Characterizations of modalities and lex modalities

A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of ∞-categories. Working in homotopy type theory, we give new characterizations of the following conditions on a reflective subuniverse L: (1) the associated subuniverse $L^{\prime }$ of L-separated types is a modality; (2) L is a modality; (3) L is a lex modality; and (4) L is a cotopological modality. In each case, we give several necessary and sufficient conditions. Our characterizations involve various families of maps associated to L, such as the L-étale maps, the L-equivalences, the L-local maps, the L-connected maps, the unit maps ${\eta }_X$, and their left and/or right orthogonal complements. More generally, our main theorem gives an overview of how all of these classes related to each other. We also give examples that show that all of the inclusions we describe between these classes of maps can be strict.     

DFT Study of the Molybdenum‐Catalyzed Deoxydehydration of Vicinal Diols

The mechanism of the molybdenum‐catalyzed deoxydehydration (DODH) of vicinal diols has been investigated using density functional theory. The proposed catalytic cycle involves condensation of the diol with an Mo^VI oxo complex, oxidative cleavage of the diol resulting in an Mo^IV complex, and extrusion of the alkene. We have compared the proposed pathway with several alternatives, and the results have been corroborated by comparison with the molybdenum‐catalyzed sulfoxide reduction recently published by Sanz et al. and with experimental observations for the DODH itself. Improved understanding of the mechanism should expedite future optimization of molybdenum‐catalyzed biomass transformations.     

Azasugar inhibitors as pharmacological chaperones for Krabbe disease

Krabbe disease is a devastating neurodegenerative disorder characterized by rapid demyelination of nerve fibers. This disease is caused by defects in the lysosomal enzyme β-galactocerebrosidase (GALC), which hydrolyzes the terminal galactose from glycosphingolipids. These lipids are essential components of eukaryotic cell membranes: substrates of GALC include galactocerebroside, the primary lipid component of myelin, and psychosine, a cytotoxic metabolite. Mutations of GALC that cause misfolding of the protein may be responsive to pharmacological chaperone therapy (PCT), whereby small molecules are used to stabilize these mutant proteins, thus correcting trafficking defects and increasing residual catabolic activity in cells. Here we describe a new approach for the synthesis of galacto-configured azasugars and the characterization of their interaction with GALC using biophysical, biochemical and crystallographic methods. We identify that the global stabilization of GALC conferred by azasugar derivatives, measured by fluorescence-based thermal shift assays, is directly related to their binding affinity, measured by enzyme inhibition. X-ray crystal structures of these molecules bound in the GALC active site reveal which residues participate in stabilizing interactions, show how potency is achieved and illustrate the penalties of aza/iminosugar ring distortion. The structure–activity relationships described here identify the key physical properties required of pharmacological chaperones for Krabbe disease and highlight the potential of azasugars as stabilizing agents for future enzyme replacement therapies. This work lays the foundation for new drug-based treatments of Krabbe disease.     

Anthracene-Porphyrin Nanoribbons

π-Conjugated nanoribbons attract interest because of their unusual electronic structures and charge-transport behavior. Here, we report the synthesis of a series of fully edge-fused porphyrin-anthracene oligomeric ribbons (dimer and trimer), together with a computational study of the corresponding infinite polymer. The porphyrin dimer and trimer were synthesized in high yield, via oxidative cyclodehydrogenation of singly linked precursors, using 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) and trifluoromethanesulfonic acid (TfOH). The crystal structure of the dimer shows that the central π-system is flat, with a slight S-shaped wave distortion at each porphyrin terminal. The extended π-conjugation causes a dramatic red-shift in the absorption spectra: the absorption maxima of the fused dimer and trimer appear at 1188 nm and 1642 nm, respectively (for the nickel complexes dissolved in toluene). The coordinated metal in the dimer was changed from Ni to Mg, using p-tolylmagnesium bromide, providing access to free-base and Zn complexes. These results open a versatile avenue to longer π-conjugated nanoribbons with integrated metalloporphyrin units.     

