The Newman--Shapiro problem

We give a negative answer to the Newman--Shapiro problem on weighted approximation for entire functions formulated in 1966 and motivated by the theory of operators on the Fock space. There exists a function in the Fock space such that its exponential multiples do not approximate some entire multiples in the space. Furthermore, we establish several positive results under different restrictions on the function in question.Comment: 28 page

Shift of Shapiro Step in High-Temperature Superconductor

Influence of the charge imbalance effect on the system of intrinsic Josephson junctions of high temperature superconductors under external electromagnetic radiation are investigated. We demonstrate that the charge imbalance is responsible for a slope in the Shapiro step in the IV-characteristic. The nonperiodic boundary conditions shift the Shapiro step from the canonical position which determined by a frequency of external radiation. We also demonstrate how the system parameters affect on the shift of Shapiro step.Comment: arXiv admin note: text overlap with arXiv:1601.0445

Half-Integer Shapiro Steps in a Short Ballistic InAs Nanowire Josephson Junction

We report on half-integer Shapiro steps observed in an InAs nanowire Josephson junction. We observed the Shapiro steps of the short ballistic InAs nanowire Josephson junction and found anomalous half-integer steps in addition to the conventional integer steps. The half-integer steps disappear as the temperature increases or transmission of the junction decreases. These experimental results agree closely with numerical calculation of the Shapiro response for the skewed current phase relation in a short ballistic Josephson junction

The Mahler measure of the Rudin-Shapiro polynomials

Littlewood polynomials are polynomials with each of their coefficients in {-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials have been known before

Slowing Down and Speeding Up PSR's Periods: A Shapiro Telescope tracing Dark Matter

Dark matter influence globally galactic Keplerian motions and individually, by microlensing, star's acromatic luminosities. Moreover gravity slows down time as probed by gravitational redshift. Therefore pulsar's (PSR's) periods may record, by slowing and speeding up, any Dark MACHO crossing along the line-of-sight. This phase delay, a Shapiro Phase Delay, is a gravitational integral one and has been probed since 1964 on planets by radar echoes. We discovered in PSRs catalog a few rare PSRs (with negative period derivatives) consistent with speeding up by Shapiro delay due to MACHOs. We propose to rediscover the Shapiro effect on our solar system (for Jupiter, Mars, Sun) to calibrate a new "Shapiro Telescope Array" made by collective monitoring of the sample of PSRs along the ecliptic plane.Comment: 5 pages,3 figures,conference IDM 98, Buxton U

Spectrum of a Rudin-Shapiro-like sequence

We show that a recently proposed Rudin-Shapiro-like sequence, with balanced weights, has purely singular continuous diffraction spectrum, in contrast to the well-known Rudin-Shapiro sequence whose diffraction is absolutely continuous. This answers a question that had been raised about this new sequence