On the Schröder–Bernstein property for modules

The well known Schröder–Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. Kaplansky asked whether direct summands subisomorphic abelian groups are always isomorphic? The question generated a great deal of interest. The study of this question for the general class of modules has been somewhat limited. We extend the study of this question for modules in this paper. We say that a module Msatisfies the Schröder–Bernstein property (S-B property) if any two direct summands of M which are subisomorphic to direct summands of each other, are isomorphic. We show that a large number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. It is also shown that over a Noetherian ring R, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings R for which every R-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain R, any two quasi-continuous subisomorphic R-modules are isomorphic if and only if R is a PID. We study other conditions related to the S-B property and obtain characterizations of certain classes of rings via those conditions. Examples which delimit and illustrate our results are provided.     

Micelle‐catalyzed reaction of ninhydrin with DL‐valine in the absence and presence of organic solvents

Kinetics of the DL ‐valine‐ninhydrin reaction has been studied spectrophotometrically under varying conditions of [CTAB], [ninhydrin], [DL ‐valine], pH, temperature, and %(v/v) organic solvents (solvents used: 1‐propanol, methylcellosolve, acetonitrile, and dimethyl sulfoxide). Addition of CTAB and increase in the proportion of organic solvents, both showed catalyzing effect on the reaction. The effect of simultaneous presence of CTAB and DMSO in the reaction mixture has also been seen. The rate profiles obtained for solutions containing from 10% to 70% DMSO exhibited clear maxima that shifted progressively to higher concentrations of CTAB. The experimental results are explained in terms of specific solvent effects and the formation of stoichiometric hydrate DMSO · 2H₂O and the inhibitory effect of DMSO on micelle formation. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 634–642, 2006     

Experimental and theoretical studies on the functionalization reactions of 4-benzoyl-1,5-diphenyl-1H-pyrazole-3-carboxylic acid and acid chloride with 2,3-diaminopyridine

The 1H-pyrazole-3-carboxylic acid 2 was converted in good yield (69%) into the corresponding 1H-pyrazole-3-carboxamide 5 via reaction of the acid chloride 3 with 2,3- diaminopyridine (4). A different product, the 3H-imidazo[4,5-b] pyridine derivative 6, was formed from the reaction of 3 with 4 and base in benzene for 5 hours. The structures of the synthesized compounds were determined spectroscopically. The mechanism of the reaction between 3 and 4 was examined theoretically.     

Sur quelques Théorémes globaux en geometrie finslérienne

Résumè Le travail suivant établit des théorèmes globaux relatifs aux variétes finslériennes completes dont le tenseur de courbure vérifie certaines hypothèses. Le théorème de Hopf-Rinow est d’abord démontrè, puis au moyen du calcul variationnel, on établit une géneralisation des théoremes de Myers, de Cartan et de L. W. Green.     

Conformational analysis of XA and AX dipeptides in water by electronic circular dichroism and 1H NMR spectroscopy

We measured the temperature-dependent electronic circular dichroism (ECD) spectra of AX, XA, and XG dipeptides in D2O. The spectra of all XA and AX peptides indicate a substantial population of the polyproline II (PPII) conformation, while the ECD spectra of LG, KG, PG, and AG were found to be quantitatively different from the alanine-based dipeptides. Additional UV absorption data indicate that the ECD spectra of the XG peptides stem from electronic coupling between the peptide and the C-terminal group, and that spectral differences reflect different orientations of the latter. We also measured the 1H NMR spectra of the investigated dipeptides to determine the 3JHalphaNH coupling constants for the C-terminal residue. The observed temperature dependence of the ECD spectra and the respective room-temperature 3JHalphaNH coupling constants were analyzed by a two-state model encompassing PPII and a beta-like conformation. The PPII propensity of alanine in the XA series is only slightly modulated by the N-terminal side chain, and is larger than 50%. As compared to AA, XA peptides containing L, P, S, K V, E, T, and I all cause a relative stabilization of the extended beta-strand conformation. The PPII fractions of XA peptides varied between 0.64 for AA and 0.58 for DA, whereas the PPII fractions of AX peptides were much lower. From the investigated AX peptides, only AL and AQ showed the expected PPII propensity. We found that AT, AI, and AV clearly prefer an extended beta-strand conformation. A quantitative comparison of AA, AAA, and AAAA revealed a hierarchy AAAA > AAA approximately AA for the PPII population, in agreement with predictions from MD calculations and results from Raman optical activity studies (McColl et al. J. Am. Chem. Soc. 2004, 126, 5076).     