An efficient microwave assisted synthesis of N′-aryl/(alkyl)-substituted N-(4-hydroxy-6-phenylpyrimidin-2-yl)guanidines: Scope and limitations

Treatment of N-[(4-hydroxy-6-phenyl)pyrimidin-2-yl]cyanamide with 1° alkyl or arylamines in isopropyl alcohol for only 10 min at 110–120 °C under microwave conditions gave the corresponding N′-alkyl(aryl)guanidine derivatives in excellent yields (65–84%). Isolated yields were greatest when >1.0 equiv. of amines were employed, but excellent results were also obtained when aryl and alkylamines were reacted with a more atom-economical loading (1.0 equiv.; 70% and 72% ave. yields, respectively). Arylamines with either highly electron withdrawing substituents (e.g. CO₂H) or pi-deficient heterocycles (e.g. variously substituted aminopyridines) did not work well under these conditions, and reaction with ureas and/or amino acids did not give detectable products. Work-up was exceedingly simple, and involved simple collection and washing of product on a sintered glass funnel. Products were obtained in analytically pure form and required approximately 1 h to prepare, start to finish.     

Explicit constructions and properties of generalized shift-invariant systems in $L^{2}(\mathbb {R})$

Generalized shift-invariant (GSI) systems, originally introduced by Hernández et al. and Ron and Shen, provide a common frame work for analysis of Gabor systems, wavelet systems, wave packet systems, and other types of structured function systems. In this paper we analyze three important aspects of such systems. First, in contrast to the known cases of Gabor frames and wavelet frames, we show that for a GSI system forming a frame, the Calderón sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calderón sum is bounded by the lower frame bound and show that under a weak assumption the condition is equivalent with the local integrability condition introduced by Hernández et al. Second, we provide explicit and general constructions of frames and dual pairs of frames having the GSI-structure. In particular, the setup applies to wave packet systems and in contrast to the constructions in the literature, these constructions are not based on characteristic functions in the Fourier domain. Third, our results provide insight into the local integrability condition (LIC).     

An improved scheme for determining heat transfer correlations from heat exchanger regression models with three unknowns

A nonlinear regression scheme for determining heat transfer correlations from overall heat exchanger measurements is presented. The proposed scheme is valid for any two-fluid heat exchanger data where only one fluid's thermal resistance varies. The resulting regression model has three unknowns; two are associated with the variable resistance, and the third is the sum of all other resistances. The model can be analyzed only by using nonlinear regression because it cannot be transformed into linear form. Unlike existing approximate methods of analysis (known as modified Wilson plots), the proposed scheme is guaranteed to converge if a solution exists. The scheme was successfully tested by analyzing data collected on a tube-fin heat exchanger with continuous plain fins.     

Gabor Frames in $${\ell }^2({\mathbb {Z}})$$ ℓ 2 ( Z ) and Linear Dependence

We prove that an overcomplete Gabor frame in \({\ell }^2({\mathbb {Z}})\) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in \({\ell }^2({\mathbb {Z}})\) with modulation parameter 1 / M and translation parameter N for some \(M,N\in {\mathbb {N}},\) and generated by a finite sequence g in \({\ell }^2({\mathbb {Z}})\) with K nonzero entries.

     

Microsoft Excel as a supplement in a high school statistics course

This study was conducted in the spring of 2002 at Arlington High School in a Probability and Statistics course for junior and senior level students. Two sections of the class were compared with respect to the use/non-use of EXCEL in the course. The experimental group (the one using EXCEL) outperformed the control group on five of the six teacher-created unit tests and on both of the criterion referenced assessments used by the district; however, the differences were not statistically significant at the 0.05 level. Furthermore, the experimental group expressed very positive feelings towards the use of the software in a survey administered at the conclusion of the course.