The Topological Structure of Subseries Convergence

Let E be an abstract set, G be an Abelian topological group, FEG and be afamily of subsets of F. Let the sequence {xj}E be subseries aF convergent. In this paper,we present a sufficient and necessary condition for {xj} to be also subseries convergent in thetopology of uniform convergence on the sets in ,     

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.     

Resultants and the algebraicity of the join pairing on Chow varieties

The Chow/Van der Waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension. The algebraicity of the join pairing on Chow varieties then follows. The approach implies a more algebraic proof of Lawson's complex suspension theorem in characteristic 0. The continuity of the action of the linear isometries operad on the group completion of the stable Chow variety is a consequence.

     

On the probability distribution of data at points in real complete intersection varieties

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multi-variate polynomial equations and convergence radius of Newton's operator for under-determined system of multi-variate polynomial equations.     

On the normality of the rings of Schubert varieties

We prove that the cone over a Schubert variety inG/P (P being a maximal parabolic subgroup of classical type) is normal by exhibiting a 2-regular sequence inR(w) (the homogeneous coordinate ring of the Schubert varietyX(w) inG/P under the canonical protective embeddingG/P ⊂→ (p (H° G/P,L)),L being the ample generator of (PicG/P), which vanishes on the singular locus ofX(w). We also prove the surjectivity ofH° (G/Q, L) H° (X(w), L), whereQ is a classical parabolic subgroup (not necessarily maximal) ofG andL is an ample line bundle onG/Q.     

Generalizations of P. Lévy's inversion theorem

We extend the Lévy inversion formula for the recovery of a bounded measure over $\text{R}$ from its Fourier-Stieltjes transform to bounded complex-valued, orthogonally scattered Hilbert space-valued, and spectral projection operator-valued measures over any first countable locally compact Abelian group. All our results are direct generalizations of known inversions for $\text{R}$.     

Symmetrizations of Markov processes

Two methods for symmetrizing Markov processes are discussed. Letu ^a(x, y) be the potential density of a Lévy process on a compact Abelian groupG. A general condition is given that guarantees thatv(x, y)=u^a(x, y)+u^a(y, x) is the potential density of a symmetric Lévy process onG. The second method arises by considering the linear space of one-potentialsU ¹ f, withf inL ², endowed with the inner product (U ¹ f,U ¹ g)=fU ¹ g+gU ¹ f. If the semigroup ofX(t) is normal, then the completionH of this space is the Dirichlet space of a symmetric processY(t). A set that is semipolar forX(t) is polar forY(t).     

Set-theoretic complete intersections in characteristic

We describe a class of toric varieties which are set-theoretic complete intersections only over fields of one positive characteristic .