We study the relationship between the Loeb measure ${}^0({}^*\mu )$ of a set $E$ and the $\mu$-measure of the set $S(E) = \{ x | {}^* x \in E \}$ of standard points in $E$. If $E$ is in the $\sigma$-algebra generated by the standard sets, then ${}^0({}^ * \mu )(E) = \mu (S(E))$. This is used to give a short nonstandard proof of Egoroffâs Theorem. If $E$ is an internal, * measurable set, then in general there is no relationship between the measures of $S(E)$ and $E$. However, if ${}^ * X$ is an ultrapower constructed using a minimal ultrafilter on $\omega$, then ${}^ * \mu (E) \approx 0$ implies that $S(E)$ is a $\mu$-null set. If, in addition, $\mu$ is a Borel measure on a compact metric space and $E$ is a Loeb measurable set, then \[ \underline \mu (S(E)) \leqslant {}^0({}^ * \mu )(E) \leqslant \overline \mu (S(E))\] where $\underline \mu$ and $\overline \mu$ are the inner and outer measures for $\mu$.
The notion of âmonadâ is generalized to infinite (i.e. non-near-standard) points in arbitrary nonstandard models of completely regular topological spaces. The behaviour of several such monad systems in finite product spaces is investigated and we prove that for paracompact spaces
The notion of âmonadâ is generalized to infinite (i.e. non-near-standard) points in arbitrary nonstandard models of completely regular topological spaces. The behaviour of several such monad systems in finite product spaces is investigated and we prove that for paracompact spaces X such that $X \times X$ is normal, the covering monad $\mu$ satisfies $\mu (x,y) = \mu (x) \times \mu (y)$ whenever x and y have the same âorder of magnitude.â Finally, monad systems, in particular non-standard models of the real line, R, are studied and we show that in a minimal nonstandard model of R exactly one monad system exists and, in fact, $\mu (x) = \{ x\}$ if x is infinite.
Abstract We discuss approaches to develop and improve habits of mind in our students. Several illustrations and strategies are offered and we demonstrate how popular culture and mystery can be used to motivate students. We offer an approach to assessing the work ethic efforts of our students based on an historical approach used at the United States Military Academy.
The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is S-thin if InfmA|A standard,* $A \supseteq E$ = 0 and Q-thin if Inf*mA|A internal, $A \supseteq E$ = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every S-thin set and that given any Q-thin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set ${\mathcal {H}} \subseteq * \left [ {0, 1} \right ]$ is constructed which is important for the study of approximate limits, derivatives etc. It is shown that every *finite sample which represents Lebesgue measure assigns inner measure zero and outer measure one to this set and that Loeb measure does the same. Finally, it is shown that Loeb measure can be extended to a $\sigma$-algebra including ${\mathcal {H}}$ in such a way that ${\mathcal {H}}$ is assigned zero measure.
Prompted by Eric Mazur's 1997 book and his promotion of the practical classroom techniques of peer instruction, many physics and engineering classrooms have evolved into activity-based studios for student learning and assessment, and Physics Education Research (PER) has emerged as a research field at many universities.This philosophical change in the way teachers think about student learning has been accompanied by new classroom technologies that included video analysis techniques, student response cards (clickers), and a robust suite of sensors that bring classrooms and laboratories to life with the ease of plug-and-play data acquisition.PASCO Systems is one such sensor suite adopted at West Point in its introductory physics and math courses.In the context of studying a vertical spring-mass system, a motion sensor that uses the echo of ultrasonic sound off of the bottom of the mass is a reasonable tool to analyze the harmonic motion and its dampening characteristics.Unfortunately, the data does not match the models used in most introductory textbooks because these simplified models omit the torsion of the spring and constrain the problem to 2-D.We have measured the impact of this omission to contribute to systematic error, and the rotation of the mass and the 3-D motion are easily observed.To complement these classroom activities, we propose using DIY-Modeling, which is a 3-D modeling-simulation program that generates realistic, game-like graphics for student visualization and experimental testing.A consortium of math, physics, and engineering educators from nine universities collaborated with a commercial software developer, Tietronix, to produce DIY-Modeling.The purpose of this paper is to address how DIY-Modeling might bridge gaps in existing classroom technologies and to develop spring-mass curricular materials that might be more generically applied to other physical systems where introductory models require more sophisticated analysis through modeling and simulation.
