Explore the vast range of titles available.

School choice programs seek to give students the option to choose their school but also close an opportunity gap. To be fair in the assignment of students, it is usually argued that the assignment of students to schools should be stable. This second concern is usually expressed in terms of proportions. As an example, in 1989, the city of White Plains, New York, required each school to have the same proportions of Blacks, Hispanics, and “others,” a term that includes Whites and Asians. Satisfying both these concerns at the same time is difficult. Prior work replaces the proportions by numbers related to the capacity of school, but this assumes each school is operating at full capacity, which is often not the case. In this paper, we treat such proportionality constraints as soft but provide ex post guarantees on how well the constraints are satisfied while preserving stability. The problem of finding stable matches that meet distributional concerns is usually formulated by imposing side constraints whose “right-hand sides” are absolute numbers specified before the preferences or number of agents on the “proposing” side are known. In many cases, it is more natural to express the relevant constraints as proportions. We treat such constraints as soft but provide ex post guarantees on how well the constraints are satisfied while preserving stability. Our technique requires an extension of Scarf’s lemma, which is of independent interest.

We are developing an approach to teaching important proportionality-based concepts to first grade students in a way that supports students’ future progress in the domain. We consider the proportionality between magnitudes as a basic relationship behind multiple cases, usually described mathematically as ratio or rate. The core of our strategy is the modelling of a situation of proportionality and its transformations by creating a compound unit. The key action is coordinated measurement (co-measurement): students work in pairs, and each student is in charge of changing one of two magnitudes, while preserving proportionality. Based on our successful experiments in Grades 2–6, we incorporate shared responsibility work organization (“joint actions”) and the idea of compound unit into Davydov’s mathematics curriculum for the first grade. We built a new module based on Davydov’s idea about the role of rule-mediated counting by sets. In this paper, we present the results of our study, showing that first graders can learn the idea of compound unit and work with two magnitudes of different kinds while preserving proportionality.

This paper studies bankruptcy problems with nontransferable utility as a generalization of bankruptcy problems with monetary estate and claims. Following the theory on TU-bankruptcy, we introduce a duality notion for NTU-bankruptcy rules and derive several axiomatic characterizations of the proportional rule and the constrained relative equal awards rule.

Proportional reasoning, the ability to use ratios in situations involving comparison of quantities, is essential for mathematical competence, especially in the middle school years, and is an important determinant of success beyond school. Research shows students find proportional reasoning and its foundational concepts difficult. Proportional reasoning does not always develop naturally, however some research suggests that with targeted teaching, its development can be promoted. This paper reports on a large Australian study involving over 130 teachers and their students. A major goal of the study was to investigate the efficacy of ongoing teacher professional development for promoting middle years students' proportional reasoning. A series of professional development workshops was designed to enhance the teachers' understanding of proportional reasoning and to extend their repertoire of teaching strategies to promote their students' proportional reasoning skills. The workshop design was informed by research literature on proportional reasoning teaching and learning as well as the results of a diagnostic instrument administered to over 2500 middle years students prior to the professional development. Between workshops, the teachers implemented a variety of targeted teaching activities. This paper reports on pre- and post- instrument student data collected at the beginning and end of the first year of the project (i.e., after completion of half of the workshops). The findings suggest that targeted professional development and explicit teaching can make a difference to students' proportional reasoning.

Digital twins have gained a lot of interest in recent years. This paper presents a survey among researchers and engineers with expertise in variation management confirming the interest of digital twins in this area. The survey shows, however, a gap between future research interest in academia and industry, identifying a larger need in industry. This indicates that there are some barriers in the industry to overcome before the benefits of a digital twin for variation management and geometry assurance can be fully capitalized on in an industrial context. To identify those barriers and challenges, an extensive interview study with engineers from eight different companies in the manufacturing sectors was accomplished. The analysis identifies industrial challenges in the areas of system-level, simulation working process, management issues, and education. One of the main challenges is to keep the 3D models fully updated, including keeping track of changes during the product development process and also feedback changes during full production to the development engineers. This is a part of what is called the digital thread, which is also addressed in this paper. Keywords: digital twin; manufacturing; tolerancing; geometry assurance; digital thread

