Short intervals asymptotic formulae for binary problems with primes and powers, I: density 3/2

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case

Short intervals asymptotic formulae for binary problems with primes and powers, II: density 1

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case

Satellite measurement of the Hannay angle

The concept of a measurement of the yet unevaluated Hannay angle, by means of an Earth-bound satellite, adiabatically driven by the Moon, is shown herein. Numerical estimates are given for the angles, the orbital displacements, the shortening of the orbital periods, for different altitudes. It is concluded that the Hannay effect is measurable in high Earth orbits, by means of atomic clocks, accurate Time & Frequency transfer system and precise positioning.Comment: Lette

Improved Soundness for QMA with Multiple Provers

We present three contributions to the understanding of QMA with multiple provers: 1) We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a PCP). 2) We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV '10], thereby improving their result from a monolithic protocol where Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers k and a soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.) 3) We make progress towards an open question of [Aaronson et al., ToC '09] about what kinds of NP-complete problems are amenable to sublinear multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature - namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom

Scalar differential invariants of symplectic Monge–Ampère equations

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère PDEs with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. A series of invariant differential forms and vector fields are also introduced: they allow one to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution to the symplectic equivalence problem for Monge-Ampère equations

Forming circumnuclear disks and rings in galactic nuclei: a competition between supermassive black hole and nuclear star cluster

We investigate the formation of circumnuclear gas structures from the tidal disruption of molecular clouds in galactic nuclei, by means of smoothed particle hydrodynamics simulations. We model galactic nuclei as composed of a supermassive black hole (SMBH) and a nuclear star cluster (NSC) and consider different mass ratios between the two components. We find that the relative masses of the SMBH and the NSC have a deep impact on the morphology of the circumnuclear gas. Extended disks form only inside the sphere of influence of the SMBH. In contrast, compact rings naturally form outside the SMBH's sphere of influence, where the gravity is dominated by the NSC. This result is in agreement with the properties of the Milky Way's circumnuclear ring, which orbits outside the SMBH sphere of influence. Our results indicate that compact circumnuclear rings can naturally form outside the SMBH sphere of influence.Comment: Accepted for publication in ApJ. 12 pages, 6 figures, 3 tables. Comments welcom

A source-free integration method for black hole perturbations and self-force computation: Radial fall

Perturbations of Schwarzschild-Droste black holes in the Regge-Wheeler gauge benefit from the availability of a wave equation and from the gauge invariance of the wave function, but lack smoothness. Nevertheless, the even perturbations belong to the C\textsuperscript{0} continuity class, if the wave function and its derivatives satisfy specific conditions on the discontinuities, known as jump conditions, at the particle position. These conditions suggest a new way for dealing with finite element integration in time domain. The forward time value in the upper node of the $(t, r^*$) grid cell is obtained by the linear combination of the three preceding node values and of analytic expressions based on the jump conditions. The numerical integration does not deal directly with the source term, the associated singularities and the potential. This amounts to an indirect integration of the wave equation. The known wave forms at infinity are recovered and the wave function at the particle position is shown. In this series of papers, the radial trajectory is dealt with first, being this method of integration applicable to generic orbits of EMRI (Extreme Mass Ratio Inspiral).Comment: This arXiv version differs from the one to be published by Phys. Rev. D for the use of British English and other minor editorial difference