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1. CMB 2012 (vol 57 pp. 178)
Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.
This property ensures that the pointwise extrema of $f$ on any nonempty
density open subset can be arbitrarily closely approximated by values of $f$
achieved on ``large'' subsets, which may be of relevance in a variety of
issues. To support this claim, we use it to characterize the common points
of continuity, or approximate continuity, of two quasiconvex functions that
coincide away from a set of measure zero.
Keywords:density topology, quasiconvex function, approximate continuity, point of continuity Categories:52A41, 26B05 |
2. CMB 2012 (vol 56 pp. 695)
Carmichael meets Chebotarev For any finite Galois extension $K$ of $\mathbb Q$
and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$,
we show that there exist infinitely many Carmichael numbers
composed solely of primes for which the associated class of Frobenius
automorphisms is $C$. This result implies that for every natural
number $n$ there are infinitely many Carmichael numbers of the form
$a^2+nb^2$ with $a,b\in\mathbb Z $.
Keywords:Carmichael numbers, Chebotarev density theorem Categories:11N25, 11R45 |
3. CMB 2011 (vol 56 pp. 161)
An Extension of the Dirichlet Density for Sets of Gaussian Integers Several measures for the density of sets of integers have been proposed,
such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and
investigate some of its properties.
Keywords:Gaussian integers, Dirichlet density Categories:11B05, 11M99, 11N99 |
4. CMB 2010 (vol 53 pp. 223)
Density of Polynomial Maps Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords:density, polynomial, endomorphism ring, PI Categories:16D60, 16S50 |
5. CMB 2001 (vol 44 pp. 97)
On the Density of Cyclic Quartic Fields An asymptotic formula is obtained for the number of cyclic quartic fields
over $Q$ with discriminant $\leq x$.
Keywords:cyclic quartic fields, density, discriminant Categories:11R16, 11R29 |