Martin's Axiom is consistent with the existence of nowhere trivial automorphisms

Martin's Axiom does not imply that all automorphisms of are somewhere trivial. An alternate method for obtaining models where every automorphism of is somewhere trivial is explained.

     

Polynomial factorization over

We describe algorithms for polynomial factorization over the binary field , and their implementation. They allow polynomials of degree up to to be factored in about one day of CPU time, distributing the work on two processors.

     

Visualization of Effect Algebras by Automorphisms

We show that every effect algebra satisfying the Riesz decomposition property can be represented as an effect algebra of automorphisms of an antilattice, and every MV-algebra can be represented as an MV-algebra of automorphisms of a linearly ordered set. Such a representation enables us to visualize effect algebras by functions. This is a variation of the Holland representation theorem for -groups and of its generalization of Glass for directed interpolation po-groups as -groups or po-groups automorphisms of linearly ordered set or of an antilattice, respectively.     

Automorphism Groups of Computably Enumerable Predicates

We study automorphism groups of two important predicates in computability theory: the predicate W_y and the graph of a universal partially computable function. It is shown that all automorphisms of the predicates in question are computable. The actions of the automorphism groups on some index sets are examined, and we establish a number of results on the structure of these. We also look into homomorphisms of the two predicates. In this case the situation changes: all homomorphisms of the universal function are computable, but in each Turing degree, homomorphisms of W_y exist.     

Commutativity of rings with polynomial constraints

Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying , respectively) where , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.     

Four-Dimensional Compact Projective Planes with a Solvable Collineation Group

We consider a four-dimensional compact projective plane whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil×R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag p W, acts transitively on and fixes no point in the set W\p. Under these conditions, we will prove that either contains a three-dimensional group of elations or acts doubly transitively on .     

Null-quadrics of codimension 4 in ℂ7

In this paper we study the automorphisms of nondegenerate quadrics of type (4, 3). In particular, we classify the so-called null-quadrics (we prove that there are exactly two null-quadrics up to equivalence), and find their automorphism groups. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 657–665, November, 1997. Translated by S. S. Anisov     

Hyperelliptic Surfaces and their Moduli

The hyperelliptic portion of the moduli space of compact Riemann surfaces of genus g2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with the lattice of maximal g-hyperelliptic group actions (classified up to topological equivalence). The resulting stratification of the hyperelliptic moduli space exhibits regularities which depend on the parity of g and can be detected at the level of groups of order 8.     

New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym-metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux-iliary tool, we compute the integral of complex powers of the generic norm on a boundedsymmetric domains using the well-known integral of Selberg. This generalizes matrix in-tegrals of Hua and leads to a special polynomial with integer or half-integer coefficientsattached to each irreducible bounded symmetric domain.     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .