Relatives of the Russell Paradox

A formula ?(x) in the one non-logical symbol ? with one free variable x is Russell if the sentence Vχ(χ ? τ ? Φ (χ)) is logically valid. This note describes a pattern common to the classical examples of Russell formulas, adds a couple of new ones, and constructs many formulas that are near-Russell.     

Some recent progress in applied mathematics (Ⅰ)

Some activities of applied mathematics are seen from different angles in a two part series. In part 1, we will emphasize on mathematical modeling, exponential prediction models. On a systemic construction of the quantitative mathematics, it is shown that there is an impassible chasm between pure and applied mathematics, that existance and structures of systems are independent of human consciousness. For the purpose of prediction, concepts of calculus are generalized to discrete time series.     

Braess's Paradox in large random graphs

Braess's Paradox is the counterintuitive fact that removing edges from a network with “selfish routing” can decreasethe latency incurred by traffic in an equilibrium flow. We prove that Braess's Paradox is likely to occur in a natural random network model: with high probability, there is a traffic rate and a set of edges whose removal improves the latency of traffic in an equilibrium flow by a constant factor. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Protein Folding and β‐Sheet Proteins

The aim of this comprehensive review is to critically evaluate the progress in research in the area of protein folding. In the first section, we discuss the various models proposed to explain the protein folding paradox. In the succeeding section of the review, a detailed account of the developments in our understanding of the folding path ways of β‐sheet proteins is provided.