of curvature (Rc). It is observed that the radius of curvature has a marginal impact on the total thermal resistance, allowing its influence to be expressed by a power-law function of (Rc/Rm)0.102, where Rm represents the minimum radius of curvature.

Coupled torsion and inflation of functionally graded and residually stressed Mooney–Rivlin hollow cylinders

Functionally graded structures have material properties continuously varying in one or more directions. Examples include human teeth, sea shells, bamboo stems, and mammal organs in which the varying volume fraction and orientation of fibers optimize their functionalities. Here we analytically study large deformations due to combined torsion and inflation of residually stressed and radially graded Mooney–Rivlin hollow circular cylinders to provide insights into how grading their material moduli according to a power law function of the radius can be advantageously used. We simulate residual stresses in a thin wall hollow cylinder by inverting it inside out and in a thick wall cylinder by assuming that a longitudinal wedge opening parallel to the cylinder axis is closed by deforming it axisymmetrically. It is found that positive integer values of the power law exponent are generally advantageous but its negative values can have deleterious effects on stress distributions. Furthermore, a hollow cylinder with residual stresses generated by one of the foregoing methods in the reference configuration should be modeled as orthotropic rather than transversely isotropic for subsequent deformations. The analytical solutions provided here should help numerical analysts verify their algorithms for large deformations of rubber-like materials that are modeled by the Mooney–Rivlin relation, and design engineers for exploiting the radial gradation of the two material moduli to reduce structure’s mass without sacrificing its performance.

Disentangling the Impacts of Environmental Factors on Evaporative Fraction Across Climate Regimes

AbstractEvaporative fraction (EF) is a useful measure for quantifying land surface energy partitioning processes and determining evaporative regimes; however, its influencing factors remain highly uncertain. Here, global data sets were compiled to disentangle the effects of environmental variables on EF variations along climate and land surface gradients. We found that (a) at annual timescales, ecosystem‐level EF could be expressed as a power law function of aridity index. The relationships of mean annual soil water content (SWC) and leaf area index (LAI) with mean annual EF resembled the traditional evaporative regime theory; (b) at daily timescales, the boosted regression tree method quantitatively revealed that the impacts of environmental variables (including meteorological variables) on EF showed equal importance, especially at humid sites, primarily due to the different response direction and magnitude of latent heat (LE) and sensible heat (H) fluxes to environmental changes. Particularly, the contrasting responses of LE (positive) and H (negative) to SWC, LAI, and relative humidity enhanced the positive effects of those influencing variables on EF; whereas, the correlations between EF and energy‐related factors (i.e., net radiation‐Rn and air temperature‐Ta) deteriorated as both LE and H showed positive response patterns to those variables; (c) meteorological factors were also found to have nonlinear effects on daily EF, further modified by climatic conditions. Rn near 150 W/m2 and Ta near 15°C appeared to be important energy‐partitioning thresholds at drier and humid sites, respectively. Moreover, changing interactions among environmental variables with climates were demonstrated to be important for better explaining EF variations.

Static, cylindrically symmetric spacetime in the coincident f(Q) gravity

In this paper we consider a static, cylindrically symmetric spacetime with coincident f(Q) gravity. Since the field equation of this spacetime in symmetric teleparallel gravity is suitable for choosing the function of f(Q) in the form of power series and exponential forms which are in coherent with cosmological observations, anisotropic perfect fluid solutions of these forms are discussed for this spacetime. Energy densities, directional pressures and energy conditions are plotted and analysed for a few different metric potentials. Although corresponding cosmic strings violate all energy conditions in both f(Q) functions, the corresponding Levi-Civita solution violates all energy conditions in the power law function of f(Q), but for exponential f(Q) gravity they are satisfied in small regions.

