Oscar E. Fernandez

Lifespan disparity, a measure of variation in ages at death, can be expressed as a linear combination of standard measures of statistical dispersion in age-at-death distributions (e.g. skewness). Using data from the Human Mortality Database (HMD) (7408 life tables from 41 countries, 1751–2019), we conduct an empirical analysis of this mathematical result and assess the extent to which the moments of age-at-death distributions determine and drive lifespan disparity and trends in it. We report four findings: (i) remaining life expectancy contributes most to lifespan disparity, but its share has declined over time as the variance, skewness and kurtosis constitute larger shares; (ii) changes in skew of the age-at-death distributions have shifted and compressed mortality: distributions including older (younger) ages show increasingly positive (negative) skewness—left-leaning (right-leaning) distributions, with recent trends towards symmetric distributions; (iii) kurtosis patterns show a persistent fraction of deaths remaining at distribution extremes suggesting a plateau in the number of centenarians; and (iv) for adult populations (e.g. aged 30+), approximately 95% of life disparity values are determined from the first four standardized moments of the associated age-at-death distributions. Incorporating skewness and kurtosis yields new insights into mortality compression and lifespan variability trends.

Previous research has derived bounds on the remaining life expectancy function e(x) that connect survivorship and remaining life expectancy at two age values and therefore can be used to, among other things, estimate life expectancy at birth when the popul

Recent empirical studies have found various patterns in the correlations between lifespan inequality and life expectancy in modern human populations. However, it is unclear how general these regularities are. Here we establish three theorems that provide theoretical foundations for such regularities. We show that for populations with a finite maximum lifespan ω , and under certain continuity assumptions, the variance in the age at death is bounded by a function of lifespan that has a maximum and tends to zero as life expectancy tends to zero and ω . We show how the change in said variance is determined by a particular interplay between the coefficient of variation and the mean age in the population. These results lead to three hypotheses—a three-phased pattern of change for the correlation between the variance and life expectancy, a particular shape of the associated variance function, and that survival curve Type is one driver of the pattern. We illustrate those hypotheses empirically via a study of the 10 countries in the Human Mortality Database with the oldest available data. Our results elucidate the emergence of the aforementioned correlation patterns and provide demographically meaningful conditions under which those correlations reverse.

In this article we develop a quantization procedure for Chaplygin Hamiltonizable nonholonomic systems—mechanical systems subject to non-integrable velocity constraints whose reduced mechanics is Hamiltonian after a suitable time reparametrization—using Poincaré transformations and geometric quantization. We illustrate the theory developed through examples and discuss potential applications to the study of the quantum mechanics of nanovehicles.

Recent work has unearthed many empirical regularities in mortality trends, including the inverse correlation between life expectancy and life span inequality, and the compression of mortality into older age ranges. These regularities have furnished important insights into the dynamics of mortality by describing, in demographic terms, how different attributes of the life table deaths distribution interrelate and change over time. However, though empirical evidence suggests that the demographically-meaningful metrics these regularities involve (e.g., life span disparity and life table entropy) are correlated to the moments of the deaths distribution (e.g., variance), the broader theoretical connections between life span inequality and the moments of the deaths distribution have yet to be elucidated. In this article we establish such connections and leverage them to furnish new insights into mortality dynamics. We prove theoretical results linking life span disparity and life table entropy to the central moments of the deaths distribution, and use these results to empirically link statistical measures of variation of the deaths distribution (e.g., variance, index of dispersion) to life span disparity and life table entropy. We validate these results via empirical analyses using data from the Human Mortality Database and extract from them several new insights into mortality shifting and compression in human populations.

This article describes the synthesis of the research on mastery grading, growth mindsets, and testing for learning to produce a new grading system—Second Chance Grading. The system and the research it is based on are described in detail. Reflections on the system's effectiveness, impact, and reception are also discussed. Finally, the appendix to the article contains an F.A.Q. section addressing some of the common questions and concerns related to the system.

We formulate a mathematical model of a rolling “molecular wheelbarrow”—a two-wheeled nanoscale molecular machine—informed by experiments on molecular machines recently synthesized in labs. The model is a nonholonomic system (briefly, a system with non-integrable velocity constraints), for which no general quantization procedure exists. Nonetheless, we successfully embed the system in a Hamiltonian one and then quantize the result using geometric quantization and other tools; we extract from the result the quantum mechanics of the molecular wheelbarrow, and derive explicit formulae for the quantized energy spectrum. We also study a few variants of our model, some of which ignore the model’s nonholonomic constraints. We show that these variants have different quantum energy spectra, indicating that in such systems one should not ignore the nonholonomic constraints, since they alter in a non-trivial way the energy spectrum of the molecule.

In this paper we study the reduction of a nonholonomic system by a group of symmetries in two steps. Using the so-called 'vertical-symmetry' condition, we first perform a 'compression' of the nonholonomic system leading to an almost symplectic manifold. Second, we perform an (almost) symplectic reduction, relying on the existence of a momentum map. In this case, we verify that the resulting (almost) symplectic reduced spaces are the leaves of the characteristic foliation of the reduced nonholonomic bracket. On each leaf we study the (Lagrangian) equations of motion, obtaining a nonholonomic version of the Lagrange-Routh equations, and we analyze the existence of a conformal factor for the reduced bracket in terms of the existence of conformal factors for the almost symplectic leaves. We also relate the conditions for the existence of a momentum map for the compressed system with gauge transformations by 2-forms. The results are illustrated in three different examples.

We report on new applications of the Poincaré and Sundman time-transformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these non-variational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results.