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## Thursday, October 4, 2018

### A New Top Quark Mass Calculation From ATLAS

There is a new paper determining the top quark mass from ATLAS experiment data from the Large Hadron Collider (LHC), although it uses only fairly early data.
The mass of the top quark is measured to be mtop=172.08±0.39(stat)±0.82(syst) GeV. A combination with previous ATLAS mtopmeasurements gives mtop=172.69±0.25(stat)±0.41(syst) GeV.
From here.

From the introduction:
The mass of the top quark mtop is an important parameter of the Standard Model (SM). Precise measurements of mtop provide crucial information for global fits of electroweak parameters [1–3] which help to assess the internal consistency of the SM and probe its extensions. In addition, the value of mtop affects the stability of the SM Higgs potential, which has cosmological implications [4–6].
Many measurements of mtop in each tt¯ decay channel were performed by the Tevatron and LHC collaborations. The most precise measurements per experiment in the tt¯ → lepton + jets channel are mtop = 172.85 ± 0.71 (stat) ± 0.84 (syst) GeV by CDF [7], mtop = 174.98 ± 0.58 (stat) ± 0.49 (syst) GeV by D0 [8], mtop = 172.33 ± 0.75 (stat) ± 1.03 (syst) GeV by ATLAS [9] and mtop = 172.35 ± 0.16 (stat) ± 0.48 (syst) GeV by CMS [10]. Combinations are performed, by either the individual experiments, or by several Tevatron and LHC experiments [11]. In these combinations, selections of measurements from all tt¯ decay channels are used. The latest combinations per experiment are mtop = 173.16 ± 0.57 (stat) ± 0.74 (syst) GeV by CDF [12], mtop = 174.95 ± 0.40 (stat) ± 0.64 (syst) GeV by D0 [13], mtop = 172.84 ± 0.34 (stat) ± 0.61 (syst) GeV by ATLAS [14] and mtop = 172.44 ± 0.13 (stat) ± 0.47 (syst) GeV by CMS [10].
In this paper, an ATLAS measurement of mtop in the tt¯ → lepton + jets channel is presented. The result is obtained from pp collision data recorded in 2012 at a centre-of-mass energy of √ s = 8 TeV with an integrated luminosity of about 20.2 fb−1 . The analysis exploits the decay tt¯ → W+W−bb¯ → νqq¯ 0bb¯, which occurs when one W boson decays into a charged lepton ( is e or µ including τ → e, µ decays) and a neutrino (ν), and the other into a pair of quarks. In the analysis presented here, mtop is obtained from the combined sample of events selected in the electron+jets and muon+jets final states. Single-top-quark events with the same reconstructed final states contain information about the top quark mass and are therefore included as signal events
The combined error in the combination is ± 0.48 GeV, which is very low. This excludes masses in excess of 173.65 GeV at the 95% confidence level, which is barely consistent with Particle Data Group indirect measurement of 173.5 GeV.

Both the new calculation and the combined measurement are lighter than previous results from all sources as of a year ago. There was a paper reviewing the most recent mass measurement in both collaborations in January of 2018. At that time, ATLAS was using only Run-1 data.

It is worth observing the range of the combined measurements from the two Tevatron experiments and the two LHC experiments: 172.44 GeV to 174.95 GeV. The spread from the midpoint of that range is ± 1.255 GeV. I think it is safe to say that somebody's error bars are probably underestimated.

It isn't at all clear why such old data is being used in a 2018 publication. As of the most recent information available ATLAS and CMS have each collected almost three times as much data as the 20.2 fb-1 relied upon in this paper.

ATLAS: 57.5 fb-1
CMS: 59.21 fb-1

Also, much of that has been at higher 13 TeV energies than the centre-of-mass energy of √ s = 8 TeV relied upon in this paper, which should also produce better data. Many papers using more recent data have been published concerning top quark physics. For example, this paper from ATLAS and this one in which:  "The data analysed correspond to 79.8 fb−1 of proton--proton collisions at a centre-of-mass energy of s√=13 TeV recorded by the ATLAS experiment at the LHC."

While the percentage error in the top quark mass determination isn't particularly high for QCD, in many circumstances, what matters is the absolute magnitude of the uncertainty, rather than the percentage uncertainty, and in terms of absolute magnitude of the error bars, the uncertainty in the top quark mass dwarfs the uncertainty in all of the other mass measurements in the Standard Model.

A May 3, 2018 paper from CMS also looks at this topic (largely identical in result and data to a January 17, 2018 paper from CMS) using more data and higher energy data:
The mass of the top quark is measured using a sample of tt events containing one isolated muon or electron and at least four jets in the final state, collected by the CMS detector using proton-proton collisions at s= 13 TeV at the CERN LHC. The events are selected from data corresponding to an integrated luminosity of 35.9 fb1. For each event the mass is reconstructed from a kinematic fit of the decay products to a \ttbar hypothesis. Using the ideogram method, the top quark mass is determined simultaneously with an overall jet energy scale factor (JSF), constrained by the mass of the W boson in qq decays. The measurement is calibrated on samples simulated at next-to-leading order matched to a leading-order parton shower. The top quark mass is found to be 172.25±0.08 (stat+JSF)±0.62 (syst) GeV. The dependence of this result on the kinematic properties of the event is studied and compared to predictions of different models of tt production, and no indications of a bias in the measurements are observed.
Other Physics News

A paper in June summed up efforts to more accurately measure the strong force coupling constant. The abstract of the paper notes that:
The latest experimental and theoretical developments in the high-precision determination of the strong coupling αs are briefly reviewed. Six groups of observables: (i) lattice QCD data, (ii) hadronic τ decays, (iii) deep-inelastic e±p data and parton distribution functions (PDF) fits, (iv) event shapes and jet rates in e+e collisions, (v) Z boson hadronic decays, and (vi) top-quark cross sections in pp collisions, are used to extract the current world-average at the Z pole mass, αs(m2Z)=0.1181±0.0011 at next-to-next-to-leading-order (NNLO), or beyond, accuracy. Additional NNLO extractions have recently appeared based on new lattice studies, the R(s) ratio in e+ehadrons, updated PDF fits, energy-energy correlations in e+e collisions, jet cross sections in e±p collisions, and the full set of pptt¯ cross sections at the LHC. Inclusion of these new data into the world-average would slightly increase its value and reduce its uncertainty to αs(m2Z)=0.1183±0.0008. Future αs extraction perspectives with permille uncertainties at future high-luminosity e+e machines -- via W and Z hadronic decays, parton fragmentation functions, and photon F2(x,Q2) structure function in γγ collisions -- are also discussed.


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show · g2c.jacobian all knowls · up · search:

The Jacobian of a genus 2 curve is an abelian surface (an abelian variety of dimension 2).

Properties of the Jacobian include its Mordell-Weil group and its endomorphism ring.

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• Last edited by John Cremona on 2018-05-24 14:50:22
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Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-22064

# Barbour, A D; Månsson, M (2000). Compound Poisson approximation and the clustering of random points. Advances in Applied Probability, 32(1):19-38.

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## Abstract

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,...). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλi be decreasing.

## Citations

4 citations in Web of Science®
5 citations in Scopus®


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# Genus of Grassmannians and Flag Manifolds

In light of the answers given to this question, I would like to pose a more general one: Do all Grassmannian spaces have genus 0? If so, do there exist any flag manifolds with non-zero genus?

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In arbitrary characteristic, the structure sheaf of any homogeneous space $G/P$ (for $G$ a semisimple group) has no higher cohomology. This is an instance of Kempf's vanishing theorem. The space of sections is 1-dimensional, so this implies that the arithmetic genus is 0.
Steven, do you mean "$\mathit{complete}$ homogeneous space", i.e. $G/P$? That's certainly implied by the question, but the answer is a but unclear. –  Victor Protsak Jun 12 '10 at 4:08


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Article | Open | Published:

# Predictability of Arctic sea ice on weather time scales

## Abstract

The field of Arctic sea ice prediction on “weather time scales” is still in its infancy with little existing understanding of the limits of predictability. This is especially true for sea ice deformation along so-called Linear Kinematic Features (LKFs) including leads that are relevant for marine operations. Here the potential predictability of the sea ice pack in the wintertime Arctic up to ten days ahead is determined, exploiting the fact that sea ice-ocean models start to show skill at representing sea ice deformation at high spatial resolutions. Results are based on ensemble simulations with a high-resolution sea ice-ocean model driven by atmospheric ensemble forecasts. The predictability of LKFs as measured by different metrics drops quickly, with predictability being almost completely lost after 4–8 days. In contrast, quantities such as sea ice concentration or the location of the ice edge retain high levels of predictability throughout the full 10-day forecast period. It is argued that the rapid error growth for LKFs is mainly due to the chaotic behaviour of the atmosphere associated with the low predictability of near surface wind divergence and vorticity; initial condition uncertainty for ice thickness is found to be of minor importance as long as LKFs are initialized at the right locations.

## Introduction

The rapid decline of Arctic sea ice has sparked considerable interest in predicting sea ice due to the projected increase in Arctic maritime operations1. As for traditional weather prediction2, skilful sea ice forecasts have the potential of saving lives, support emergency management, prevent economic losses from hazardous environmental conditions; and they may also create substantial financial revenue. Such forecasts are also needed to optimize shipping routes, which could be relevant for minimizing pollution.