The aeroelastic stability characteristics of a ring-stiffened conical three-layered sandwich shell with an FG auxetic honeycomb core utilizing zig-zag shell theory

This paper is devoted to the supersonic flutter analysis of a ring-stiffened conical three-layered sandwich shell consisting of an auxetic honeycomb (AH) core fabricated from functionally graded material (FGM). The face sheets are manufactured of FGM as well in which the volume fraction of the ceramic increases from zero at the inner surface to one at the outer surface based on a power-law function. The sandwich shell is mathematically modeled based on the Murakami's zig-zag shell theory and the aerodynamic pressure is modeled utilizing piston theory. The derivation of governing equations along with compatibility and boundary conditions are performed using Hamilton's principle and are solved through a semi-analytical solution consisting of an exact solution in the circumferential direction followed by an approximate solution in the meridional direction. The flutter boundaries are attained to investigate the variations of the natural frequencies and damping ratios to find the critical aerodynamic pressure (CAP). The impacts of several factors on the CAP are examined such as the power-law index, geometric characteristics of the cells in the FGAH, thickness of the honeycomb core, location of the ring, and boundary conditions. It is observed that an optimal location can be found for the ring support somewhere between the middle length and large radius of a conical shell that provides the best aeroelastic stability.

Pink-noise dynamics in an evolutionary game on a regular graph.

We consider a multiplayer prisoner's dilemma game on a square lattice and regular graphs based on the pairwise-Fermi update rule, and we obtain heatmaps of the fraction of cooperators and the correlation of neighboring pairs. In the heatmap, we find a mixed region where cooperators and defectors coexist, and the correlation between neighbors is significantly enhanced. Moreover, we observe pink-noise behavior in the mixed region, where the power spectrum can be fitted by a power-law function of frequency. We also find that the pink-noise behavior can be reproduced in a simple random-walk model. In particular, we propose a modified random-walk model which can reproduce not only the pink-noise behavior but also the deviation from it observed in a low-frequency region.

Constraining the Earth-mass Primordial Black Hole Mergers Model of the Non-repeating FRBs Using the First CHIME/FRB Catalog

In this paper, we upgrade the constraints for the Earth-mass primordial black hole mergers model based on the first Canadian Hydrogen Intensity Mapping Experiment (CHIME)/fast radio burst (FRB) catalog. Assuming the null hypothesis that the observed non-repeating FRBs originate from Earth-mass primordial black hole mergers, we find that how the charges were distributed in the primordial black hole population is well described by a double power-law function with typical charge value of qc/10−5=1.60−0.28+0.28 , where the power-law index α1=2.33−0.18+0.15 for q < q c and α2=4.56−0.26+0.30 for q ≥ q c. Here, q represents the charge of the black hole in units of GM , where M is the mass of the black hole. Furthermore, we infer the local event rate of the bursts is 8.8−2.1+5.7×104Gpc−3yr−1 , which indicates that an abundance of the primordial black hole population f ≳ 10−4 is needed to account for the observed FRBs by CHIME. The results of this paper lay the basis for further research on the electromagnetic radiation background generated by the merger of primordial black hole mergers.

GRB 231129C: Another Thermal Emission Dominated Gamma-Ray Burst

This study presents detailed time-integrated and time-resolved spectral analysis of the Fermi Gamma-ray Burst Monitor observations of the bright GRB 231129C. The results reveal its distinct spectral characteristics, featuring a hard low-energy spectral index (α) and soft high-energy spectral index (β), similar to GRB 090902B, suggesting a possible dominance of thermal emission. Further analysis indicates that 92% of the spectral indices exceed the synchrotron “line of death,” with the hardest index at α ∼ +0.44. Simultaneously, 53% of the spectra can be well fitted by the nondissipative photosphere model, supporting a potential origin from a nondissipative photosphere. Additionally, we observe strong correlations between the spectral index α and peak energy E p with flux. For the α−F relationship, we employ F = F 0 e (3.00±0.10)α to describe it, whereas the E p−F relationship requires a smoothly bending power-law function. Based on the framework proposed by Hascoët et al. and Gao & Zhang, the jet characteristics of this burst were studied, revealing that both methods support the suitability of a pure fireball model for this GRB at small initial jet radii.

