On the Galois groups of Legendre polynomials

Ever since Legendre introduced the polynomials that bear his name in 1785, they have played an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers. In this paper, assuming Stieltjes’ conjecture, we formulate a conjecture about the Galois groups of Legendre polynomials, to the effect that they are “as large as possible,” and give theoretical and computational evidence for it.     

Synthesis of Novel 1,2,3-Triazole-dihydro[3,2-c]chromenones as Acetylcholinesterase Inhibitors

A novel series of 1,2,3-triazole-dihydro[3,2-c]chromenone derivatives were synthesized through an efficient three-step reaction starting from 4-hydroxybenzaldehyde. All the newly synthesized compounds were characterized by infrared and NMR spectroscopy as well as elemental analysis and evaluated for their acetylcholinesterase inhibitory activity.     

Multi‐faceted Reactivity of Alkyltellurophenols Towards Peroxyl Radicals: Catalytic Antioxidant Versus Thiol‐Depletion Effect

Hydroxyaryl alkyl tellurides are effective antioxidants both in organic solution and aqueous biphasic systems. They react by an unconventional mechanism with ROO^. radicals with rate constants as high as 10⁷ M ^?1 s^?1 at 303 K, outperforming common phenols. The reactions proceed by oxygen atom transfer to tellurium followed by hydrogen atom transfer to the resulting RO^. radical from the phenolic OH. The reaction rates do not reflect the electronic properties of the ring substituents and, because the reactions occur in a solvent cage, quenching is more efficient when the OH and TeR groups have an ortho arrangement. In the presence of thiols, hydroxyaryl alkyl tellurides act as catalytic antioxidants towards both hydroperoxides (mimicking the glutathione peroxidases) and peroxyl radicals. The high efficiency of the quenching of the peroxyl radicals and hydroperoxides could be advantageous under normal cellular conditions, but pro‐oxidative (thiol depletion) when thiol concentrations are low.     

Synthesis and Evaluation of Coumarin–Resveratrol Hybrids as 15-Lipoxygenaze Inhibitors

A series of coumarin–resveratrol hybrids, 3-arylcoumarin derivatives 3a–u, were synthesized through the intermolecular condensation reaction of various salicylaldehydes and phenylacetic acids in the presence of 1,4-diazabicyclo[2.2.2]octane under solvent-free conditions. All the synthesized compounds were screened for their inhibitory potency against soybean 15-lipoxygenase. Among them, three compounds (3c, 3j, and 3q) showed good enzyme-inhibitory activities.     

Synthesis of Isoindolo[2,1‐a]quinazoline‐5,11‐dione Derivatives via the Reductive One‐Pot Reaction of N‐Substituted 2‐Nitrobenzamides and 2‐Formylbenzoic Acids

Various isoindolo[2,1‐a]quinazoline‐5,11‐dione derivatives 3 were synthesized in good yields by means of the reductive reaction of N‐substituted 2‐nitrobenzamides 1 and 2‐formylbenzoic acids 2 in the presence of SnCl₂?2 H₂O under reflux in EtOH (Scheme, Table). The procedure needed two steps, the reduction of the nitro group of the 2‐nitrobenzamide and ring closure by nucleophilic addition of the NH₂ group to both the formyl and carboxylic acid C?O groups.     

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h³) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.     

The necessary length of carbon nanotubes required to optimize solar cells

Background  

In recent years scientists have been trying both to increase the efficiency of solar cells, whilst at the same time reducing dimensions and costs. Increases in efficiency have been brought about by implanting carbon nanotubes onto the surface of solar cells in order to reduce the reflection of sunrays, as well as through the insertion of polymeric arrays into the intrinsic layer for charge separation.     

Detector-Based Calibration Method for High-Accuracy Solar UV Measurements

A novel method for calibrating the absolute responsivity of solar UV spectroradiometers has been developed and tested. The method is based on calibrated filter radiometers constructed from a detector, a precision aperture, a band-pass filter and devices for temperature stabilization. The filter radiometers utilize a trap detector with very low reflectance. The filter and the detector can therefore be characterized separately. As an example the detector-based calibration is compared at 312 nm wavelength with lamp-based calibration by measuring the irradiance of a deuterium lamp with both the filter radiometer and the lamp-calibrated spectroradiometer. The agreement between the results is at the level of 1%, well within the estimated uncertainties of both methods.     

Microwave Assisted Mannich Reaction of Terminal Alkynes on Alumina

Summary.  Terminal alkynes, secondary amines, and formaldehyde undergo a Mannich reaction at room temperature in the presence of CuCl on Al₂O₃ without any organic solvent as reaction medium. The reaction can be promoted by microwave irradiation and is complete within one minute. Received May 2, 2001. Accepted (revised) June 25, 2001     

Synthesis of Novel Pyrazino[2,1‐a]isoindolediones via Intramolecular Hydroamination of 2,3‐Dihydro‐3‐oxo‐2‐(prop‐2‐yn‐1‐yl)‐1H‐isoindole‐1‐carboxamides

Novel derivatives of pyrazino[2,1‐a]isoindolediones were synthesized through 6‐exo‐dig intramolecular hydroamination of 2,3‐dihydro‐3‐oxo‐2‐(prop‐2‐yn‐1‐yl)‐1H‐isoindole‐1‐carboxamides followed by 1,3‐H shift, in the presence of sodium hydride in DMF at 80°. All products were obtained in good yields (60 – 80%) within short reaction time (40 – 60 min).