Recent years have seen substantial improvements in our understanding of Arctic sea ice predictability on seasonal time scales3,4,5,6,7,8. Outcomes from the Sea Ice Outlook, launched in 2008 as an open forum for predicting September Arctic sea ice extent9, are a prominent example of the progress made. For marine operations, sea ice forecasts on shorter, daily (“weather”) time scales are also critical, especially when it comes to preparation and decision making on the risk of exceptional situations and for operational decision making. Very little is known, however, when it comes to short-term predictability of Arctic sea ice10,11, one reason being that previous predictability research was focussing mostly on the atmosphere and non-polar regions1.

When it comes to sea ice, past predictability studies have considered mostly the pan-Arctic sea ice extent, or the more user-relevant location of the ice edge4. However, sea ice deformation is of critical importance for decision-making in marine operations within the ice pack because deformation is related to sea ice pressure and leads. The initialization and short-term forecasting of deformation features has thus been identified as a promising as well as challenging research task12. Until recently, it was believed that classical sea ice models, employing viscous-plastic rheologies, would not be able to represent deformation-related aspects such as leads which localize along so-called Linear Kinematic Features (LKFs). Here, we determine the limits of predictability of Arctic sea ice on “weather time scales” (i.e. from 1–10 days), exploiting the fact that sea ice-ocean models at very high horizontal resolutions are actually capable of representing observed aspects related to sea ice deformation1,13,14,15. Furthermore, we compare outcomes for sea ice parameters traditionally considered, such as sea ice concentration, with the predictability of LKFs. Given the challenges that come with verifying LKFs due to their sharp and elongated structures, we also consider various verification methods that differ in the degree to which small errors in predicting LKF location are penalized.

This paper is organized as follows: The methods, including the experimental setup, the detection of LKF and the determination of potential predictability are described in Section 2. This is followed by a presentation of the results and a discussion and conclusions section.

## Methods

### Experiments

Our study is based on simulations with the Massachusetts Institute of Technology general circulation model (MITgcm)16,17 with a viscous-plastic sea ice component18. We use a standard viscous-plastic rheology19 in our simulations that relates the sea ice strain rates, concentration, and thickness to the internal stresses. The ice strength depends linearly on the ice thickness and exponentially on ice concentration and consequently the sea ice state is a controlling factor in the generation of LKFs. We use an Arctic-wide grid at a horizontal resolution of 4.5 km with 50 vertical layers in the ocean component, spanning one face of the cubed sphere, centered on the North Pole. The open boundaries in the Pacific and the Atlantic are prescribed using monthly mean data taken from a global simulation20. Atmospheric forcing fields are taken from either the ERA-Interim Reanalysis21 or from individual members of the ECMWF Ensemble Prediction System (EPS)22 as detailed below.

A control simulation from 2001 to 2015 forced with ERA-Interim reanalysis data provides initial conditions for all ensemble prediction experiments. Two sets of 15-member ensemble prediction experiments are considered: In the first set (AtmU hereafter), we account only for atmospheric forcing uncertainty associated with the chaotic atmospheric nature: Different sea ice-ocean model realizations are driven with different members of the ECMWF Ensemble Prediction System using the same initial conditions for the sea ice-ocean system. In the second set (AtmU + IcU hereafter), we additionally take into account the uncertainty of the initial conditions for sea ice. This is achieved by perturbing the initial sea ice thickness fields $$h$$ as follows:

$$h^{\prime} =h\cdot \exp (f\cdot \varepsilon )\,,$$
(1)

where $$h^{\prime}$$ is the perturbed sea ice thickness; $$f$$ is a perturbation strength parameter; and $$\varepsilon$$ is noise with zero mean and unit variance, derived from white noise that was spatially smoothed with a Gaussian filter with 50 km standard deviation. We choose $$f=0.1$$ such that the resulting perturbations in Arctic sea ice thickness amount to a maximum of about 0.5 m (see Fig. 1a,b for an example of the perturbed field for a particular day)—an estimate which is consistent with the observational uncertainty for sea ice thickness23. Note that by perturbing thickness only, we leave the initial locations of LKFs essentially unchanged.

To reduce the impact of sampling uncertainty, each of the two ensemble experiments was conducted for six different cases by initializing forecasts on the first and fifteenth of the months February, March, and April in 2005. We focus our analysis on February, March and April, when the LKF density distributions are relatively homogenous, noting that LKF predictability might be different in other seasons. We focus on the months around the maximum ice extent because (i) a continuum sea ice model with a 4.5-km grid spacing is thought to best represent a dense ice pack with large floes, (ii) a larger area exhibits LKFs in winter and spring compared to summer and autumn, and (iii) this avoids difficulties in terms of interpretation that might arise from the seasonal cycle in LKF characteristics. The year 2005 was chosen because at that time the operational EPS was technically very similar to the ERA-Interim reanalysis system. Note that for each of the configurations and each of the starting dates, fifteen ensemble members were integrated for ten days (i.e. 10-day forecasts).

We follow a “perfect-model” approach in the sense that we do not quantify forecast errors with respect to actual observations, but study how differences between individual ensemble members evolve, with one of the ensemble members being considered the “truth”. Our analysis thus provides an estimate of potential predictability, which typically is regarded as an upper bound for the skill that can be achieved in real operational forecasts4.

### LKF detection

Our analysis requires an algorithm that diagnoses linear kinematic features (LKFs) from model output. It is well established to derive LKFs from sea ice deformation fields. The LKFs, indentified in this way, are highly correlated with the spatial pattens of leads24. Sea ice shear and divergence also correspond to leads25. Various methods have been proposed to detect geomorphological features in sea ice over the past two decades, but there is no unique detection algorithm26,27,28,29. The most straightforward approach is to use a fixed threshold for distinguishing linear (kinematic) features from background fields24,26,27,30. Obviously, the result depends on the threshold value. Given that zones of shear as well as convergence and divergence of sea ice are concentrated along LKFs, we opt for a simple method that is based solely on the total deformation rate of sea ice: Regions with sea ice concentration below 80% are excluded from the analysis. After taking the logarithm of total deformation, we subtract the local background which is determined by applying a 2D Gaussian filter with a standard deviation of five grid cells (20 km). Grid cells are considered to belong to an LKF if total deformation exceeds the background deformation by at least a factor of $${10}^{0.2}\approx 1.6$$ (Fig. 1d). Note that this approach takes into account that atmospheric winds and coastal geometry impose spatial variations in background deformation of the sea ice. The character of sea ice deformation and the diagnosed LKFs can be inferred from Fig. 1c and d for 1 February 2005.

### Measuring potential predictability

To quantify the difference between pairs of ensemble members (i.e. “forecast error”), we use the simple spatial Pearson correlation as well as the more sophisticated Modified Hausdorff Distance [MHD31]. Applied to a pair of LKF patterns, the MHD determines the average distance between any LKF grid cell to the closest such grid cell in the other pattern. In contrast to the spatial correlation, which strongly penalizes even small errors in the predicted location of the LKFs, the MHD can discriminate between small and large differences in LKF location. Given that MHD is constrained to binary fields, we use it to determine the predictability of LKFs only. In contrast, spatial correlations are determined for LKFs as well as several continuous fields including sea ice total deformation anomaly, sea ice concentration anomaly, and the divergence of near-surface winds.

To determine potential predictability PP, we relate the above metrics, averaged over all possible pairs of ensemble members and cases, to their climatological saturation levels. For the spatial correlation this can be expressed as follows:

$${{\rm{PP}}}_{\rho }=1-\frac{1-{\rho }_{E}}{1-{\rho }_{C}},$$
(2)

where $${\rho }_{E}$$ denotes the mean spatial correlation between ensemble members, and $${\rho }_{C}$$ is the climatological spatial correlation, that is, the spatial correlation between pairs of states from the same time of the year but taken from different years of the control run, averaged over all possible pairs of the years 2001 to 2015 from February to April. Similarly, potential predictability for MHD is determined as follows:

$${{\rm{PP}}}_{{\rm{MHD}}}=1-\frac{{{\rm{MHD}}}_{E}}{{{\rm{MHD}}}_{C}}\,\mathrm{.}$$
(3)

Equations (2) and (3) are consistent with skill scores used elsewhere by the forecast verification community32

### Data Availability

The datasets generated during the current study are available from the corresponding author on request.

## Results

To provide a more intuitive insight into the predictability of LKFs, we first consider an individual forecast case. Figure 2 shows 1-day and 4-day predictions of LKFs, initialized on 1 February 2005, from two ensemble members with atmospheric uncertainty only (AtmU). There is evidence for relatively high potential predictability of LKFs during the first day of the forecast (Fig. 2a). In weather prediction this time range spans what is usually referred to as nowcasting and early short-range forecasting. Visual inspection of Fig. 2b reveals a much larger difference between the two ensemble members 4 days into the forecast, suggesting that the potential predictability in the early medium-range, at least for some of the simulated LKFs, is already quite small.

More quantitative estimates of predictability are provided by combining all ensemble experiments for the 6 different initialization times (Fig. 3). From the average spatial correlation coefficients for the pan-Arctic area (north of 54°N) a rapid drop in skill with increasing lead time can be inferred (Fig. 3a), with correlation coefficients dropping below 0.6 (i.e. less than 36% of the variance explained) by day 2; by day 4 the correlation has dropped further to about 0.4. A comparison between the experiment with atmospheric forcing uncertainty only (AtmU) and with both atmospheric and initial sea ice uncertainty (AtmU-IcU) shows that it is primarily atmospheric uncertainty that is responsible for the rapid error growth. This is especially true for lead times beyond 2 days or so.

