Microring-Assisted Coupling Between Dissimilar Waveguides

The use of microring resonators to assist in the evanescent field coupling between dissimilar waveguides is proposed and analyzed. Theoretical analysis based on the coupled mode theory and nu-merical example show that complete cross power transfers can be obtained near the microring resonances. Applications of the device include power dividers, low-power thermo-optic or electro-optic switches, and modulators.     

<Emphasis Type="Italic">The Mathematical Intelligencer</Emphasis> Flunks the Olympics

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals.     

Toward a History of Mathematics Focused on Procedures

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.     

Cauchy’s Infinitesimals,His Sum Theorem,and Foundational Paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.     

The DIX domain targets dishevelled to actin stress fibres and vesicular membranes

Colorectal cancer results from mutations in components of the Wnt pathway that regulate beta-catenin levels. Dishevelled (Dvl or Dsh) signals downstream of Wnt receptors and stabilizes beta-catenin during cell proliferation and embryonic axis formation. Moreover, Dvl contributes to cytoskeletal reorganization during gastrulation and mitotic spindle orientation during asymmetric cell division. Dvl belongs to a family of eukaryotic signalling proteins that contain a conserved 85-residue module of unknown structure and biological function called the DIX domain. Here we show that the DIX domain mediates targeting to actin stress fibres and cytoplasmic vesicles in vivo. Neighbouring interaction sites for actin and phospholipid are identified between two helices by nuclear magnetic resonance spectroscopy (NMR). Mutation of the actin-binding motif abolishes the cytoskeletal localization of Dvl, but enhances Wnt/beta-catenin signalling and axis induction in Xenopus. By contrast, mutation of the phospholipid interaction site disrupts vesicular association of Dvl, Dvl phosphorylation, and Wnt/beta-catenin pathway activation. We propose that partitioning of Dvl into cytoskeletal and vesicular pools by the DIX domain represents a point of divergence in Wnt signalling.