The large sieve with sparse sets of moduli

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao

Building galaxy models with Schwarzschild method and spectral dynamics

Tremendous progress has been made recently in modelling the morphology and kinematics of centers of galaxies. Increasingly realistic models are built for central bar, bulge, nucleus and black hole of galaxies, including our own. The newly revived Schwarzschild method has played a central role in these theoretical modellings. Here I will highlight some recent work at Leiden on extending the Schwarzschild method in a few directions. After a brief discussion of (i) an analytical approach to include stochastic orbits (Zhao 1996), and (ii) the ``pendulum effect'' of loop and boxlet orbits (Zhao, Carollo, de Zeeuw 1999), I will concentrate on the very promising (iii) spectral dynamics method, with which not only can one obtain semi-analytically the actions of individual orbits as previously known, but also many other physical quantities, such as the density in configuration space and the line-of-sight velocity distribution of a superposition of orbits (Copin, Zhao & de Zeeuw 1999). The latter method also represents a drastic reduction of storage space for the orbit library and an increase in accuracy over the grid-based Schwarzschild method.Comment: 11 pages including 3 ps figures, Contribution to IAU Colloquium 172 on ``Impact of Modern Dynamics in Astronomy'', July 6-11, 1998, Namur, Belgium; ed. S. Ferraz-Mello (Dordrecht:Kluwer

Some Wolstenholme type congruences

In this paper we give an extension and another proof of the following Wolstenholme's type curious congruence established in 2008 by J. Zhao. Let s and l be two positive integers and let p be a prime such that p ls + 3. Then H(fsgl; p1) S(fsgl; p1) 8>>< >>: s(ls + 1)p2 2(ls + 2) Bpls2 (mod p3) if 2 - ls (1)l1 sp ls + 1 Bpls1 (mod p2) if 2 j ls: APs an application, for given prime p 5, we obtain explicit formulae for the sum 1 k1< <kl p1 1=(k1 kl) (mod p3) if k 2 f1; 3; : : : ; p 2g, and for the sum P 1 k1< <kl p1 1=(k1 kl) (mod p2) if k 2 f2; 4; : : : ; p 3