اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدVector spaces as unions of proper subspaces
115
0
0.0
(
0
)
تأليف
Apoorva Khare
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces.
قيم البحث
اقرأ أيضاً
Given a $K$-vector space $V$, let $sigma(V,K)$ denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers $V$. Analogously, define $sigma(M,R)$ for a module $M$ over a unital commutative ring $R$; this incl
udes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare-Tikaradze [Comm. Algebra, in press] showed for several classes of rings $R$ and $R$-modules $M$ that $sigma(M,R)=min_{mathfrak{m}in S_M} |R/mathfrak{m}| + 1$, where $S_M$ is the set of maximal ideals $mathfrak{m}$ such that $dim_{R/mathfrak{m}}(M/mathfrak{m}M)geq 2$. (That $sigma(M,R)leqmin_{mathfrak{m}in S_M}|R/mathfrak{m}|+1$ is straightforward.) Our first main result extends this equality to all $R$-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce a topological counterpart for finitely generated $R$-modules $M$ over rings $R$, whose some residue fields are infinite, which we call the Zariski covering number $sigma_tau(M,R)$. To do so, we first define the induced Zariski topology $tau$ on $M$, and now define $sigma_tau(M,R)$ to be the smallest (cardinal) number of proper $tau$-closed subsets of $M$ whose union covers $M$. We first show that our choice of topology implies that $sigma_tau(M,R)leqsigma(M,R)$, the covering number. We then show our next main result: $sigma_tau(M,R)=min_{mathfrak{m}in S_M} |R/mathfrak{m}|+1$, for all finitely generated $R$-modules $M$ for which (a) the dual Goldie dimension is finite, and (b) $mathfrak{m} otin S_M$ whenever $R/mathfrak{m}$ is finite. As a corollary, this alternately recovers the above formula for the covering number $sigma(M,R)$ of the aforementioned finitely generated modules. We also extend these topological studies to general finitely generated $R$-modules, using the notion of $kappa$-Baire spaces.
Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that if $h_2^* leq h_1^*$, then $P$ is IDP. More generally, we show the corresponding statements for semi-standard graded Cohen-Macaulay domains over algebraically closed fields.
In this paper, we will describe the space spanned by the angle-sums of polytopes, recorded in the alpha-vector. We will consider the angles sums of simplices and the angles sums and face numbers of simplicial polytopes and general polytopes. We will
construct families of polytopes whose angle sums span the spaces of polytopes defined by the Gram and Perles equations, analogues of the Euler and Dehn-Sommerville equations. We show that the dimensions of the affine span of the space of angle sums of simplices is floor[(d-1)/2] + 1, and that of the angle sums and face numbers of simplicial polytopes and general polytopes are d-1 and 2d-3 respectively.
We prove that a sufficiently large subset of the $d$-dimensional vector space over a finite field with $q$ elements, $ {Bbb F}_q^d$, contains a copy of every $k$-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.
A sequence which is a finite union of interpolating sequences for $H^infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that
the multiplication $f mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also provide a particularly simple new proof of one of the known characterizations. One of the new characterizations is that they are interpolating sequences for a more general interpolation problem.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
