Regiochemical control in the diels-alder reactions of substituted naphthoquinones: orientation in the synthesis of benz[a]anthraquinones

The regioselectivity in the Diels-Alder reactions of styrene with certain substituted 1,4-naphthoquinones was explored. The results were consistent with predictions based upon analysis of dione polarization.     

A Markov partition that reflects the geometry of a hyperbolic toral automorphism

We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the -torus. The transition matrix is the transpose of the matrix induced by the automorphism in -dimensional homology, provided this is non-negative. (Here denotes the expanding dimension.) That condition is satisfied, at least for some power of the original automorphism, under a certain non-degeneracy condition on the Galois group of the characteristic polynomial. The rectangles are constructed by an iterated function system, and they resemble the product of the projection of a -dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal -dimensional face onto the stable subspace.

     

Miller–Urey Revisited: When Lightning Strikes the Earth

A discharge is arced to different solutions containing the simple molecular species water, methanol, and ammonia. We show that the impact of the discharge at the discharge-solution interface produces a range of organic molecules, including amino acids and polymers containing carboxylate, amine, imine, and cyano groups. The classic Miller-Urey experiment, of which there are hundreds of variations, changing parameters such as gas composition, pressures, and voltage have been tested, involves the production of simple amino acids and other species in the gas phase in an arc discharge. This works emphasis is the production of various chemical species at a discharge-liquid interface. The analysis of the product is conducted by IR, UV–VIS, LC–MS and MALDI–MS. As a laboratory exercise or a demonstration, this simple derivation of the Miller-Urey experiment can be used in a variety of teaching settings, from high school through advanced undergraduate research classes to demonstrate the basic hypothesis of how life on Earth may have started.     

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.