Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

Thieno[3,4-b]pyrazines and their applications to low band gap organic materials

The application of fused-ring thieno[3,4-b]pyrazines in conjugated organic polymers has been found to be a powerful approach to the production of low band gap materials. While thieno[3,4-b]pyrazine-based materials date back to the early 1990s, significant advances in the preparation and scope of thieno[3,4-b]pyrazine-based materials have been reported in recent years, primarily in response to the increasing demand for reduced band gap materials in photovoltaic devices. In this review, we provide an overview of thieno[3,4-b]pyrazines and their application to conjugated materials, highlighting in particular the recent advances in the breadth of thieno[3,4-b]pyrazine building blocks and the promise of tuning materials to achieve optimal properties for specific applications.     

Structural effects on the electronic properties of extended fused-ring thieno[3,4-b]pyrazine analogues

The synthesis and characterization of the extended thieno[3,4-b]pyrazine analogues acenaphtho[1,2-b]thieno[3,4-e]pyrazine (3a), 3,4-dibromoacenaphtho[1,2-b]thieno[3,4-e]pyrazine (3b), 3-octylacenaphtho[1,2-b]thieno[3,4-e]pyrazine (3c), dibenzo[f,h]thieno[3,4-b]quinoxaline (4), and thieno[3',4':5,6]pyrazino[2,3-f][1,10]phenanthroline (5) are reported. Comparison of structural, electrochemical, and photophysical properties to those of simple thieno[3,4-b]pyrazines are provided in order to provide structure-function relationships within this series of compounds.     

Thieno[3,4‐b]pyrazines and Its Extended Analogs: Important Buildings Blocks for Conjugated Materials

This review describes the synthesis and characterization of thieno[3,4‐b]pyrazines and its extended fused‐ring analogs as important building blocks for the production of low‐band gap conjugated materials.