Comparing LKF forecasts with a metric like the spatial correlation coefficient, which is widely used in forecast verification, has its limitation, because LKF forecasts represent binary information (either there is an LKF or not), and all location errors, whether small or large, are penalized alike. It is therefore useful to also consider a metric such as the MHD that is better suited for binary information and that can discriminate between small and large displacement errors. The MHD for all forecast cases along with the corresponding potential predictability is shown in Fig. 3b for the pan-Arctic region. Evidently, the MHD-based potential predictability of LKFs remains above 0.6 for longer than that based on the spatial correlation coefficients. Even for longer lead times up to 10 days the MHD metric indicates some potential skill in predicting LKFs. As for the spatial correlation coefficient, atmospheric uncertainty dominates over initial sea ice uncertainty, especially beyond 3 days into the forecast.

So far, the discussion has focussed on the entire Arctic. However, it could be argued that the nature and hence predictability of LKFs in the interior of the Arctic, where relatively thick ice prevails and the effect of surrounding coastlines decreases, is different from regions near the sea ice margins. Figure 3c shows the MHD and corresponding potential predictability for the central Arctic, which here is simply defined as being north of 80° N and between 120°E–240°E. It turns out that the potential predictability in the central Arctic is slightly larger than for the pan-Arctic region, which we attribute to the presence of thicker, hence stronger and less mobile sea ice. At the same time, however, the difference between the different ensemble experiments (thin lines in Fig. 3) is larger and noisier. One explanation for this is that fewer LKFs are found in the interior of the Arctic (Fig. 1) compared to the sea ice margins [e.g.13, their Fig. 2], so that the metrics are subject to more sampling variability; thus a larger number of cases are required to obtain more robust statistics.

It is informative to put the predictability of LKFs into the context of that of other variables. The potential predictability for (logarithmized) pan-Arctic sea ice deformation anomalies as determined by means of spatial correlation can be inferred from Fig. 3d. Here anomalies instead of full fields were considered in order to avoid that the model’s ability, to simulate climatological patterns of sea ice deformation, contributes to the correlation (by analogy with the widely used 500 hPa geopotential height anomaly correlation). Evidently, the sea ice deformation anomalies are much more predictable than LKFs, with potential predictability going down to 0.6 by day 10 only. This is consistent with the fact that the full sea ice deformation fields include larger spatial scales compared to LKFs (Fig. 1c and d). Note in this context that LKFs are obtained from sea ice deformation by effectively applying a spatial high-pass filter. Interestingly, when the full deformation fields are considered instead of LKFs, initial sea ice perturbations affect the predictions compared to atmospheric uncertainty even less. This suggests that the prediction of larger-scale sea ice aspects is relatively insensitive to perturbations to the initial sea ice thickness at the considered time scale. For anomalies in sea ice concentration very similar results are found (Fig. 3e) with prospects for skilful forecasts up to 10 days and beyond.

To provide an additional point of reference, we computed the pairwise Integrated Ice Edge Error [IIEE4] for our ensembles. Averaged over the experiments AtmU and AtmU + IcU and normalised with the climatological IIEE, the corresponding predictability after 10 days is 0.72, which is marginally higher than the predictability of sea-ice concentration anomalies of 0.69 (Fig. 3e).

Atmospheric winds are the most important driver of sea ice changes on daily to subseasonal time scales. It is instructive, therefore, to compare the predictability of sea ice with that of the atmospheric flow. The potential predictiability of the near surface wind divergence, a parameter that is of particular relevance for the generation of leads, shows that the skill for the atmosphere and LKFs is comparable up to about 4 days into the future (Fig. 3f). Thereafter, the predictability of LKFs is larger than that of the divergent wind, suggesting that sea ice provides additional memory that translates into predictability. In general, however, a comparison of atmospheric and sea ice predictability is difficult to carry out due to the strong dependence on the exact variable, metric, and spatial scale being considered [Fig. 3a,d and e;33,34].

## Discussion and Conclusions

Experiments with a high-resolution sea ice-ocean model, forced with weather forecasts from the ECWMF ensemble prediction system, have been used to determine the potential predictability of Arctic sea ice on “weather time scales” (1–10 day forecasts), a time range for which skilful forecasts are of critical importance for marine stakeholders in tactical decision making. Our study augments previous work, most of which focussed on longer-term seasonal and interannual prediction3. Our results highlight that the predictability of Arctic sea ice on short time scales is critically dependent on the variable being considered. Sea ice concentration anomalies, for example, show high levels of predictability throughout the whole forecast range considered in this study, whereas linear kinematic features (LKFs) such as leads are much less well predictable, especially beyond the first 2–4 days of the forecast. In addition, a skillful prediction of actual LKFs is a challenging task that requires a sophisticated forecasting system with new data assimilation capabilities to obtain accurate initial conditions for sea-ice failure12.

Our results suggest that atmospheric uncertainty is more important in generating sea ice forecast error than initial sea ice uncertainty. In this context, we have conducted additional experiments in which we varied the parameters of the initial perturbations. From these experiments we conclude that smaller-scale perturbations introduce spurious additional LKFs, and that the chosen set of parameters appears appropriate. Still, it cannot be excluded that our initial sea ice perturbations are sub-optimal, in the way they reflect real uncertainties and in the way they project onto possible directions of growing forecast error. However, our results are consistent with the notion that sea ice dynamics are primarily determined by atmospheric winds. The higher predictability of wind divergence in the central Arctic (Fig. 3f) might also contribute to higher predictability of sea ice in the central Arctic.

Our study shows that the verification of forecasts of small-scale sea ice features (including determining exact limits of predictability) poses challenges. Traditional measures such as the anomaly correlation coefficient, for example, are quite sensitive to small errors in location. Hence, other less well-established methods such as Modified Hausdorff Distance need to be exploited as well when it comes to user-relevant forecast verification. Future work on sea ice verification can benefit from the expertise gained from the verification of other sharp features such as atmospheric fronts35. Furthermore, estimates of deterministic and probabilistic predictability may differ. Therefore, deterministic verification of forecasts, as done in this study essentially by comparing pairs of ensemble members, should be augmented by probabilistic verification where ensemble-derived probabilities are taken into account.

Individual LKFs may not be the only important predictands for shipping or marine operations. Integral properties such as ridge and lead density may also be relevant and probably easier to predict but would require a different type of analysis and potentially different model parameterizations. For example, a ridge density parameterization36 or an ice thickness distribution37,38 open new fields of application in the context of predictability. With our current model configuration, these questions cannot be addressed.

Simulated small-scale sea ice features depend heavily on the rheology used for the representation of sea ice dynamics. It seems natural, therefore, to ask how model dependent the results from this potential predictability study are. The high intermittency of sea ice deformation found in temporal scaling analyses of drift observations could not be reproduced with viscous-plastic sea ice models at coarse grid resolutions39 and only partly for models at higher grid resolution like the one used here15. In this light, the potential predictability reported here may be actually slightly overestimated. Repeating our experiments with a new generation of rheologies that produce more intermittency in models40,41,42 may provide enhanced estimates of Arctic sea ice predictability on weather time scales.

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Rabatel, M., Rampal, P., Carrassi, A., Bertino, L. & Jones, C. K. R. T. Impact of rheology on probabilistic forecasts of sea ice trajectories: application for search and rescue operations in the Arctic. The Cryosphere, 12, 935–953 (2018).

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Wang, Q., Danilov, S., Jung, T., Kaleschke, L. & Wernecke, A. Sea ice leads in the Arctic Ocean: Model assessment, interannual variability and trends. Geophysical Research Letters 43, 7019–7027 (2016).

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Spreen, G., Kwok, R., Menemenlis, D. & Nguyen, A. T. Sea ice deformation in a coupled ocean–sea ice model and in satellite remote sensing data. The Cryosphere 11, 1553–1573 (2017).

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Hutter, N., Losch, M. & Menemenlis, D. Scaling properties of Arctic sea ice deformation in a high-resolution viscous-plastic sea ice model and in satellite observations. Journal of Geophysical Research 123, 672–687 (2018).

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Marshall, J., Adcroft, A., Hill, C., Perelman, L. & Heisey, C. A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. Journal of Geophysical Research: Oceans 102, 5753–5766 (1997).

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Losch, M., Menemenlis, D., Campin, J.-M., Heimbach, P. & Hill, C. On the formulation of sea ice models. Part 1: Effects of different solver implementations and parameterizations. Ocean Modelling 33, 129–144 (2010).

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## Acknowledgements

This study has been supported through the AWI strategy fund. Acknowledgement is made for the use of ECMWF’s computing and archive facilities under the special project “Potential sea ice predictability with a high resolution Arctic sea ice-ocean model”. H.F.G. acknowledges the financial support by the Federal Ministry of Education and Research of Germany in the framework of SSIP (grant 01LN1701A). T.J. acknowledges the funding from the European Union’s Horizon 2020 Research & Innovation programme through grant agreement No. 727862 APPLICATE. The content of the paper is the sole responsibility of the authors and it does not represent the opinion of the European Commission, and the Commission is not responsible for any use that might be made of information contained. We are grateful for comments by Sergey Danilov.

## Author information

H.F.G., M.L. and T.J. conceived the research, M.M.A. and H.F.G. developed the methods, N.H. contributed to the methods with results and experience from his previous work, M.L. and M.M.A. configured the MITgcm setup, M.M.A. conducted the simulations, analyzed the results, and prepared the figures. All authors discussed the results and contributed to the writing of the manuscript.

