On the Canonical Decomposition of Quiver Representations

Kac introduced the notion of the canonical decomposition for a dimension vector of a quiver. Here we will give an efficient algorithm to compute the canonical decomposition. Our study of the canonical decomposition for quivers with three vertices gives us fractal-like pictures.     

Moduli Problems of Sheaves Associated with Oriented Trees

To every oriented tree we associate vector bundle problems. We define semistability concepts for these vector bundle problems and establish the existence of moduli spaces. As an important application, we obtain an algebraic construction of the moduli space of holomorphic triples.     

Coarse Cohomology and lp-Cohomology

We prove that, for any exact category M, any element of K₁(M)can be described in terms of a pair of admissible monomorphisms A X, B Y and an isomorphism :A X/A Y B Y/B X.     

A geometric approach to the Kronecker problem I: The two row case

Given two irreducible representations μ, v of the symmetric group S _d , the Kronecker problem is to find an explicit rule, giving the multiplicity of an irreducible representation, λ, of S _d , in the tensor product of μ and v. We propose a geometric approach to investigate this problem. We demonstrate its effectiveness by obtaining explicit formulas for the tensor product multiplicities, when the irreducible representations are parameterized by partitions with at most two rows.     

The transition matrix between PBW basis and semicanonical basis of U+(sl_n(C))

We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.     

GIT-equivalence and semi-stable subcategories of quiver representations

In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [14]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.     

Unit and Proper Tube Orders

In 2005, we defined the n-tube orders, which are the n-dimensional analogue of interval orders in 1 dimension, and trapezoid orders in 2 dimensions. In this paper we consider two variations of n-tube orders: unit n-tube orders and proper n-tube orders. It has been proven that the classes of unit and proper interval orders are equal, and the classes of unit and proper trapezoid orders are not. We prove that the classes of unit and proper n-tube orders are not equal for all n ≥ 3, so the general case follows the situation in 2 dimensions.     

Projective Representations of Quivers #

In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A _∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A _∞, which will allow us to characterize the projective representations of A _∞. This will improve some previous results and make more accurate the statement made in Benson (1991 Benson , D. J. ( 1991 ). Representations and Cohomology I . Cambridge Studies in Advanced Mathematics , Vol. 30 . Cambridge : Cambridge University Press . [Google Scholar]). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module.     

基于顺序统计量的几何分在特征的进一步结果

用顺序统计量来刻划几何分布的特征已有不少结果，但仍有问题有待解决．本文在文［１］的基础上，针对Ａｒｎｏｌｄ［２］所提问题进行了进一步研究，得到了一些进一步的结果．这些结果同时给出了几何分布的基于顺序统计量的特征．     

McKay箭图与覆盖空间

本文研究在自然扩张和嵌入下特殊线性群和一般线性群的有限子群的McKay 箭图间的关系. 我们证明在特定条件下, 一般线性群GL(m;C) 的有限子群G的McKay 箭图是其正规子群G∩SL(m;C)的McKay 箭图的正则覆盖, 而当把G 嵌入SL(m+1;C) 时, 新的McKay 箭图由在原来的McKay 箭图的每一顶点加上一个由其Nakayama 平移到其自身的箭向得到. 作为例子, 我们指出如何用这些方法得到一些有趣的McKay 箭图.     

几何分布的几个性质

本文给出几何分布与指数分布相同及相异的几个性质．     

Lattice-finite Rings

We study a one-dimensional analogue of representation-finite rings. For a left Noetherian semilocal ring R, we define an R-lattice to be a finitely generated R-module with zero socle. We call R lattice-finite if the number of isomorphism classes of indecomposable R-lattices is finite. Under this assumption, we give several equivalent criteria for the existence of Auslander–Reiten sequences in the category of R-lattices. A necessary condition is that the maximal left quotient ring of R is semisimple, and the main sufficient criterion states that R admits a semiperfect semiprime Asano left overorder. Presented by I. Reiten Mathematics Subject Classifications (2000) Primary: 16G70, 16G30; secondary: 16G60.     

Linear versus set valued Kronecker representations

A set valued representation of the Kronecker quiver is nothing but a quiver. We apply the forgetful functor from vector spaces to sets and compare linear with set valued representations of the Kronecker quiver.     

关于结构弯扭屈曲理论的修正

本文讨论和指出了目前结构弯扭屈曲理论中所存在的问题．从而得出结论:目前的理论限制了结构的弯担变形过程,使得变形过程受制于线位移与转角位移按一定的先后顺序产生．为解决这一问题,提出了一种描述结构实际弯扭屈曲过程的新思路,从而导出了新的几何关系、建立了新的势能变分方程和新的中性平衡微分方程．实例说明了修正后的理论与原理论的不同．     

关于带有关系箭图的路余代数

带有关系箭图的路余代数在余代数理论研究中起着重要作用.研究带有关系的路余代数,得到了一个有趣的结果,这对余代数结构的研究会有所裨益.     

星形箭图偏周期预投射代数的Hilbert级数

设△是一个有限无圈的箭图.引入了由Δ所决定的偏周期预投射代数,它是一个定义在周期为p的稳定平移箭图Z△/(τ~p)上的代数,记为Π_(Q(Δ,p),J).当周期p=1时,偏周期预投射代数就是偏预投射代数.我们推广了Eting和Eu的方法并得到无圈的星形箭图△所决定的偏周期预投射代数Π_((Q(Δ,p)),J)的Hilbert级数的计算公式.     

Music as Embodied Mathematics: A Study of a Mutually Informing Affinity

The argument examined in this paper is that music – when approached through making and responding to coherent musical structures,facilitated by multiple, intuitively accessible representations – can become a learning context in which basic mathematical ideas can be elicited and perceived as relevant and important. Students' inquiry into the bases for their perceptions of musical coherence provides a path into the mathematics of ratio,proportion, fractions, and common multiples. Ina similar manner, we conjecture that other topics in mathematics – patterns of change,transformations and invariants – might also expose, illuminate and account for more general organizing structures in music. Drawing on experience with 11–12 year old students working in a software music/math environment, we illustrate the role of multiple representations, multi-media, and the use of multiple sensory modalities in eliciting and developing students' initially implicit knowledge of music and its inherent mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Asymptotic relations betweenL- andM-estimators in the linear model

We obtain Bahadur-type representations for one-stepL-estimators,M- and one-stepM-estimators in the linear model. The order of the remainder terms in these representations depends on the smooth-ness of the weight function forL-estimators and on the smoothness of the ψ-function forM- and one-stepM-estimators. We use the representations to investigate the asymptotic relations between these estimators. In particular, we show that asymptotically equivalentL- andM-estimators of the slope parameter exist even when the underlying distribution is asymmetric. It is important to consider the asymmetric case for both practical and robustness reasons: first, there is no compelling argument which precludes asymmetric distributions from arising in practice, and, secondly, even if a symmetric model can be posited, it is important to allow for the possibility of mild (and therefore difficult to detect) departures from the symmetric model.