Extreme-value modelling of the brightest galaxies at z ≳ 9

Abstract Data from the James Webb Space Telescope have revealed an intriguing population of bright galaxies at high redshifts. In this work, we use extreme-value statistics to calculate the distribution (in UV magnitude) of the brightest galaxies in the redshift range 9 ≲ z ≲ 16. We combine the Generalised Extreme Value (GEV) approach with modelling of the galaxy luminosity function. We obtain predictions of the brightest galaxies for a suite of luminosity functions, including the Schechter and double power law functions, as well as a model parametrised by the stellar formation efficiency f*. We find that the JWST data is broadly consistent with f* of $5~{{\ \rm per\ cent}}-10~{{\ \rm per\ cent}}$, and that the brightest galaxy at z ∼ 16 will have $M_{\rm UV}\approx -23.5^{0.8}_{0.4}$. If f* is dependent on halo mass, we predict $M_{\rm UV}\approx -22.5^{0.5}_{1.5}$ for such an object. We show that extreme-value statistics not only predicts the magnitude of the brightest galaxies at high redshifts, but may also be able to distinguish between models of star formation in high-redshift galaxies.

Multi-scaling allometry in human development, mammalian morphology, and tree growth

Various animal and plant species exhibit allometric relationships among their respective traits, wherein one trait undergoes expansion as a power-law function of another due to constraints acting on growth processes. For instance, the acknowledged consensus posits that tree height scales with the two-thirds power of stem diameter. In the context of human development, it is posited that body weight scales with the second power of height. This prevalent allometric relationship derives its nomenclature from fitting two variables linearly within a logarithmic framework, thus giving rise to the term “power-law relationship.” Here, we challenge the conventional assumption that a singular power-law equation adequately encapsulates the allometric relationship between any two traits. We strategically leverage quantile regression analysis to demonstrate that the scaling exponent characterizing this power-law relationship is contingent upon the centile within these traits’ distributions. This observation fundamentally underscores the proposition that individuals occupying disparate segments of the distribution may employ distinct growth strategies, as indicated by distinct power-law exponents. We introduce the innovative concept of “multi-scale allometry” to encapsulate this newfound insight. Through a comprehensive reevaluation of (i) the height-weight relationship within a cohort comprising 7, 863, 520 Japanese children aged 5–17 years for which the age, sex, height, and weight were recorded as part of a national study, (ii) the stem-diameter-height and crown-radius-height relationships within an expansive sample of 498, 838 georeferenced and taxonomically standardized records of individual trees spanning diverse geographical locations, and (iii) the brain-size-body-size relationship within an extensive dataset encompassing 1, 552 mammalian species, we resolutely substantiate the viability of multi-scale allometric analysis. This empirical substantiation advocates a paradigm shift from uni-scaling to multi-scaling allometric modeling, thereby affording greater prominence to the inherent growth processes that underlie the morphological diversity evident throughout the living world.

FGM concept used in damage analysis of heat-treated elbows under out-of-plane combined bending moment

The elbow is presented in pressurized tubular structures as a connecting element, generally subjected to bending moments either out-of-plane or in-plane, in closure or opening. Another mode of bending moment that can also occur in these tubular structures is the combined bending moment between out-of-plane and in-plane in closure, or out-of-plane and in-plane in opening. In this work, to better present the potential severity caused by this loading mode, an elbow structure attached by straight tubular sections made of X60 steel is analyzed and its damage under this combined bending moment with orientations between the plane and out-of-plane of the structure is discussed. A reinforcement proposal will be made in the second part of this work. Indeed, the structure is pretreated thermally at the elbow, resulting in a graded structure along the thickness with two different material constituents: the base metal and the thermally affected zone (HAZ). Based on the concept of Functionally Graded Materials (FGM), the gradation is by volume fraction between the base metal and the HAZ, following a power law function of a volume fraction index (n). This graded property is introduced along the thickness by finite element layers. The elastic-plastic behavior of the HAZ-base metal mixture is modeled using the Voce model and the Von Mises equivalent stress flow theory. Using the XFEM technique(extended finite element method), the damage of the pressurized and thermally treated structure under a combined out-of-plane bending moment shows that these parameters condition the response and the level of damage.