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### Competing Interests

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• ### Data fusion and data assimilation of ice thickness observations using a regularisation framework

• , K. Andrea Scott
•  & David A. Clausi

Tellus A: Dynamic Meteorology and Oceanography (2019)

• ### Leads and ridges in Arctic sea ice from RGPS data and a new tracking algorithm

• Nils Hutter
• , Lorenzo Zampieri
•  & Martin Losch

The Cryosphere (2019)

• ### Medium-range predictability of early summer sea ice thickness distribution in the East Siberian Sea based on the TOPAZ4 ice–ocean data assimilation system

• Takuya Nakanowatari
• , Jun Inoue
• , Kazutoshi Sato
• , Laurent Bertino
• , Jiping Xie
• , Mio Matsueda
• , Akio Yamagami
• , Takeshi Sugimura
• , Hironori Yabuki
•  & Natsuhiko Otsuka

The Cryosphere (2018)

• ### Bright Prospects for Arctic Sea Ice Prediction on Subseasonal Time Scales

• Lorenzo Zampieri
• , Helge F. Goessling
•  & Thomas Jung

Geophysical Research Letters (2018)


text:
## Anyone know how to this?

• I'm working on your other question......i'll upload the detailed solution in sometime...plz dont rate anyone else.....thank you (rate atleast 3 stars)
• Anonymous commented

• By definition, the resonance frequency is, in this context, the frequency at which the impedance of the equivalent impedance is real.

The branch with the R and L has an impedance of: R+j?L

The branch with the C has an impedance of: 1j?C

Now, the equivalent impedance is just:

(R+j?L)||1j?C

The resonance frequency is the frequency for which the above is purely real, i.e., set the imaginary part of the above equal to zero and solve for the frequency.

When you do this correctly, you should get

or equivalently, as the Wikipedia link Renan gives:

?0=1LC?(RL)2??????????


text:
# 7 Reporting and Results

Scoring of Dynamic Learning Maps® (DLM®) assessments is based on linkage levels mastered across all assessed Essential Elements (EEs). Performance levels describe overall achievement in each subject based on the total linkage levels mastered, with students achieving At Target or Advanced demonstrating proficiency. This chapter summarizes assessment results for the DLM Alternate Assessment System and presents evidence of their appropriateness.

The chapter begins by presenting the number of students who participated in the assessment in 2021–2022, followed by student performance data. Performance data are described in terms of the percentage of students achieving at each performance level both overall and by student group. The chapter then describes how students demonstrated linkage level mastery from among three scoring rules as well as the distribution of the highest linkage level mastered by each student and preliminary evidence about how educators define skill mastery. Because writing testlets are scored by the test administrator, evidence of interrater reliability of writing sample scoring is evaluated. The chapter then summarizes evidence that students who meet the DLM alternate achievement standards are prepared to pursue postsecondary opportunities. Finally, the chapter describes the data files, score reports, guides for interpreting score reports, and quality-control procedures for data files and score reports.

## 7.1 Student Participation

During spring 2022, assessments were administered to 88,404 students in 14 states. Counts of students tested in each state are displayed in Table 7.1. The assessments were administered by 23,210 educators in 12,150 schools and 3,844 school districts. A total of 1,423,995 test sessions were administered during the spring assessment window. One test session is one testlet taken by one student. Only test sessions that were complete at the close of the spring assessment window counted toward the total sessions.

Table 7.1: Student Participation by State (N = 88,404)
State Students (n)
Illinois 13,917
Maryland   5,030
New Hampshire      630
New Jersey 11,266
New Mexico   2,035
New York 18,219
Oklahoma   5,292
Pennsylvania 16,650
Rhode Island      905
Utah   3,944
West Virginia   1,445
Wisconsin   4,460

Table 7.2 summarizes the number of students assessed in each grade. In grades 3–8, over 11,140 students participated in each grade. In high school, the largest number of students participated in grade 11, and the smallest number participated in grade 12. The differences in high school grade-level participation can be traced to differing state-level policies about the grade(s) in which students are assessed.

Table 7.2: Student Participation by Grade (N = 88,404)
3 11,145
4 11,435
5 11,528
6 11,699
7 12,101
8 12,120
9   5,921
10   2,907
11   8,720
12      828

Table 7.3 summarizes the demographic characteristics of the students who participated in the spring 2022 administration. The distribution of students across the different subgroups was fairly consistent with prior years’ distributions. The majority of participants were male (68%) and white (56%). About 7% of students were monitored or eligible for English learning services.

Table 7.3: Demographic Characteristics of Participants (N = 88,404)
Subgroup n %
Gender
Male 59,978 67.8
Female 28,349 32.1
Nonbinary/undesignated       77   0.1
Race
White 49,713 56.2
African American 18,967 21.5
Two or more races 11,719 13.3
Asian   5,055   5.7
American Indian   2,351   2.7
Native Hawaiian or Pacific Islander      421   0.5
Hispanic ethnicity
Non-Hispanic 66,401 75.1
Hispanic 22,003 24.9
English learning (EL) participation
Not EL eligible or monitored 82,497 93.3
EL eligible or monitored   5,907   6.7

In addition to the spring assessment window, instructionally embedded assessments are also made available for educators to optionally administer to students during the year. Results from the instructionally embedded assessments do not contribute to final summative scoring but can be used to guide instructional decision-making. Table 7.4 summarizes the number of students who completed at least one instructionally embedded assessment by state. A total of 2,418 students in 8 states took at least one instructionally embedded testlet during the 2021–2022 academic year.

Table 7.4: Students Completing Instructionally Embedded Testlets by State (N = 2,418)
State n
Maryland 1,358
New Jersey    583
New Mexico        7
New York      74
Oklahoma    295
Utah      12
West Virginia        1
Note. Maryland required administration of instructionally embedded assessments during fall 2021.

Table 7.5 summarizes the number of instructionally embedded testlets taken in ELA and mathematics. Across all states, students took 19,171 ELA testlets and 19,069 mathematics testlets during the instructionally embedded window.

Table 7.5: Number of Instructionally Embedded Testlets by Grade and Subject (N = 38,240)
3      951      791
4   1,346   1,308
5   1,589   1,512
6   1,577   1,236
7   2,287   1,970
8   3,775   3,341
9      308      435
10       62       75
11   5,175   6,231
12   2,101   2,170
Total 19,171 19,069

## 7.2 Student Performance

Student performance on DLM assessments is interpreted using cut points, For a description of the standard setting process used to determine the cut points, see Chapter 6 of this manual. which describe student achievement using four performance levels. A student’s performance level is determined based on the total number of linkage levels mastered across the assessed EEs.

For the spring 2022 administration, student performance was reported using four performance levels:

• The student demonstrates Emerging understanding of and ability to apply content knowledge and skills represented by the EEs.
• The student’s understanding of and ability to apply targeted content knowledge and skills represented by the EEs is Approaching the Target.
• The student’s understanding of and ability to apply content knowledge and skills represented by the EEs is At Target. This performance level is considered to be meeting achievement expectations.
• The student demonstrates Advanced understanding of and ability to apply targeted content knowledge and skills represented by the EEs.

### 7.2.1 Overall Performance

Table 7.6 reports the percentage of students achieving at each performance level from the spring 2022 administration for ELA and mathematics. In ELA, the percentage of students who achieved at the At Target or Advanced levels (i.e., proficient) ranged from approximately 18% to 30%. In mathematics, the percentage of students meeting or exceeding At Target expectations ranged from approximately 11% to 37%.

Table 7.6: Percentage of Students by Grade and Performance Level
English language arts
3 (n = 11,134) 60.6 15.2 22.5   1.8 24.2
4 (n = 11,425) 60.7 21.5 16.1   1.7 17.8
5 (n = 11,510) 52.8 17.1 24.4   5.7 30.1
6 (n = 11,687) 48.8 26.4 19.7   5.2 24.8
7 (n = 12,080) 41.9 29.2 24.0   4.9 28.9
8 (n = 12,083) 43.1 31.4 25.0   0.5 25.5
9 (n = 5,907) 37.7 36.5 22.4   3.4 25.8
10 (n = 2,903) 36.9 38.9 23.7   0.4 24.1
11 (n = 8,681) 34.4 35.7 25.4   4.4 29.9
12 (n = 822) 42.1 34.2 20.3   3.4 23.7
Mathematics
3 (n = 11,103) 63.7 12.8 14.5   9.0 23.5
4 (n = 11,400) 51.5 11.8 25.1 11.6 36.7
5 (n = 11,481) 49.3 22.6 14.0 14.2 28.2
6 (n = 11,653) 58.9 22.6 11.2   7.3 18.5
7 (n = 12,057) 66.6 19.5   7.9   6.0 14.0
8 (n = 12,086) 57.7 30.6   6.9   4.8 11.7
9 (n = 5,894) 50.4 27.6 17.8   4.2 22.0
10 (n = 2,899) 63.9 25.1   9.7   1.3 11.0
11 (n = 8,668) 50.9 22.2 26.0   0.9 27.0
12 (n = 811) 59.3 18.1 22.1   0.5 22.6

### 7.2.2 Subgroup Performance

Data collection for DLM assessments includes demographic data on gender, race, ethnicity, and English learning status. Table 7.7 and Table 7.8 summarize the disaggregated frequency distributions for ELA and mathematics, respectively, collapsed across all assessed grade levels. Although state education agencies each have their own rules for minimum student counts needed to support public reporting of results, small counts are not suppressed here because results are aggregated across states and individual students cannot be identified.

