let $r=k[x_1,x_2,cdots, x_n]$ be a polynomial ring over a field $k$. we prove that for any positive integers $m, n$, $text{reg}(i^mj^nk)leq mtext{reg}(i)+ntext{reg}(j)+text{reg}(k)$ if $i, j, ksubseteq r$ are three monomial complete intersections ($i$, $j$, $k$ are not necessarily proper ideals of the polynomial ring $r$), and $i, j$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...

A representation for a group or a monoid is called a representation with Borel mold if it can be normalized to a representation in upper triangular matrices whose image of the group or monoid generates the algebra of upper triangular matrices. In [Na2] the moduli of representations with Borel mold has been constructed for each group or monoid. The moduli of representations with Borel mold has s...

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not univ...

Abstract. The dynamical Borel-Cantelli lemma for some interval maps is considered. For expanding maps whose derivative has bounded variation, any sequence of intervals satisfies the dynamical Borel-Cantelli lemma. If a map has an indifferent fixed point, then the dynamical Borel-Cantelli lemma does not hold even in the case that the map has a finite absolutely continuous invariant measure and s...

Let Γ be a Borel class, or a Wadge class of Borel sets, and 2≤d≤ω a cardinal. We study the Borel subsets of R that can be made Γ by refining the Polish topology on the real line. These sets are called potentially Γ. We give a test to recognize potentially Γ sets. 2010 Mathematics Subject Classification. Primary: 03E15, Secondary: 54H05, 28A05, 26A21

In this paper we extend previous studies of selection principles for families of open covers of sets of real numbers to also include families of countable Borel covers. The main results of the paper could be summarized as follows: (1) Some of the classes which were different for open covers are equal for Borel covers – Section 1; (2) Some Borel classes coincide with classes that have been studi...

We show that any infinite-dimensional Banach (or more generally, Fréchet) space contains linear subspaces of arbitrarily high Borel complexity which admit separable complete norms giving rise to the inherited Borel structure. © 2007 Elsevier Inc. All rights reserved.

We show that the continuum hypothesis implies that every measure preserving near-action of a group on a standard Borel probability space (X,μ) has a pointwise implementation by Borel measure preserving automorphisms.