Strong and coalition-proof political equilibria under plurality and runoff rule

We analyze the effects of voter coordination on the equilibrium of voting games. Specifically, we analyze the concepts of strong equilibrium and coalition-proof equilibrium for plurality rule and runoff rule elections. We characterize these equilibria in the three candidate case, and provide results for important special cases in the general multicandidate case.      

Access to Indoles via Diels–Alder Reactions of 5‐Methylthio‐2‐vinylpyrroles with Maleimides

Diels–Alder reactions of 5‐methylthio‐2‐vinyl‐1H‐pyrroles with maleimides followed by isomerization gave tetrahydroindoles in moderate to good yield. Aromatization using activated MnO₂ in refluxing toluene gave the corresponding 2‐methylthioindoles in good yields, and demethylthioation using Raney nickel gave the 2‐H indoles in excellent yields. The protection of the adducts produced aromatization in improved yield, demonstrating the effectiveness of the methylthio group as a protecting group for pyrroles; however, 5‐methylthio‐2‐vinylpyrrole was shown to perform with slightly less efficiency than 2‐vinylpyrrole in Diels–Alder reactions, indicating the protective group was more deactivating than desired. This route toward indoles offers high convergency and conveniently available starting materials that are easily purified. Bis‐methylthioated vinylpyrroles were shown to have potential as highly activated Diels–Alder dienes.     

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.     

Dynamic control of 3D chemical profiles with a single 2D microfluidic platform

Dynamic control of three-dimensional (3D) chemical patterns with both high precision and high speed is important in a range of applications from chemical synthesis, flow cytometry, and multi-scale biological manipulation approaches. A central challenge in controlling 3D chemical patterns is the inability to create rapidly tunable 3D profiles with simple and direct approaches that avoid complicated microfabrication. Here, we present the ability to rapidly and precisely create 3D chemical patterns using a single two-dimensional (2D) microfluidic platform. We are not only able to create these 3D patterns, but can rapidly switch from one mode to another (e.g. from a focused to a defocused pattern in less than 1 second) via simple changes in inlet pressures. A feedback control scheme with a pressure modulation mechanism controls the pressure changes. In addition to experiments, we conducted computational simulations for guiding the optimum design of the channels as well as revealing the sensitivity of the patterns to the channel dimensions; these simulations have high experimental correlations. We also show that microvortices play an important role in creating these tunable 3D patterns in this microfluidic platform. We quantitatively determine the degrees of the focused patterns in 2D cross-sections using a focus index with a 2D Gaussian function. Our integrated approach combining feedback control with simple microfluidics will be useful for researchers in diverse disciplines including chemistry, engineering, physics, and biology.