The well-known Stolz-Cesaro lemma is due to the mathematicians Ernesto Cesaro (1859-1906) and Otto Stolz (1842-1905). The aim of this article is to give new forms of Stolz-Cesaro lemma involving the limit [image omitted].

A standard example used in introductory combinatoric courses is to count the number of five-card poker hands possible from a straight deck of 52 distinct cards. A more interesting problem is to count the number of distinct hands possible from a Pinochle deck in which there are multiple, but obviously limited, copies of each type of card (two…

Contemporary educational accountability systems, including state-level systems prescribed under No Child Left Behind as well as those envisioned under the "Race to the Top" comprehensive assessment competition, rely on school-level summaries of student test scores. The precision of these score summaries is almost always evaluated using models that…

One of the often overlooked aspects of modern statistics is the analysis of time series data. Modern introductory statistics courses tend to rush to probabilistic applications involving risk and confidence. Rarely does the first level course linger on such useful and fascinating topics as time series decomposition, with its practical applications…

When the LHC started running at the end of March 2010--after a 14-month shutdown for major repairs--one of the main objectives was reaching a luminosity of 10[superscript 32] cm [superscript -2] s [superscript -1] by the end of 2010. On 13 October 2010 that goal was achieved. One important parameter to take into account to reach this luminosity is…

Sir Nevil Maskelyne is remembered by many as the main rival to clockmaker John Harrison in the famous "Longitude" affair of the 18th century. However, his contributions to science go well beyond this often misreported drama; three of his many observations and developments are considered here in the year marking the 200th anniversary of his death.…

There is a remarkable difference between formal knowledge and true understanding of the subject. While the former helps students earn top grades, the latter is crucial to the solution of real-world problems. An excellent example is the computation of capacitance, with which some students have difficulty. Also, most textbooks limit problem analysis…

Going "green" is a slogan that is very contemporary, both with industry and in the political arena. Choosing more energy-efficient devices is one way homeowners can "go green." A simple method is to change home lighting from hot incandescent bulbs to compact fluorescent lights (CFLs). But do they really save energy? How do their illuminations…

This research provides a demonstration of the utility of mixture Rasch models. Specifically, a model capable of estimating a mixture partial credit model using joint maximum likelihood is presented. Like the partial credit model, the mixture partial credit model has the beneficial feature of being appropriate for analysis of assessment data…

We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer "Mathematica" code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist,…

What abilities are entailed in being numerate? Certainly, one is the ability to hold the exact quantity of a set in mind, even as it changes, and even after its members can no longer be perceived. Is counting language necessary to track and reproduce exact quantities? Previous work with speakers of languages that lack number words involved…

Rating scale items have been widely used in educational and psychological tests. These items require people to make subjective judgments, and these subjective judgments usually involve randomness. To account for this randomness, Wang, Wilson, and Shih proposed the random-effect rating scale model in which the threshold parameters are treated as…

Compared to parametric models, nonparametric and semiparametric approaches to modeling nonlinearity between latent variables have the advantage of recovering global relationships of unknown functional form. Bauer (2005) proposed an indirect application of finite mixtures of structural equation models where latent components are estimated in the…

Computerized adaptive testing (CAT) is an active current research field in psychometrics and educational measurement. However, there is very little software available to handle such adaptive tasks. The R package "catR" was developed to perform adaptive testing with as much flexibility as possible, in an attempt to provide a developmental and…

The latent growth curve modeling (LGCM) approach has been increasingly utilized to investigate longitudinal mediation. However, little is known about the accuracy of the estimates and statistical power when mediation is evaluated in the LGCM framework. A simulation study was conducted to address these issues under various conditions including…