Partition identities involving gaps and weights

We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers-Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.

     

A Critique of Recent Semi‐Classical Spin‐Half Quantum Plasma Theories

Certain recent semi‐classical theories of spin‐half quantum plasmas are examined with regard to their internal consistency, physical applicability and relevance to fusion, astrophysical and condensed matter plasmas. It is shown that the derivations and some of the results obtained in these theories are internally inconsistent and contradict well‐established principles of quantum and statistical mechanics, especially in their treatment of fermions and spin. Claims of large semi‐classical effects of spin magnetic moments that could dominate the plasma dynamics are found to be invalid both for single‐particles and collectively. Larmor moments dominate at high temperature while spin moments cancel due to Pauli blocking at low temperatures. Explicit numerical estimates from a variety of plasmas are provided to demonstrate that spin effects are indeed much smaller than many neglected classical effects. The analysis presented suggests that the aforementioned ‘Spin Quantum Hydrodynamic’ theories are not relevant to conventional laboratory or astrophysical plasmas. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Single-mode low-loss chalcogenide glass waveguides for the mid-infrared

We demonstrate the design, fabrication, and characterization of single-mode low-loss waveguides for mid-infrared (MIR) wavelengths. Planar waveguide structures were fabricated from multilayer thin films of arsenic-based chalcogenide glasses followed by the creation of channel waveguides by using the photodarkening effect. Propagation losses as low as 0.5 dB/cm were measured for a quantum cascade laser end-fire coupled into the waveguides. This is a first step toward the design and fabrication of integrated optical components for MIR applications.     

Reaction of β-keto ethyleneacetals with hydroxylamine: A correction

A reaction of 2-(2-nitrobenzoylmethyl)-1,3-dioxolane ( 3 ) with hydroxylamine, followed by acid catalyzed cyclization, produced 5-(2-nitrophenyl)isoxazole ( 5 ) as the only isolable product, whereas 2-(benzoylmethyl)-1,3-dioxolane ( 9 ) under identical conditions produced a 2.5:1 mixture of 3-phenyl and 5-phenylisoxazoles 10 and 11 . These findings contradict the literature report that β-keto ethyleneacetals on treatment with hydroxylamine produce exclusively 3-substituted isoxazoles. As an additional proof, 3-(2-nitrophenyl)isoxazole ( 8 ) was prepared by an unambiguous method via the nitrile oxide route for comparison. The intermediate obtained on treatment of 2-(2-nitrobenzoylmethyl)-1,3-dioxolane ( 3 ) with hydroxylamine was found to be an isomeric mixture of 5-hydroxy-5-(2-nitrophenyl)-2-isoxazoline ( 4 ) and the syn and anti mono-oximes 19 (at least in solution), either of which could give 5-(2-nitrophenyl)isoxazole ( 5 ) on acid treatment. A mechanistic rationale is provided to explain the anomalous results.     

Man-made plutonium in freshwater and marine environments

Significant concentrations of plutonium isotopes are found in lake and coastal marine sediments accumulating at rates of 1–5 mm/yr. The activity levels of^{239, 240}Pu range between 0·7–3·3 dpm/g for post 1960 sediments. The measured and the calculated integrated activities of²³⁹Pu in the lake sediments are in agreement. This suggests that plutonium is chemically reactive in freshwater environments and it is removed to sediments from the water column shortly after its injection. A delay of about three years has been observed between the peak of plutonium fallout and its incorporation in the varved sediment from Santa Barbara basin. Based on this observation the mean size of particles transporting plutonium to the basin sediments is estimated to be about three microns. The usefulness of plutonium isotopes as a tracer nuclide for estimating sedimentation rates for the last two decades have been evaluated.     

On the decomposition of the Feynman propagator

The Feynman propagator, in momentum representation, is a four-dimensional transform over space and time variables. If the space and time integrations are performed separately, the propagator can be decomposed into two parts, one corresponding to positive and the other to negative energy intermediate state. By the use of this decomposed propagator, the relative contributions of the positive and negative energy intermediate states to the matrix element can be estimated. For example in Compton scattering it leads to the apparently paradoxical result that in the “nonrelativistic approximation” it is only the negative energy intermediate state that contributes to the matrix element.     

A new approach to the cascade theory

The cascade theory of cosmic rays is investigated from a new point of view; we deal with the number of particles produced in a certain thickness of matter, the energy of each particle being greater than E at the point of its production. This new approach leads to an elegant asymptotic theory for large thicknesses and it is better adapted to the interpretation of experiments in nuclear emulsions.