Table 7.7: ELA Performance Level Distributions by Demographic Subgroup (N = 88,232)
Emerging
Approaching
At Target
At Target +
Subgroup n % n % n % n % n %
Gender
Male 28,843 48.2 15,590 26.0 13,417 22.4 2,011 3.4 15,428 25.8
Female 13,480 47.6   7,583 26.8   6,316 22.3    915 3.2   7,231 25.6
Nonbinary/undesignated       40 51.9       17 22.1       17 22.1        3 3.9       20 26.0
Race
White 23,433 47.2 12,859 25.9 11,549 23.3 1,773 3.6 13,322 26.9
African American   8,856 46.8   5,278 27.9   4,195 22.2    598 3.2   4,793 25.3
Two or more races   5,890 50.3   3,098 26.5   2,372 20.3    339 2.9   2,711 23.2
Asian   2,891 57.3   1,134 22.5      886 17.5    138 2.7   1,024 20.3
American Indian      966 41.1      686 29.2      626 26.6      71 3.0      697 29.7
Native Hawaiian or Pacific Islander      209 50.1       99 23.7      100 24.0        9 2.2      109 26.1
Alaska Native      118 66.7       36 20.3       22 12.4        1 0.6       23 13.0
Hispanic ethnicity
Non-Hispanic 31,711 47.8 17,307 26.1 15,080 22.8 2,178 3.3 17,258 26.0
Hispanic 10,652 48.5   5,883 26.8   4,670 21.3    751 3.4   5,421 24.7
English learning (EL) participation
Not EL eligible or monitored 39,519 48.0 21,513 26.1 18,509 22.5 2,790 3.4 21,299 25.9
EL eligible or monitored   2,844 48.2   1,677 28.4   1,241 21.0    139 2.4   1,380 23.4
Table 7.8: Mathematics Performance Level Distributions by Demographic Subgroup (N = 88,052)
Emerging
Approaching
At Target
At Target +
Subgroup n % n % n % n % n %
Gender
Male 33,317 55.8 12,291 20.6   9,122 15.3 4,980   8.3 14,102 23.6
Female 16,801 59.4   6,164 21.8   3,797 13.4 1,503   5.3   5,300 18.8
Nonbinary/undesignated       56 72.7       12 15.6          8 10.4        1   1.3          9 11.7
Race
White 27,964 56.4 10,484 21.2   7,442 15.0 3,649   7.4 11,091 22.4
African American 10,647 56.4   4,044 21.4   2,739 14.5 1,455   7.7   4,194 22.2
Two or more races   6,924 59.3   2,445 21.0   1,641 14.1    658   5.6   2,299 19.7
Asian   3,124 62.3      815 16.2      641 12.8    438   8.7   1,079 21.5
American Indian   1,134 48.4      561 23.9      396 16.9    253 10.8      649 27.7
Native Hawaiian or Pacific Islander      245 58.3       87 20.7       59 14.0      29   6.9       88 21.0
Alaska Native      136 76.4       31 17.4          9   5.1        2   1.1       11   6.2
Hispanic ethnicity
Non-Hispanic 37,750 57.1 13,976 21.1   9,701 14.7 4,714   7.1 14,415 21.8
Hispanic 12,424 56.7   4,491 20.5   3,226 14.7 1,770   8.1   4,996 22.8
English learning (EL) participation
Not EL eligible or monitored 46,958 57.2 17,224 21.0 12,035 14.6 5,945   7.2 17,980 21.9
EL eligible or monitored   3,216 54.6   1,243 21.1      892 15.1    539   9.2   1,431 24.3

## 7.3 Mastery Results

As described above, the student performance levels are determined by applying cut points to the total number of linkage levels mastered in each subject. In this section, we summarize student mastery of assessed EEs and linkage levels, including how students demonstrated mastery from among three scoring rules and the highest linkage level students tended to master.

### 7.3.1 Mastery Status Assignment

Take, for example, a student who tested only on the Target linkage level of an EE. If the student demonstrated mastery of the Target linkage level, as defined by the .80 posterior probability of mastery cutoff or the 80% correct rule, then all linkage levels below and including the Target level would be categorized as mastered. If the student did not demonstrate mastery on the tested Target linkage level, then mastery would be assigned at two linkage levels below the tested linkage level (i.e., the Distal Precursor), rather than showing no evidence of mastery at all. The two-down rule is based on linkage level ordering evidence and the underlying learning map structure (see Chapter 2 of this manual), and it is designed to not penalize students who test at higher linkage levels. That is, in the example above, the student is assigned mastery of the Distal Precursor, rather than being assigned no mastery for the EE.

To evaluate the degree to which each mastery assignment rule contributed to students’ linkage level mastery status during the 2021–2022 administration of DLM assessments, the percentage of mastery statuses obtained by each scoring rule was calculated, as shown in Figure 7.1. Posterior probability was given first priority. That is, if multiple scoring rules agreed on the highest linkage level mastered within an EE (e.g., the posterior probability and 80% correct both indicate the Target linkage level as the highest mastered), the mastery status was counted as obtained via the posterior probability. If mastery was not demonstrated by meeting the posterior probability threshold, the 80% scoring rule was imposed, followed by the two-down rule. This means that EEs that were assessed by a student at the lowest two linkage levels (i.e., Initial Precursor and Distal Precursor) are never categorized as having mastery assigned by the two-down rule. This is because the student would either master the assessed linkage level and have the EE counted under the posterior probability or 80% correct scoring rule, or all three scoring rules would agree on the score (i.e., no evidence of mastery), in which case preference is given to the posterior probability. Across grades and subjects, approximately 65%–80% of mastered linkage levels were derived from the posterior probability obtained from the modeling procedure. Approximately <1%–8% of linkage levels were assigned mastery status by the percentage correct rule. The remaining approximately 15%–32% of mastered linkage levels were determined by the minimum mastery, or two-down rule.

Because correct responses to all items measuring the linkage level are often necessary to achieve a posterior probability above the .80 threshold, the percentage correct rule overlapped considerably (but was second in priority) with the posterior probabilities. The percentage correct rule did, however, provide mastery status in those instances where correctly responding to all or most items still resulted in a posterior probability below the mastery threshold. The agreement between these two methods was quantified by examining the rate of agreement between the highest linkage level mastered for each EE for each student. For the 2021–2022 operational year, the rate of agreement between the two methods was 87%. However, in instances in which the two methods disagreed, the posterior probability method indicated a higher level of mastery (and therefore was implemented for scoring) in 68% of cases. Thus, in some instances, the posterior probabilities allowed students to demonstrate mastery when the percentage correct was lower than 80% (e.g., a student completed a four-item testlet and answered three of four items correctly).

Figure 7.2 displays the percentage of students who mastered each linkage level as the highest linkage level across all assessed EEs for ELA and mathematics. For example, across all grade 3 mathematics EEs, the Initial Precursor level was the highest level that students mastered 44% of the time. The percentage of students who mastered the Target or Successor linkage level ranged from approximately 42% to 55% in ELA and ranged from approximately 16% to 45% in mathematics.

This section describes additional scoring evidence for DLM assessments, including preliminary evidence from a pilot survey collecting information on how educators define skill mastery, interrater reliability of test administrator scoring of student writing samples, and evidence that students who achieve At Target on DLM assessments are on track to pursue postsecondary opportunities.

### 7.4.1 Pilot Survey Defining Student Mastery

In 2019, DLM staff piloted a survey to obtain information about how educators view skill mastery for students with significant cognitive disabilities who take DLM assessments. The survey was completed by 95 educators across six states and included an open-ended item asking educators to describe how they define student mastery. The survey responses revealed that many educators defined mastery as 75%–80% successful trials on the skills, which is common in special education, but other educators shared ideas around consistent skill demonstration over a longer period of time, generalization and transfer, independence and speed of demonstration, and students’ ability to explain the concept . While these definitions differ from how mastery is determined on DLM assessments, they provide some insight as to how educators may interpret mastery on DLM assessments.

### 7.4.2 Writing Sample Scoring

#### 7.4.2.1 Preparing the Data

Educators administer writing testlets at two levels: emergent and conventional. Emergent testlets measure nodes at the Initial Precursor and Distal Precursor levels, while conventional testlets measure nodes at the Proximal Precursor, Target, and Successor levels. The number of items that evaluate the writing sample per grade-level testlet is summarized in Table 7.9.

Testlets include one to six items evaluating the sample, administered as either multiple-choice or multiple-select multiple-choice items. Because each answer option could correspond to a unique linkage level and/or EE, writing items are dichotomously scored at the option level. Each item, which includes four to nine answer options, was scored as a separate writing item. For this reason, writing items are referred to as writing tasks in the following sections, and the options were scored as individual items. The dichotomous option responses (i.e., each scored as an item) were the basis for the evaluation of interrater agreement.

Table 7.9: Number of Items That Evaluate the Writing Product per Testlet by Grade
3 * 3
4 1 4
5 * 2
6 * 4
7 1 4
8 * 4
9 1 8
10 1 8
11 1 8
Note. Items varied slightly by blueprint model; the maximum number of items per testlet is reported here.
* The testlet at this grade included only items evaluating the writing process, with no evaluation of the sample.