Pediatric abdominal masses: diagnostic accuracy of diffusion weighted MRI

Purpose

To retrospectively identify apparent diffusion coefficient (ADC) values of pediatric abdominal mass lesions, to determine whether measured ADC of the lesions and signal intensity on diffusion-weighted (DW) images allow discrimination between benign and malignant mass lesions.

Materials and Methods

Approval for this retrospective study was obtained from the institutional review board. Children with abdominal mass lesions, who were examined by DW magnetic resonance imaging (MRI) were included in this study. DW MR images were obtained in the axial plane by using a non breath-hold single-shot spin-echo sequence on a 1.5-T MR scanner. ADCs were calculated for each lesion. ADC values were compared with Mann–Whitney U test. Receiver operating characteristic curve analysis was performed to determine cut-off values for ADC. The results of visual assessment on b800 images and ADC map images were compared with chi-square test.

Results

Thirty-one abdominal mass lesions (16 benign, 15 malignant) in 26 patients (15 girls, 11 boys, ranging from 2 days to 17 years with 6.9 years mean) underwent MRI. Benign lesions had significantly higher ADC values than malignant ones (P<.001). The mean ADCs of malignant lesions were 0.84±1.7×10⁻³ mm²/s, while the mean ADCs of the benign ones were 2.28±1.00×10⁻³ mm²/s. With respect to cutoff values of ADC: 1.11×10⁻³ mm²/s, sensitivity and negative predictive values were 100%, specificity was 78.6% and positive predictive value was 83.3%. For b800 and ADC map images, there were statistically significant differences on visual assessment. All malignant lesions had variable degrees of high signal intensity whereas eight of the 16 benign ones had low signal intensities on b800 images (P<.001). On ADC map images, all malignant lesions were hypointense and most of the benign ones (n=11, 68.7%) were hyperintense (P<.001).

Conclusion

DW imaging can be used for reliable discrimination of benign and malignant pediatric abdominal mass lesions based on considerable differences in the ADC values and signal intensity changes.     

PBL as a pedagogical approach for integrated STEM: Evidence from prospective teachers

Problem-based learning (PBL) and science, technology, engineering, and mathematics (STEM) are two acronyms widely visible in education literature today. However, few studies have explored these in connection with one another, specifically with regard to teacher preparation. This study investigated how 47 prospective elementary teachers developed PBL units and how they integrated STEM and other disciplines into those units. It also addressed the affordances and constraints of integrated STEM as perceived by the prospective elementary teachers. Data sources in this multimethod study included PBL units and interviews. Findings revealed that all of the units integrated at least two of the STEM disciplines, as well as literacy, in a variety of ways. The prospective teachers articulated perceived benefits of integrated STEM, such as: making connections across content areas, preparing students for the real world, teaching students that failure is not a bad thing, and providing future opportunities. They also addressed perceived barriers of integrated STEM, such as: having limited experience with the content, diminishing the effect of individual content areas, and needing better curriculum alignment. Overall, this study provides evidence that PBL can be a pedagogical approach to integrate STEM. Implications for teachers, teacher educators, and curriculum specialists are discussed.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

Perturbations Results for Some Classes Related to Browder Linear Relations and Applications

This paper is devoted to the investigation of the perturbation problem of right (left) Browder linear relations and lower (upper) semi-Browder linear relations under commuting compact linear relations. Further, our results are used to show the invariance of Browder’s spectrum.     

Oscillatory and Periodic Solutions of Impulsive Differential Equations with Piecewise Constant Argument

We study the existence of oscillatory and periodic solutions of a class of first order scalar impulsive delay differential equations with piecewise constant argument.