Identity principles for commuting holomorphic self-maps of the unit disc

Some identity principles for holomorphic functions are investigated

Identity principles for commuting holomorphic self-maps of the unit disc

3Some identity principles for holomorphic functions are investigated.nonemixedF. BRACCI; R. TAURASO; VLACCI, FABIOF., Bracci; R., Tauraso; Vlacci, Fabi

Congruences concerning Jacobi polynomials and Ap\'ery-like formulae

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.Comment: to appear in Int. J. Number Theor

MOONLIGHT: A NEW LUNAR LASER RANGING RETROREFLECTOR INSTRUMENT

Since 1969 Lunar Laser Ranging (LLR) to the Apollo Cube Corner Reflector (CCR) arrays has supplied several significant tests of gravity: Geodetic Precession, the Strong and Weak Equivalence Principle (SEP, WEP), the Parametrized Post Newtonian (PPN) parameter , the time change of the Gravitational constant (G), 1/r2 deviations and new gravitational theories beyond General Relativity (GR), like the unified braneworld theory (G. Dvali et al., 2003). Now a new generation of LLR can do better using evolved laser retroreflectors, developed from tight collaboration between my institution, INFN–LNF (Istituto Nazionale di Fisica Nucleare – Laboratori Nazionali di Frascati), and Douglas Currie (University of Maryland, USA), one of the fathers of LLR. The new lunar CCR is developing and characterizing at the "Satellite/Lunar laser ranging Characterization Facility" (SCF), in Frascati, performing our new industry standard space test procedure, the "SCF-Test"; this work contains the experimental results of the SCF-Test applied to the new lunar CCR, and all the new payload developments, including the future SCF tests. The International Lunar Network (ILN) research project considers our new retroreflector as one of the possible "Core Instruments

Fundamental physics and absolute positioning metrology with the MAGIA lunar orbiter

MAGIA is a mission approved by the Italian Space Agency (ASI) for Phase A study. Using a single large-diameter laser retroreflector, a large laser retroreflector array and an atomic clock onboard MAGIA we propose to perform several fundamental physics and absolute positioning metrology experiments: VESPUCCI, an improved test of the gravitational redshift in the Earth–Moon system predicted by General Relativity; MoonLIGHT-P, a precursor test of a second generation Lunar Laser Ranging (LLR) payload for precision gravity and lunar science measurements under development for NASA, ASI and robotic missions of the proposed International Lunar Network (ILN); Selenocenter (the center of mass of the Moon), the determination of the position of the Moon center of mass with respect to the International Terrestrial Reference Frame/System (ITRF/ITRS); this will be compared to the one from Apollo and Lunokhod retroreflectors on the surface; MapRef, the absolute referencing of MAGIA's lunar altimetry, gravity and geochemical maps with respect to the ITRF/ITRS. The absolute positioning of MAGIA will be achieved thanks to: (1) the laboratory characterization of the retroreflector performance at INFN-LNF; (2) the precision tracking by the International Laser Ranging Service (ILRS), which gives two fundamental contributions to the ITRF/ITRS, i.e. the metrological definition of the geocenter (the Earth center of mass) and of the scale of length; (3) the radio science and accelerometer payloads; (4) support by the ASI Space Geodesy Center in Matera, Italy. Future ILN geodetic nodes equipped with MoonLIGHT and the Apollo/Lunokhod retroreflectors will become the first realization of the International Moon Reference Frame (IMRF), the lunar analog of the ITRF

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Constraining spacetime torsion with LAGEOS

We compute the corrections to the orbital Lense-Thirring effect (or frame-dragging) in the presence of spacetime torsion. We derive the equations of motion of a test body in the gravitational field of a rotating axisymmetric massive body, using the parametrized framework of Mao, Tegmark, Guth and Cabi. We calculate the secular variations of the longitudes of the node and of the pericenter. We also show how the LAser GEOdynamics Satellites (LAGEOS) can be used to constrain torsion parameters. We report the experimental constraints obtained using both the nodes and perigee measurements of the orbital Lense-Thirring effect. This makes LAGEOS and Gravity Probe B (GPB) complementary frame-dragging and torsion experiments, since they constrain three different combinations of torsion parameters

PROBING GRAVITY IN NEO'S WITH HIGH-ACCURACY LASER-RANGED TEST MASSES

Received 9 August 2006Communicated by S. G. TuryshevGravity can be studied in detail in near Earth orbits NEO's using laser-ranged testmasses tracked with few-mm accuracy by ILRS. The two LAGEOS satellites have beenused to measure frame dragging (a truly rotational effect predicted by GR) with a 10%error. A new mission and an optimized, second generation satellite, LARES (I. CiufoliniPI), is in preparation to reach an accuracy of 1% or less on frame dragging, to measuresome PPN parameters, to test the