The scoring process for DLM writing testlets is as follows. Data are extracted from the database that houses all DLM data. For writing items, the response-option identifiers are treated as item identifiers so that each response option can be scored as correct or incorrect for the EE and linkage level it measures. Also, response-option dependencies are built in, based on scoring direction provided by the ELA test development team, to score as correct response options that are subsumed under other correct response options. Once the data structure has been transformed and response-option dependencies are accounted for, the writing data are combined with all other data to be included in the scoring process. For more information on the scoring process for linkage level mastery, see Section 7.3.2.

#### 7.4.2.2 Recruitment

Recruitment for the evaluation of interrater agreement of writing samples included the submission of student writing samples and direct recruitment of educators to serve as raters.

##### 7.4.2.2.1 Samples

Educators are annually asked to submit student writing samples. Requested submissions included papers that students used during testlet administration, copies of student writing samples, or photographs of student writing samples. To allow the sample to be matched with test administrator response data, each sample was submitted in the online system to enable direct matching to the administered testlet, including original test administrator responses.

In 2022, DLM staff evaluated interrater reliability for writing samples collected during the 2019–2020 and 2020–2021 administrations. A total of 1,914 student writing samples were submitted from districts in 12 states across the spring assessment windows in 2020 and 2021. In several grades, the emergent writing testlet does not include any tasks that evaluate the writing sample (as shown in Table 7.9); therefore, emergent samples submitted for these grades were not included in the interrater reliability analysis (e.g., Grade 3 emergent writing samples). To lessen the burden on the raters, we sampled approximately 20 writing samples from each grade and writing level (excluding the emergent levels in certain grades described above). This resulted in the assignment of 309 writing samples to raters for evaluation of interrater agreement.

##### 7.4.2.2.2 Raters

Recruited educators were required to have experience administering and rating DLM writing testlets to ensure they had already completed required training and were familiar with how to score the writing samples. Participants were screened and selected using the Accessible Teaching, Learning, and Assessment Systems (ATLAS) recruitment database platform in MemberClicks. Volunteers complete a demographic survey to be considered for ATLAS events, and this information is stored in the database for recruitment purposes. In total, 18 were selected to participate.

Raters had a range of teaching experience, as indicated in Table 7.10. Most had taught ELA and/or students with the most significant cognitive disabilities for at least 6 years. Furthermore, eight raters (44%) reported experience as DLM external reviewers (see Chapter 3 for a description of external review).

Table 7.10: Raters’ Teaching Experience (N = 18)
1–5 years
6–10 years
> 10 years
Teaching experience n % n % n %
English language arts 2 11.1 2 11.1 14 77.8
Students with significant cognitive disabilities 1 5.6 4 22.2 13 72.2

Demographic information for participants in this study is summarized in Table 7.11. Participating raters were mostly female (78%), white (78%), and non-Hispanic/Latino (89%). Raters came from a variety of teaching settings.

Table 7.11: Raters’ Demographic Information (N = 18)
Subgroup n %
Gender
Female 14 77.8
Male   4 22.2
Race
White 14 77.8
Black or African American   2 11.1
American Indian   1   5.6
Asian   1   5.6
Hispanic ethnicity
Non-Hispanic/Latino 16 88.9
Hispanic/Latino   2 11.1
Teaching setting
Rural   7 38.9
Urban   6 33.3
Suburban   5 27.8

#### 7.4.2.3 Sample Ratings

All ratings occurred during a virtual event in July 2022. Raters were provided with deidentified PDF versions of student writing samples via an online Moodle course, which they could no longer access following the completion of ratings. They were also provided a link to a Qualtrics survey that included the writing tasks corresponding to the grade and level (i.e., emerging or conventional) of the assigned writing sample. Raters submitted all ratings online.

Writing samples were assigned to raters in batches of 13–24, using a partially crossed matrix design to assign each sample to a total of three raters. Thus, educators rated between 60 and 83 writing samples. Table 7.12 summarizes the number of samples that were rated at each grade and level.

Table 7.12: Student Writing Samples With Ratings by Grade (N = 309)
Number of writing samples
Grade Emergent Conventional Total number of samples
3 *   28   28
4 31   33   64
5 *   19   19
6 *   24   24
7 24   22   46
8 *   25   25
9 14   13   27
10 13   15   28
11 23   25   48
Total 105 204 309
* The testlet at this grade included only items evaluating the writing process, with no evaluation of the sample.

Ratings submitted in Qualtrics were combined with the original spring 2020 and spring 2021 student data from when the writing sample was rated by the student’s educator, resulting in four ratings for each of the 309 student writing samples.

#### 7.4.2.4 Interrater Reliability

Because each writing sample was evaluated by multiple and different raters, interrater reliability was summarized by Fleiss’s kappa and intraclass correlation (ICC) values. The purpose of Fleiss’s kappa is to provide a measure of absolute agreement across two or more raters. Fleiss’s kappa is defined as

$$$\kappa=\frac{\bar{P} - \bar{P_e}}{1 - \bar{P_e}} \tag{7.1}$$$

where the denominator gives the degree of absolute agreement attainable above chance, and the numerator gives the degree of absolute agreement actually achieved above chance.

The purpose of the ICC is to provide a means for measuring rater agreement and consistency. For interrater reliability studies, rater agreement is of most interest. For this study, a one-way, random-effects model using the average kappa rating was selected because each writing sample was rated by a rater who was randomly selected from the pool of available raters. Using this model, only absolute agreement is measured by the ICC.

To summarize global agreement across all student writing samples, educators’ original ratings (from spring 2022 operational administration) were compared against the additional three ratings obtained during the July 2022 event. Interrater agreement results are presented in Table 7.13. Results are also provided separately for emergent and conventional testlets.

Based on the guidelines specified by Cicchetti (1994), ICC agreement fell in the excellent range ($$\geq$$ .75), and Fleiss’s kappa fell in the good (.60 – .74) to excellent range. Agreement was slightly higher for conventional testlets.

Table 7.13: Interrater Agreement for Writing Samples (N = 309)
Group n ICC ICC lower bound ICC upper bound Fleiss’s $$\kappa$$
Overall 309 .967 .966 .969 .881
EW 105 .901 .887 .914 .695
CW 204 .969 .967 .970 .886
Note. ICC = intraclass correlation; EW = emergent writing; CW = conventional writing.

Educator-administered testlets measuring reading and mathematics were not included in the study. Also, although student writing samples were evaluated, the student writing process was not. Additional data collection related to educator fidelity, including fidelity in educator-administered testlets in each subject, is provided in the Test Administration Observations section of Chapter 4 of this manual.

### 7.4.3 Alignment of At Target Achievement With Postsecondary Opportunities

DLM academic achievement standards are intended to reflect rigorous expectations for students with the most significant cognitive disabilities. During 2021–2022, evidence was collected to evaluate the extent to which the DLM alternate academic achievement standards are aligned to ensure that a student who meets these standards is on track to pursue postsecondary education or competitive integrated employment.

Chapter 2 of this manual provides evidence of vertical alignment for the alternate academic achievement standards to reflect how academic content builds across grades to reflect high expectations for the student population. Here we describe the relationship of DLM alternate academic achievement standards (specifically the At Target performance level) to the knowledge, skills, and understandings needed to pursue postsecondary opportunities.

We developed two hypotheses about the expected relationship between meeting DLM alternate academic achievement standards and being prepared for a variety of postsecondary opportunities.

1. Nearly all academic skills will be associated with At Target performance level descriptors at a variety of grades between Grade 3 and high school. Few, if any, academic skills will first occur before Grade 3 At Target or after high school At Target.
2. Because academic skills may be associated with multiple opportunities and with soft skills needed for employment and education, we expected Hypothesis 1 to hold for academic skills associated with employment opportunities, education opportunities, and soft skills.

Similar to academic education for all students, academic skills for students with significant cognitive disabilities develops across grades. Individuals use academic skills at varying levels of complexity, depending on specific employment or postsecondary education settings. Therefore, academic skills associated with achieving At Target in lower grades demonstrate where students are able to apply the least complex version of the skill. Given the vertical alignment of DLM content and achievement standards, students are expected to continue learning new skills in subsequent grades and be prepared for more complex applications of the academic skills by the time they transition into postsecondary education and employment.

A panel of experts on secondary transition and/or education of students with significant cognitive disabilities identified postsecondary competitive integrated employment and education opportunities. Their goal was to identify an extensive sampling of opportunities rather than an exhaustive list. Panelists also considered the types of educational and employment opportunities currently available to students with significant cognitive disabilities as well as opportunities that may be more aspirational (i.e., opportunities that may become available in the future). Panelists identified 57 employment opportunities and 7 postsecondary education opportunities. Employment opportunities spanned sectors including agriculture, arts, business, education, health sciences, hospitality, information technology, manufacturing, and transportation.

Panelists next identified the knowledge, skills, and understandings needed to fulfill the responsibilities for the employment opportunities as well as eight common responsibilities across all postsecondary education opportunities. Finally, the panel identified the knowledge, skills, and understandings within soft skills (e.g., social skills, self-advocacy) applicable across multiple postsecondary settings. Subject-matter experts in ELA and mathematics reviewed and refined the academic skill statements to provide clarity and consistency across skills. This resulted in 50 ELA academic skills and 41 mathematics academic skills to be used in the next phase of the study.

The second set of panels, one for each subject, examined the relationship between the academic skills and the types of academic knowledge, skills, and understandings typically associated with meeting the DLM alternate academic achievement standards (i.e., achieving At Target). By identifying the lowest grade where a student achieving At Target is likely to consistently demonstrate the academic skill, the second panel identified the first point where students would be ready to pursue postsecondary opportunities that required the least complex application of the skill.

