A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

Semiconductor Emitters in Entropy Sources for Quantum Random Number Generation

Random number generation (RNG) is needed for a myriad of applications ranging from secure communication encryption to numerical simulations to sports and games. However, generating truly random numbers can be elusive. Pseudorandom bit generation using computer algorithms provides a high random bit generation rate. Nevertheless, the reliance on predefined algorithms makes it deterministic and predictable once initial conditions are known. Relying on physical phenomena (such as measuring electrical noise or even rolling dice) can achieve a less predictable sequence of bits. Furthermore, if the physical phenomena originate from quantum effects, they can be truly random and completely unpredictable due to quantum indeterminacy. Traditionally, physical RNG is significantly slower than pseudorandom techniques. To meet the demand for high-speed RNG with perfect unpredictability, semiconductor light sources are adopted as parts of the sources of randomness, i.e., entropy sources, in quantum RNG (QRNG) systems. The high speed of their noise, the high efficiency, and the small scale of these devices make them ideal for chip-scale QRNG. Here, the applications and recent advances of QRNG are reviewed using semiconductor emitters. Finally, the performance of these emitters is compared and discuss their potential in future technologies.     

A new naturally derived photosensitizer and its phototoxicity on head and neck cancer cells

In our screening for photosensitizers from natural resources, 15(1)-hydroxypurpurin-7-lactone ethyl methyl diester (compound 1) was isolated for the first time from an Araceae plant. To evaluate the efficacy of compound 1 as a photosensitizer for head and neck cancers, compound 1 was studied in reference to a known photosensitizer pheophorbide-a (Pha), in terms of photophysical properties, singlet oxygen generation and in in vitro experiments (intracellular uptake and phototoxicity assays) in two oral (HSC2 and HSC3) and two nasopharyngeal (HK1 and C666-1) cancer cell lines. In this study, compound 1 exhibited higher intracellular uptake over 24 h compared with Pha in both HSC3 and HK1 cells. When activated by ≥4.8 J cm(-2) of light, compound 1 was slightly more potent as a photosensitizer than Pha by consistently having marginally lower IC(50) values across different cell lines. In flow cytometry experiments to study the mechanism of photoactivated cell death in HSC3, compound 1 was observed to induce more pronounced apoptosis compared with Pha, which may have been driven by the transient G(2)/M cell cycle block which was also observed. These promising results on compound 1 warrant its further investigation as a clinically useful photodynamic therapy agent for head and neck cancer.     

Molecular dynamics calculation of activation volumes

We propose a novel approach toward calculating activation and reaction volumes based on MD simulations of reaction systems. The accuracy of the calculated volumes, verified by the comparison to the experimental data, is suitable for quantitative analysis of the experimental volumes of activations in terms of structural parameters of the corresponding transition states.     

Characterization of porosity in a 19th century painting ground by synchrotron radiation X-ray tomography

The study of the early oeuvre of the Swiss painter Cuno Amiet (1868–1961) has revealed that, up to 1907, many of his grounds were hand applied and are mainly composed of chalk, bound in protein. These grounds are not only lean and absorbent, but also, as Synchrotron radiation X-ray microtomography has shown, porous. Our approach to the characterization of pore structure and quantity, their connectivity, and homogeneity is based on image segmentation and application of a clustering algorithm to high-resolution X-ray tomographic data. The issues associated with the segmentation of the different components of a ground sample based on X-ray imaging data are discussed. The approach applied to a sample taken from “Portrait of Max Leu” (1899) by Amiet revealed the presence of three sublayers within the ground with distinct porosity features, which had not been observed optically in cross-section. The upper and lower layers are highly porous with important connectivity and thus prone to water uptake/storage. The middle layer however shows low and nonconnected porosity at the resolution level of the X-ray tomography images, so that few direct water absorption paths through the entire sample exist. The potential of the method to characterize porosity and to understand moisture-related issues in paint layer degradation are discussed.     

Does the Realistic Interpretation of a Mathematical Superposition of States Imply a Mixture?

Contrary to conventional view, it is shown that, for an ensemble of either single-particle systems or multi-particle systems, the realistic interpretation of a mathematical superposition of states that mathematically describes the ensemble does not imply that the ensemble is a mixture. Therefore it cannot be argued, as is conventionally done, that the realistic interpretation is wrong on the basis that some predictions derived from the mixture are different from the corresponding predictions derived from the mathematical superposition of states.     

An application of theta functions to the band problem

The utility of theta function-modulated plane wave bases — the modulation is essentially by site-centered Gaussians — in the band problem is shown. The Kronig-Penney model is used as a test system.