Core allocation to minimize total flow time in a multicore system in the presence of a processing time constraint

Data centers are a vital and fundamental infrastructure component of the cloud. The requirement to execute a large number of demanding jobs places a premium on processing capacity. Parallelizing jobs to run on multiple cores reduces execution time. However, there is a decreasing marginal benefit to using more cores, with the speedup function quantifying the achievable gains. A critical performance metric is flow time. Previous results in the literature derived closed-form expressions for the optimal allocation of cores to minimize total flow time for a power-law speedup function if all jobs are present at time 0. However, this work did not place a constraint on the makespan. For many diverse applications, fast response times are essential, and latency targets are specified to avoid adverse impacts on user experience. This paper is the first to determine the optimal core allocations for a multicore system to minimize total flow time in the presence of a completion deadline (where all jobs have the same deadline). The allocation problem is formulated as a nonlinear optimization program that is solved using the Lagrange multiplier technique. Closed-form expressions are derived for the optimal core allocations, total flow time, and makespan, which can be fitted to a specified deadline by adjusting the value of a single Lagrange multiplier. Compared to the unconstrained problem, the shortest job first property for optimal allocation is maintained; however, a number of other properties require revising and other properties are only retained in a modified form (such as the scale-free and size-dependence properties). It is found that with a completion deadline the optimal solution may contain groups of simultaneous completions. In general, all possible patterns of single- and group-completion need to be considered, producing an exponential search space. However, the paper determines analytically that the optimal completion pattern consists of a sequence of single completions followed by a single group of simultaneous completions at the end, which reduces the search space dimension to being linear. The paper validates the Lagrange multiplier approach by verifying constraint qualifications.

The crowd in MOOCs: a study of learning patterns at scale

ABSTRACT The increasing availability of learning activity data in Massive Open Online Courses (MOOCs) enables us to conduct a large-scale analysis of learners' learning behavior. In this paper, we analyze a dataset of 351 million learning activities from 0.8 million unique learners enrolled in over 1.6 thousand courses within two years. Specifically, we mine and identify the learning patterns of the crowd from both temporal and course enrollment perspectives leveraging mutual information theory and sequential pattern mining methods. From the temporal perspective, we find that the time intervals between consecutive learning activities of learners exhibit a mix of power-law and periodic cosine function distribution. By qualifying the relationship between course pairs, we observe that the most frequently co-enrolled courses usually fall in the same category or the same university. We demonstrate these findings can facilitate manifold applications including recommendation tasks on courses. A simple recommendation model utilizing the course enrollment patterns is competitive to the baselines with 200× faster training time.

A New Approach for Understanding International Hospital Bed Numbers and Application to Local Area Bed Demand and Capacity Planning.

Three models/methods are given to understand the extreme international variation in available and occupied hospital bed numbers. These models/methods all rely on readily available data. In the first, occupied beds (rather than available beds) are used to measure the expressed demand for hospital beds. The expressed occupied bed demand for three countries was in the order Australia > England > USA. Next, the age-standardized mortality rate (ASMR) has dual functions. Less developed countries/regions have low access to healthcare, which results in high ASMR, or a negative slope between ASMR versus available/occupied beds. In the more developed countries, high ASMR can also be used to measure the 'need' for healthcare (including occupied beds), a positive slope among various social (wealth/lifestyle) groups, which will include Indigenous peoples. In England, a 100-unit increase in ASMR (European Standard population) leads to a 15.3-30.7 (feasible range) unit increase in occupied beds per 1000 deaths. Higher ASMR shows why the Australian states of the Northern Territory and Tasmania have an intrinsic higher bed demand. The USA has a high relative ASMR (for a developed/wealthy country) because healthcare is not universal in the widest sense. Lastly, a method for benchmarking the whole hospital's average bed occupancy which enables them to run at optimum efficiency and safety. English hospitals operate at highly disruptive and unsafe levels of bed occupancy, manifesting as high 'turn-away'. Turn-away implies bed unavailability for the next arriving patient. In the case of occupied beds, the slope of the relationship between occupied beds per 1000 deaths and deaths per 1000 population shows a power law function. Scatter around the trend line arising from year-to-year fluctuations in occupied beds per 1000 deaths, ASMR, deaths per 1000 population, changes in the number of persons hidden in the elective, outpatient and diagnostic waiting lists, and local area variation in births affecting maternity, neonatal, and pediatric bed demand. Additional variation will arise from differences in the level of local funding for social care, especially elderly care. The problems associated with crafting effective bed planning are illustrated using the English NHS as an example.