Panels consisted of general educators and special educators who administered DLM assessments from across DLM states. Most panelists had expertise across multiple grade bands, and some had certification in both an academic subject and special education. Panels completed training and calibration activities prior to making independent ratings. Panels discussed ratings until consensus when there was not an initial majority agreement.

Panels identified the lowest grade in which students who achieve At Target on the DLM alternate assessment are at least 80% likely to be able to demonstrate each skill, showing the first point of readiness to pursue postsecondary opportunities that require the least complex application of academic skills. In ELA, students achieving At Target are expected to first demonstrate 66% of those skills by Grade 5 and 27% in middle grades (Grades 6–8). In mathematics, students meeting achievement standards are expected to first demonstrate 81% of the academic skills by Grade 5 and 19% of skills in middle grades.

Overall, findings from panels indicate that most academic skills needed to pursue postsecondary opportunities are first associated with meeting the DLM academic achievement expectations in earlier grades (i.e., 3–5). Given the vertical alignment of the DLM academic achievement standards, students who achieve At Target in early grades further develop these skills so that, by the time they leave high school, they are ready to pursue postsecondary opportunities that require more complex applications of the academic skills.

Panelists also participated in focus groups to share their perceptions of opportunities, skills, and expectations for students with significant cognitive disabilities. Panelists believed the academic skills were important to postsecondary opportunities for all students, not only those who take the DLM assessment. Panelists indicated that students who were At Target in high school on the DLM assessment were likely to possess the needed academic knowledge, skills, and understandings to pursue a range of postsecondary opportunities.

Evaluation of panelists’ experiences from both panels and DLM TAC members’ review of the processes and evaluation of results provide evidence that the methods and processes used achieved the goals of the study. See Karvonen et al. (2022) for the full version of the postsecondary opportunities technical report. Overall, this evidence shows that students who demonstrate proficiency on DLM assessments are on track to pursue postsecondary opportunities.

## 7.5 Data Files

DLM assessment results were made available to DLM state education agencies following the spring 2022 administration. The General Research File (GRF) contained student results, including each student’s highest linkage level mastered for each EE and final performance level for the subject for all students who completed any testlets. In addition to the GRF, the states received several supplemental files. The special circumstances file provided information about which students and EEs were affected by extenuating circumstances (e.g., chronic absences), as defined by each state. State education agencies also received a supplemental file to identify exited students. The exited students file included all students who exited at any point during the academic year. In the event of observed incidents during assessment delivery, state education agencies are provided with an incident file describing students impacted; however, no incidents occurred during 2021–2022.

State education agencies were provided with a 2-week window following data file delivery to review the files and invalidate student records in the GRF. Decisions about whether to invalidate student records are informed by individual state policy. If changes were made to the GRF, state education agencies submitted final GRFs via Educator Portal. The final GRF was used to generate score reports.

## 7.6 Score Reports

Assessment results were provided to state education agencies to report to parents/guardians, educators, and local education agencies. Individual Student Score Reports summarized student performance on the assessment by subject. Several aggregated reports were provided to state and local education agencies, including reports for the classroom, school, district, and state.

### 7.6.1 Individual Student Score Reports

Individual Student Score Reports were developed through a series of focus groups, including a set conducted in partnership with The Arc, a community-based organization advocating for and serving people with intellectual and developmental disabilities and their families. First, several groups focused on parent/guardian perceptions of existing alternate assessment results and score reports . These findings informed the development of prototype DLM score reports. Prototypes were reviewed by the DLM Governance Board and revised based on multiple rounds of feedback. Refined prototypes were shared with parents/guardians, advocates, and educators through additional focus groups before finalizing the 2015 reports.

Individual Student Score Reports are comprised of two parts: (1) the Performance Profile, which aggregates linkage level mastery information for reporting performance in the subject overall and on each conceptual area, and (2) the Learning Profile, which reports specific linkage levels mastered for each assessed EE. There is one Individual Student Score Report per student per subject. Figure 7.3 and Figure 7.4 display a sample page from the Performance Profile and Learning Profile for ELA, respectively.

Additional interviews and focus groups with DLM educators provided evidence of score report effectiveness and yielded additional information for refining score reports . These studies indicated educators were able to appropriately interpret score report contents, found them instructionally useful, and used them to reflect on their instructional decisions. Educators also suggested additional improvements, for instance, to improve clarity of language. Minor adjustments to score report structure is implemented on an annual basis based on feedback from the field and input from the DLM scoring and reporting ad hoc committee and the larger DLM Governance Board. Sample reports for 2021–2022 are displayed in the Scoring and Reporting Resources webpage (http://dynamiclearningmaps.org/srr/ye-2022). This page contains an overview of scoring, score report delivery, and data files. Because of continued impacts of the COVID-19 pandemic on instruction and assessment, during 2021–2022, state education agencies were given the option to add a cautionary statement to Individual Student Score Reports, which indicated that the results may reflect the continued effects of the COVID-19 pandemic on student performance. Four states opted to include the cautionary statement on their individual score reports.

### 7.6.2 Aggregate Reports

Student results are also aggregated into several other types of reports. At the classroom and school levels, roster reports list individual students with the number of EEs assessed, number of linkage levels mastered, and final performance level. District- and state-level reports provide frequency distributions, by grade level and overall, of students assessed and achieving at each performance level in ELA and mathematics. Sample aggregate reports are provided in the Scoring and Reporting Resources webpage (http://dynamiclearningmaps.org/srr/ye-2022).

### 7.6.3 Interpretation Resources

At the onset of the DLM assessment, the Theory of Action for the assessment set forth four tenets for score interpretation and use:

1. Scores represent what students know and can do.
2. Performance level descriptors provide useful information about student achievement.
3. Inferences regarding student achievement and progress can be drawn for individual conceptual areas.
4. Assessment scores provide information that can be used to guide instructional decisions.

To these ends, multiple supports are provided to aid score interpretation:

• The Parent Interpretive Guide is designed to provide definition and context to student score reports.
• Parent/guardian letter templates are designed to be used by educators and state superintendents to introduce the student reports to parents/guardians.
• The Talking with Parents Guide is designed to support educators’ discussions and build understanding for parents/guardians and other stakeholders.
• The Scoring and Reporting Guide for Administrators targets building- and district-level administrators.
• A set of four short score-report interpretation videos, known as helplets, are available to help key stakeholders interpret assessment results as intended.
• All of the resources listed above and described in further detail below are compiled on the Scoring and Reporting Resources webpage (http://dynamiclearningmaps.org/srr/ye-2022). The score report videos are available on the “DLM Score Report Videos for Year-End States” webpage (https://dynamiclearningmaps.org/score-report-videos-ye).

#### 7.6.3.1 Parent Interpretive Guide

The Parent Interpretive Guide uses a sample individual student report and text boxes to explain that the assessment measures student performance on alternate achievement standards for students with the most significant cognitive disabilities—the DLM EEs. The guide first provides a high-level explanation of the DLM alternate assessment before delving into specifics about interpreting the Performance Profile and the Learning Profile.

#### 7.6.3.2 Parent Letters

Templates were developed for explanatory letters that educators and state superintendents could use to introduce parents/guardians to the student reports. These letters provided context for the reports, including a description of the DLM assessment, when the assessment was administered, and what results indicate about student performance.

The letter from the state superintendent emphasized that setting challenging and achievable academic goals for each student is the foundation for a successful and productive school year. The letter acknowledged that students have additional goals that parents/guardians and the students’ Individualized Education Plan teams have established.

#### 7.6.3.3 Talking with Parents Guide

An interpretive guide was provided for educators who would discuss results with parents/guardians or other stakeholders. The guide walked educators through directions for getting ready for a parent/guardian meeting, discussing the score report, and explaining other aspects of the DLM assessment.

#### 7.6.3.4 Guide to Scoring and Reporting for Data Managers

The guide designed for local administrators, such as superintendents, district test coordinators, and other district staff, covered how assessment results were calculated and each type of report provided for DLM assessments (including class- and school-level results as well as district- and state-level results) and explained how reports would be distributed. The guide explained the contents of each report and provided hints about interpretation.

#### 7.6.3.5 Score Report Videos

Short videos, called helplets, found on the DLM website describe score report contents and ways educators can use results to inform instruction. The video content went through multiple rounds of internal review, followed by TAC review, and external review by educators from six states. Feedback from each group was incorporated into the final videos. The available helplets are:

• What do the DLM Alternate Assessments Measure?
• What is Skill Mastery?
• What Information is Contained in a Score Report?
• How can Score Reports be Used?

## 7.7 Quality-Control Procedures for Data Files and Score Reports

Quality-control procedures are implemented for all three data file types. To ensure that formatting and the order of columns are identical, column names in each file were compared with the data dictionary that is provided to states. Additional file-specific checks are conducted to ensure accuracy of all data files.

To allow quality-control checks to be performed more rapidly and efficiently, R programs were developed to perform quality-control procedures on the GRF and on Individual Student Score Reports.

The first program written to perform automated checks is designed to perform quality-control on the GRF. This program conducts a series of checks that can be organized into four main steps.

1. Check the data for reasonableness (checks described below).
2. Ensure the number of linkage levels mastered for each student is less than or equal to the maximum possible value for that grade and subject.
3. Check all EE scores against the original scoring file.
4. Ensure all necessary fields are present and in the correct order. The automated program checks each row of data in the GRF and generates errors for review by the psychometrics team.

The reasonableness checks ensure that the GRF column names accurately match the data dictionary provided to states and check the columns to ensure that data match defined parameters. If invalid values are found, they are corrected as necessary by DLM staff and/or state education agencies during their 2-week review period.

An automated program is used to support manual review of all Individual Student Score Reports. The program was written to check key values used to generate Individual Student Score Reports. As the program generates reports, it creates a proofreader file containing the values that are used to create each score report. These values are then checked against the GRF to ensure that they are being accurately populated into score reports.

Demographic values including student name, school, district, grade level, state, and state student identifier are checked to ensure a precise match. Values of skills mastered, performance levels, conceptual areas tested and mastered, and EEs mastered and tested are also checked to ensure the correct values are populated. Values referring to the total number of skills, EEs, or conceptual areas available are checked to ensure they are the correct value for that grade and subject. In addition to the data files, individual student and aggregated score reports are generated within Kite® Educator Portal and checked for quality. Given the large number of score reports generated, a random sample of approximately 5%–10% of the score reports generated are checked manually.

For this sample, both the Performance Profile and the Learning Profile portions of the Individual Student Score Reports are checked for accuracy. Performance Profiles are checked to make sure the correct performance level displayed and matched with the value in the GRF. The percentage of skills mastered in the Performance Profile is compared against the GRF and the Learning Profile portion of the student score report to ensure that all three contain the same values. Additionally, the number of conceptual areas listed in the Performance Profile is compared with the blueprint. For each EE on the student’s Learning Profile, the highest linkage level mastered is compared with the value for the EE in the GRF. For both the Performance Profile and Learning Profile, the number of EEs listed on the report is compared against the number listed in the blueprint for that subject and grade. Demographic information in the header of the Performance Profile and Learning Profile is checked to ensure that it matches values in the GRF. Formatting and text within each report is given an editorial review as well.

Aggregated reports undergo similar checks, including the comparison of header information to GRF data and verification that all students rostered to an educator or school (for class and school reports, respectively) are present and that no extraneous students are listed. Performance levels (for class and school reports) and the number of students with a given performance level (for district and state reports) are checked against the GRF.

Once all reports are checked, all files to be disseminated to states undergo a final set of checks to ensure that all files are present. This last set of checks involves higher-level assurances that the correct number of student files are present for each type of report according to the expected number calculated from the GRF, that file naming conventions are followed, and that all types of data files are present.

Any errors identified during quality-control checks are corrected prior to distribution of data files and score reports to states.

## 7.8 Conclusion

Results for DLM assessments include students’ overall performance levels and linkage level mastery decisions for each assessed EE and linkage level. During spring 2022, assessments were administered to 88,404 students in 14 states. Between 11% and 37% of students achieved at the At Target or Advanced levels across all grades and subjects. Of the three scoring rules, linkage level mastery status was most frequently assigned by the diagnostic classification model, and students tend to demonstrate mastery of the Target or Successor level at higher rates in ELA than in mathematics. Additional scoring evidence indicates that writing tasks are reliably scored by test administrators, and students who achieve at the At Target performance level are on track to pursue postsecondary opportunities.

Following the spring 2022 administration, three data files were delivered to state education agencies: GRF, special circumstance code file, and exited students file. Score reports summarized assessment results at the individual student, class, school, district, and state levels. Supplementary materials to support score report interpretation and use are available on the DLM website.


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0

This question is evidence that the scientific method has sort of gotten lost in the way the ML world communicates about models, and it causes students to get confused when they enter industry jobs. Let's illustrate the difference through the metaphor of a soccer team training for the upcoming season. Most of the time will be spent doing drills for core ...

1

You are correct. They are not independent and they are positively correlated. Let $A$ and $B$ (or $X$ and $Y$) be two events for stating general theorems and $M$ and $T$ be the events "customer purchases mozzarella" and "customer purchases tomatoes" in this specific example. We will use $\wedge$ to mean "and" so that $M \wedge T$...

2

Usually you first split your dataset into train-/test- set, and then if your model training process requires validation set, you can further split your train-set into the final train-set and the validation-set. Simple rule is test set never shows up in your model development process, including when you develop your data preprocessing steps (such as your data ...

0

You can use the testing data to perform hyperparameters optimization to see which hyperparameters of your model pipeline work the best. The validation data is then only used once to see how the whole model pipeline works on out of sample data. For this process the test dataset cannot be used again as this data was already used to select the best ...

0

Removing outliers in a high-dimensional scenario can for example be done after dimension reduction by principal component analysis. In the dimension-reduced space either boxplots (1 dimension), bagplots (2 dimension) or gemplots (3 dimensions) can be applied to detect outliers. For details please look at Kruppa, J., & Jung, K. (2017). Automated ...

1

Given you have two categorical variables and the associated contingency table, one option is to calculate the joint and marginal probabilities: $$P = \frac{count}{total}$$ Probabilities Tomatoes No Tomatoes Row Mozzarella 0.4 0.1 0.5 No Mozzarella 0.2 0.3 0.5 Column 0.6 0.4 0.1 The probabilities then can be used to answer questions about the data - If ...

1

Market Basket Analysis is the way to go about it. Market Basket Analysis is a technique which identifies the strength of association between pairs of products purchased together and identify patterns of co-occurrence. A co-occurrence is when two or more things take place together. Market Basket Analysis creates If-Then scenario rules, for example, if item A ...

0

The article you have shared answers the question correctly - your algorithm could input both int and float, and its internals (weights, biases, etc) will all be float matrices. A common example that you will find is the "customer churn" datasets, in which you need to classify whether a customer will stay/leave a company to go to a competitor. The ...

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Market Basket Analysis is a technique which identifies the strength of association between pairs of products purchased together and identify patterns of co-occurrence. A co-occurrence is when two or more things take place together. Market Basket Analysis creates If-Then scenario rules, for example, if item A is purchased then item B is likely to be ...

1

These string data, called categorical data can be converted to numerical data using many Categorical Encoding Techniques. Encoding categorical data is a process of converting categorical data into integer format so that the data with converted categorical values can be provided to the different models. Types of Categorical Techniques: Backward Difference ...

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You can multiple things here : Converting them to numerical introduces some sense of ordering For example if you say slovenia is 1 and USA is 2 ans ordering is introduced instead you can use one hot encoding. Pandas getdummies function will do it for you If one of your string has a lot of values say 1000 one hot encoding does not makes sense. In those ...

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It seems that the function tries to call the SPLAT! command line using srtm2sdf on all files in in_path. Trying to run a command line program using subprocess.run when the command line program doesn't exist (i.e. returns a 'command not found' error when trying to run a command on the command line) gives a FileNotFoundError instead of the actual error you get ...

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The message tells you that there is no variable mcq_grade. Most likely when you ran the code yourself you had already loaded some data from a file and assigned it to mcq_grade, it could look like this: mcq_grade <- read.table(filename) All the code that you put in your R markdown source document must be independent because it's executed in a new ...

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Yes, you can train XGBoost in parallel using the Dask backend. Short Solution Training XGBoost in parallel with Dask requires 2 changes in your code: substitute dtrain = xgb.DMatrix(X_train, y_train) with dtrain = xgb.dask.DaskDMatrix(X_train, y_train) substitute xgb.train(params, dtrain, ...) with xgb.dask.train(client, params, dtrain, ...)

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The problem seems to be that you are dividing by 10 (k) at each iteration, I can think to try to calculate the average, this is incorrect and probably it is what is causing you to see a very low metric value. It would be simpler and correct, to only store the values for the metric in each iteration at the validation set and finally just calculate the average ...

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text:
1. stochastic particle movement probabilities

consider a particle that moves along the set of integers. if it is presently at i then it next moves to i+1 with probability p and i-1 with probability 1-p. starting at 0, let a denote the probability that the particle ever reaches 1.

a. show a=p+(1-p)a^2
b. show: a= 1 if p>= .5
p/(1-p) if p<.5

c. calculate the probability that the particle reaches n, for n>0
d. Suppose p<.5 and the particle does reach n, n>0. if the particle is at i, i<n, and n has not yet been reached show particle will move to i+1 with probability 1-p. OR
P(next at i+1 | at i and will reach n)= 1-p

2. Hi,

what part of the problem have you done?, what have you tried?

for a., if you haven't done it yet, you have to decompose the event according to the first step: either the first step goes to 1, and we are done, or it goes down to -1, hence the particle needs to first go back to 0 and then from 0 to 1. But going from -1 to 0 is "the same" as going from 0 to 1... I let you try to make some sense out of this and conclude.


text:
07-10E

# 07-10E

By - Posted on 08 February 2012

UDC:
532; 533; 533.9:530.182; 536.75; 536-12.01.
PACS:
05.60.+w, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi

Influence of long-range correlated surface and near the surface disorder on adsorption of long-flexible polymer chains

Z.Usatenko
J.-U. Sommer

We investigate the influence of quenched long-range correlated surface and decaying near surface disorder on the process of adsorption of long-flexible polymer chains with excluded volume interactions on inert and attractive walls in the framework of renormalization group field theoretical approach up to first order of perturbation theory in a double ($\epsilon$,$\delta$)- expansion for the semi-infinite $|\phi |^4$ $O(m,n)$ model in the limit $m,n\to 0$. We obtained series for bulk and the whole set of surface critical exponents, characterizing the process of adsorption of polymer chains. The scaling behavior of the parallel and perpendicular part of the mean-squared end-to-end distance as well as the behavior of monomer density profiles near the surface and the fraction of adsorbed monomers are studied.

Year:
2007
Pages:
29