Students' Accuracy of Measurement Estimation: Context,Units, and Logical Thinking

This study examined students' accuracy of measurement estimation for linear distances, different units of measure, task context, and the relationship between accuracy estimation and logical thinking. Middle school students completed a series of tasks that included estimating the length of various objects in different contexts and completed a test of logical thinking ability. Results found that the students were not able to give accurate estimations for the lengths of familiar objects. Students were also less accurate in estimating in metric units as compared to English or novel units. Estimation accuracy was dependent on the task context. There were significant differences in estimation accuracy for two‐ versus three‐dimensional estimation tasks. There were no significant differences for estimating objects with different orientations or embedded objects. For the tasks requiring the students to estimate in English units, the embedded task and the three‐dimensional tasks were correlated with logical thinking. For estimation tasks with novel units, three‐dimensional and two‐dimensional estimation tasks were significantly correlated with the logical thinking. In order to interact effectively with our environment it is essential to possess an intuitive grasp of both dimension and scale and to be able to manipulate such information. Estimation, approximating and measuring are all components of such intuition ( Forrester, Latham, & Shire, 1990 , p. 283).     

Lindelöf spaces which are indestructible, productive, or D

We discuss relationships in Lindelöf spaces among the properties “indestructible”, “productive”, “D”, and related properties.     

Metal-organic framework materials with ultrahigh surface areas: is the sky the limit?

We have synthesized, characterized, and computationally simulated/validated the behavior of two new metal-organic framework (MOF) materials displaying the highest experimental Brunauer-Emmett-Teller (BET) surface areas of any porous materials reported to date (～7000 m(2)/g). Key to evacuating the initially solvent-filled materials without pore collapse, and thereby accessing the ultrahigh areas, is the use of a supercritical CO(2) activation technique. Additionally, we demonstrate computationally that by shifting from phenyl groups to "space efficient" acetylene moieties as linker expansion units, the hypothetical maximum surface area for a MOF material is substantially greater than previously envisioned (～14600 m(2)/g (or greater) versus ～10500 m(2)/g).     

High-throughput examination of fluorescence resonance energy transfer-detected metal-ion response in mammalian cells

Fluorescence resonance energy transfer (FRET)-based genetically encoded metal-ion sensors are important tools for studying metal-ion dynamics in live cells. We present a time-resolved microfluidic flow cytometer capable of characterizing the FRET-based dynamic response of metal-ion sensors in mammalian cells at a throughput of 15 cells/s with a time window encompassing a few milliseconds to a few seconds after mixing of cells with exogenous ligands. We have used the instrument to examine the cellular heterogeneity of Zn(2+) and Ca(2+) sensor FRET response amplitudes and demonstrated that the cluster maps of the Zn(2+) sensor FRET changes resolve multiple subpopulations. We have also measured the in vivo sensor response kinetics induced by changes in Zn(2+) and Ca(2+) concentrations. We observed an ～30 fold difference between the extracellular and intracellular sensors.     

Highly Monodisperse M(III)-Based soc-MOFs (M = In and Ga) with Cubic and Truncated Cubic Morphologies

In this work, we carry out an investigation on shape-controlled growth of In(III)- and Ga(III)-based square-octahedral metal-organic frameworks (soc-MOFs). In particular, controllable crystal morphological evolution from simple cubes to complex octadecahedra has been achieved, and resultant highly uniform crystal building blocks promise new research opportunities for preparation of self-assembled MOF materials and related applications.     

Designing higher surface area metal-organic frameworks: are triple bonds better than phenyls?

We have synthesized, characterized, and computationally validated the high Brunauer-Emmett-Teller surface area and hydrogen uptake of a new, noncatenating metal-organic framework (MOF) material, NU-111. Our results imply that replacing the phenyl spacers of organic linkers with triple-bond spacers is an effective strategy for boosting molecule-accessible gravimetric surface areas of MOFs and related high-porosity materials.     

On a core concept of Arhangel'ski?

Arhangel'ski? [A.V. Arhangel'ski?, Locally compact spaces of countable core and Alexandroff compactification, Topology Appl. 154 (2007) 625-634] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper.     

The homogeneous q-difference operator

We introduce a q-differential operator D_xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.     

Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain

We establish Kamenev-type criteria and interval criteria for oscillation of the second-order scalar differential equation (p(t)x^Δ(t))^Δ+q(t)x(σ(t))=0 on a measure chain. Our results cover those for differential equations and provide new oscillation criteria for difference equations. Several examples are given to show the significance of the results.