Ramanujan and the Regular Continued Fraction Expansion of Real Numbers

In some recent papers, the authors considered regular continued fractions of the form $$[a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ],$$ where $a_{0} \geq 0$, $a \geq 2$ and $m \geq 1$ are integers. The limits of such continued fractions, for general $a$ and in the cases $m=1$ and $m=2$, were given as ratios of certain infinite series. However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases $m=1$ and $m=2$ and also use known results about other $q$-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions.Comment: 14 page

Geomagnetically Trapped Radiation Produced by a High-Altitude Nuclear Explosion on July 9, 1962

Geomagnetically trapped radiation produced by a high altitude nuclear explosio

Diversity RF receiving system with improved phase-lock characteristics

Improved diversity receiving system automatically utilizes the combined output from its two independent receiving channels /with cross- polarized receiving antennas/ to increase the reliability of maintaining the requisite phase lock for optimum signal reception. It is adapted for use with AM, PM, or narrow band FM signals

Note on the Electron Energy Spectrum in the Inner Van Allen Belt

Electron energy spectrum in the inner van allen bel

The use of standard electrode potentials to predict the taste of solid metals

Not all metals taste equally metallic when placed in the mouth. While much work has been done to examine the metallic taste sensations arising from metal ions in solutions, there is comparatively less known about the taste of solid metals. In this study seven metals in the form of spoons were used to compare the perception of taste arising from solid utensils placed inside the mouth. 32 participants tasted seven spoons of identical dimensions plated with each of the following metals: gold, silver, zinc, copper, tin, chrome and stainless steel. More negative standard electrode potentials were found to be good predictors of solid metals that had tastes scoring highest for the taste descriptors strong, bitter, and metallic. Thus, it was found that both gold and chrome (having the most positive standard electrode potentials) were considered the least metallic, least bitter and least strong tasting of the spoons. Zinc and copper (having the most negative standard electrode potentials) were the strongest, most metallic, most bitter, and least sweet tasting of the spoons. We conclude that gold and chrome have tastes that are less strong than metals with lower standard electrode potential

The Energetic Costs of Cellular Computation

Cells often perform computations in response to environmental cues. A simple example is the classic problem, first considered by Berg and Purcell, of determining the concentration of a chemical ligand in the surrounding media. On general theoretical grounds (Landuer's Principle), it is expected that such computations require cells to consume energy. Here, we explicitly calculate the energetic costs of computing ligand concentration for a simple two-component cellular network that implements a noisy version of the Berg-Purcell strategy. We show that learning about external concentrations necessitates the breaking of detailed balance and consumption of energy, with greater learning requiring more energy. Our calculations suggest that the energetic costs of cellular computation may be an important constraint on networks designed to function in resource poor environments such as the spore germination networks of bacteria.Comment: 9 Pages (including Appendix); 4 Figures; v3 corrects even more typo