Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

On the Iwasawa -invariants of real abelian fields

For a prime number and a number field , let denote the projective limit of the -parts of the ideal class groups of the intermediate fields of the cyclotomic -extension over . It is conjectured that is finite if is totally real. When is an odd prime and is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where divides the degree of , we also obtain a rather simple criterion.

     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.     

On the Volume of Flowers in Space Forms

Let f(x ₁,..., x _N ) be a lattice polynomial in N variables, in which each variable occurs exactly once, B ₁,..., B _N be smoothly moving balls in the hyperbolic, Euclidean, or spherical space. Introducing a suitable modification of the Dirichlet–Voronoi decomposition, we prove a formula for the derivative of the volume of the domain f(B ₁,..., B _N ). As an application of the formula, we show that the volume increases if the balls move continuously in such a way that the functions _ij d _ij increase for all 1 i < j N, where _ij is a sign depending on f, d _ij is the distance between the centers of B _I and B _j .     

The Lu Qi-Keng conjecture fails for strongly convex algebraic complete Reinhardt domains in

In this note, we give an example of a strongly convex algebraic complete Reinhardt domain which is not Lu Qi-Keng in for any

     

On the Brocard–Ramanujan Diophantine Equation n! + 1 = m2

In 1876, H. Brocard posed the problem of finding all integral solutions to n! + 1 = m². In 1913, unaware of Brocard's query, S. Ramanujan gave the problem in the form, The number 1 + n! is a perfect square for the values 4, 5, 7 of n. Find other values. We report on calculations up to n = 10⁹ and briefly discuss a related problem.     

Integral points, divisibility between values of polynomials and entire curves on surfaces

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first examples, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariski-dense. Some of our results are connected with divisibility problems, i.e. the problem of describing the integral points in the plane where the values of some given polynomials in two variables divide the values of other given polynomials.     

Abelian monopoles and complex geometry

Three remarks concerning the constructive properties of smooth 4-manifolds with nontrivial Seiberg-Witten invariants are presented. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 420–428, March, 1999.     

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.