We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.

We prove some tradeoos between the size and depth of algebraic formulae. In particular, we show that, for any xed > 0, any algebraic formula of size S can be converted into an equivalent formula of depth O(log S) and size O(S 1+). This result is an improvement over previously-known results where, to obtain the same depth bound, the formula-size is (S), with 2.

We give a partial answer to the effective non-vanishing problem for algebraic surfaces in positive characteristic, and also give counterexamples for the Kawamata-Viehweg vanishing and the logarithmic semipositivity on ruled surfaces in positive characteristic.

In this report, we first review some algebraic and topological structures relevant to some difference sequence spaces which were constructed in comparison with the classical spaces 0 c , c and ∞ l .Then we review Cesàro and lacunary summability methods for difference sequences. Lastly, we review algebraic duals for several different kinds of difference sequence spaces.

If an algebraic torus T acts on a complex projective algebraic variety X then the affine scheme Spec H∗ T (X;C) associated to the equivariant cohomology is often an arrangement of linear subspaces of the vector space H 2 (X;C). In many situations the ordinary cohomology ring of X can be described in terms of this arrangement.

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C n which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact we establish a slightly stronger version which applies to a wider class of boundary 1-cycles. Arguments here are based on the Hahn-Banach Theorem and some geometric...

In [3] and [5] the authors ask how many primes are of the form xY + yX, where gcd (x, y) = 1 and x, y 2: 2. Moreover, Jose Castillo (see [2]) asks how many primes are of the Smarandache form xil + X2 X3 + ... + Xn Xl , where n > 1, Xl, X2, ••• , Xn > 1 and gcd (Xl, X2, ••• , X n ) = 1 (see [9]). In this article we announce a lower bound for the size of the largest prime divisor of an expression...

We obtain a formula for the S-class number of an algebraic torus defined over a number field in terms of the étale and Galois cohomology groups of its character module. As applications, we give different proofs of some classical class number formulas of Shyr, Ono, Katayama and Morishita.

We prove that for any Set-endofunctor F the category SetF of F -coalgebras is distributive if F preserves preimages, i.e. pullbacks along an injective map, and that the converse is also true whenever SetF has finite products.

Let Z be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform BriançonSkoda theorem for the local ring OZ ; a result which is previously proved by Huneke by algebraic methods. For ideals with few generators we also get some sharper results.