We consider the problem of constructing a complete set of parameters that account for all of the degrees of freedom for point-biserial variation. We devise an algorithm where sort as an intrinsic property of both numbers and labels, is used to generate the parameters. Algebraically, point-biserial variation is represented by a Cartesian product of statistical parameters for two sets of R 1 data, and the difference between mean values ( δ ) corresponds to the representation of variation in the center of mass coordinates, ( δ , μ ). The existence of alternative effect size measures is explained by the fact that mathematical considerations alone do not specify a preferred coordinate system for the representation of point-biserial variation. We develop a novel algorithm for estimating the nonoverlap proportion ( ρ pb ) of two sets of R 1 data. ρ pb is obtained by sorting the labeled R 1 data and analyzing the induced order in the categorical data using a diagonally symmetric 2 × 2 contingency table. We examine the correspondence between ρ pb and point-biserial correlation ( r pb ) for uniform and normal distributions. We identify the R 2 , P 1 , and S + 1 representations for Pearson product-moment correlation, Cohen’s d , and r pb . We compare the performance of r pb versus ρ pb and the sample size proportion corrected correlation ( r pbd ), confirm that invariance with respect to the sample size proportion is important in the formulation of the effect size, and give an example where three parameters ( r pbd , μ , ρ pb ) are needed to distinguish different forms of point-biserial variation in CART regression tree analysis. We discuss the importance of providing an assessment of cost-benefit trade-offs between relevant system parameters because ‘substantive significance’ is specified by mapping functional or engineering requirements into the effect size coordinates. Distributions and confidence intervals for the statistical parameters are obtained using Monte Carlo methods.

This paper presents a new method for the simultaneous estimation of system states and unknown inputs in fractional-order Takagi–Sugeno (FO-TS) systems with unmeasurable premise variables (UPVs), by introducing a fractional-order proportional-integral unknown input observer (FO-PIUIO) based on partial state augmentation. This approach permits the estimation of both states and unknown inputs, which are essential for system monitoring and control. Partial state augmentation allows the integration of unknown inputs into a partially augmented model, ensuring accurate estimates of both states and unknown inputs. The state estimation error is formulated as a perturbed system. The convergence conditions for the state estimation errors between the system and the observer are derived using the second Lyapunov method and the L2 approach. Compared to traditional integer-order unknown input observers or fuzzy observers with measurable premise variables, in our method, fractional-order dynamics are combined with partial state augmentation uniquely for the persistent estimation of states along with unknown inputs in unmeasurable premise variable systems. Such a combination allows for robust estimation even under uncertainties in systems and long memory phenomena and is a significant step forward from traditional methods. Finally, a numerical example is provided to illustrate the performance of the proposed observer.

The need to model proportional data is common in a range of disciplines however, due to its bimodal nature, U- or J-shaped data present a particular challenge. In this study, two parsimonious mixture models are proposed to accurately characterise this proportional U- and J-shaped data. The proposed models are applied to loss given default data, an application area where specific importance is attached to the accuracy with which the mean is estimated, due to its linear relationship with a bank’s regulatory capital. In addition to using standard information criteria, the degree to which bias reduction in the estimation of the distributional mean can be achieved is used as a measure of model performance. The proposed models outperform the benchmark model with reference to the information criteria and yield a reduction in the distance between the empirical and distributional means. Given the special characteristics of the dataset, where a high proportion of observations are close to zero, a methodology for choosing a rounding threshold in an objective manner is developed as part of the data preparation stage. It is shown how the application of this rounding threshold can reduce bias in moment estimation regardless of the model choice.

In this paper, we propose a general class of bivariate proportional hazard distributions, which is based on the family of asymmetric proportional hazard distributions and the bivariate Pareto copula. Distributional properties of the bivariate proportional hazard distribution are derived. We specialize the bivariate proportional hazard family of distributions to the normal case, and so we introduce the bivariate proportional hazard normal distribution. Parameter estimation by the maximum likelihood method of the bivariate proportional hazard normal distribution is then discussed. Finally, an application of the new bivariate distribution to real data is considered for illustrative purposes.

Background Value-based pricing (VBP) of new drugs has been suggested both as a way to control health expenditures and to maximize health benefits based on the available resources. The purpose of this work is to present a simple mathematical proof showing that prices of new drugs are set in a mathematically consistent way when the sum of intervention and downstream costs is proportional to the size of health benefits. Such proportional relationship underlies the efficiency-frontier method used by the German Institute for Quality and Efficiency in Health Care (IQWiG). Methods A proof by contradiction is presented that is based upon the following three premises: 1) total costs (intervention plus downstream costs) of existing non-dominated drugs and interventions are acceptable to decision-making bodies; 2) new drugs with health benefits in-between those of the most and second most effective existing interventions are not automatically excluded from reimbursement and are acceptable if prices are sufficiently low; and 3) value is measured on a cardinal scale. Result The proof shows that a proportional rule sets reimbursement prices of new drugs in a mathematically consistent way. Conclusion Based on the proof and the underlying assumptions a proportional relationship between costs and health benefits ensures mathematical consistency in VBP of drugs.