Stanley Depth of Quotient of Monomial Complete Intersection Ideals

We compute the Stanley depth for a particular but important case of the quotient of complete intersection monomial ideals. Also, in the general case, we give sharp bounds for the Stanley depth of a quotient of complete intersection monomial ideals. In particular, we prove the Stanley conjecture for quotients of complete intersection monomial ideals.     

Upper and Lower Bounds for the Stanley Depth of Certain Classes of Monomial Ideals and Their Residue Class Rings

We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.     

The Stanley Depth in the Upper Half of the Koszul Complex

Let R = K[X₁, ?c, X_n] be a polynomial ring over some field K. In this article, we prove that the kth syzygy module of the residue class field K of R has Stanley depth n ? 1 for ?n/2? ≤k < n, as it had been conjectured by Bruns et al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than 1. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

Colourful Simplicial Depth

Inspired by Barany’s Colourful Caratheodory Theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d² + 1 and that the maximum is d^d+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.     

Depth notions for orthogonal regression

Global depth, tangent depth and simplicial depths for classical and orthogonal regression are compared in examples, and properties that are useful for calculations are derived. The robustness of the maximum simplicial depth estimates is shown in examples. Algorithms for the calculation of depths for orthogonal regression are proposed, and tests for multiple regression are transferred to orthogonal regression. These tests are distribution free in the case of bivariate observations. For a particular test problem, the powers of tests that are based on simplicial depth and tangent depth are compared by simulations.     

统计深度的几何推广

改进统计深度的定义,并将点的深度概念推广到直线与平面的深度,由此得到深度计算的基本定理和深度的一系列性质.最后讨论应用展望.     

统计深度函数及其应用

次序统计量在一维统计数据分析中起着很重要的作用．多年来,人们一直在商维数据处理和分析中寻找“次序统计量”,却没有得到很满意的结果．由于缺少自然而有效的高维数据排序方法,因而象一维“中位数”的概念很难推广到高维．统计深度函数则提供了高维数据排序的一种工具,其主要思想是提供了一种从高维数据中心(最深点)向外的排序方法．不仅如此,统计深度函数已经在探索性高维数据分析,统计判决等方面带给我们一种全新的前景,并在工业、工程、生物医学等诸多领域得到很好的应用．本文介绍了统计深度函数概念及其应用,讨论了位置深度函数的标准,介绍了几种常用的统计深度函数．给出了由深度函数特别是由投影深度函数所诱导的位置和散布阵估计,介绍了它们的诸多优良性质,如极限分布,稳健性和有效性．由于在大多数场合下,高崩溃点的估计不是较有效的估计,而由统计深度函数所诱导的估计具有多元仿射不变性,并能提供理想的稳健性与有效性之间的平衡,本文还讨论了基于深度的统计检验和置信区域,介绍了统计深度函数的其他应用,如多元回归、带有变量误差模型、质量控制等,以及实际计算问题．指出了统计深度函数领域有关进一步的工作和研究方向．     

Depth of modular invariant rings

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper¹ we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].     

Multivariate Regression Depth

   Abstract. The regression depth of a hyperplane with respect to a set of n points in \Real ^d is the minimum number of points the hyperplane must pass through in a rotation to vertical. We generalize hyperplane regression depth to k -flats for any k between 0 and d-1 . The k=0 case gives the classical notion of center points. We prove that for any k and d , deep k -flats exist, that is, for any set of n points there always exists a k -flat with depth at least a constant fraction of n . As a consequence, we derive a linear-time (1+ɛ) -approximation algorithm for the deepest flat. We also show how to compute the regression depth in time O(n ^d-2 +nlog n) when 1≤ k≤ d-2 .     

Sliding Functor and Polarization Functor for Multigraded Modules

We define sliding functors, which are exact endofunctors of the category of multigraded modules over a polynomial ring. They preserve several invariants of modules, especially the (usual) depth and Stanley depth. In a similar way, we can also define the polarization functor. While this idea has appeared in papers of Bruns–Herzog and Sbarra, we give slightly different approach. Keeping these functors in mind, we treat simplicial spheres of Bier–Murai type.     

Depth in an Arrangement of Hyperplanes

A collection of n hyperplanes in ^d forms a hyperplane arrangement. The depth of a point is the smallest number of hyperplanes crossed by any ray emanating from θ . For d=2 we prove that there always exists a point θ with depth at least . For higher dimensions we conjecture that the maximal depth is at least . For arrangements in general position, an upper bound on the maximal depth is also established. Finally, we discuss algorithms to compute points with maximal depth. Received December 1, 1997, and in revised form June 6, 1998.     

Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods

Continuity of procedures based on the halfspace (Tukey) depth (location and regression setting) is investigated in the framework of continuity concepts from set-valued analysis. Investigated procedures are depth contours (upper level sets) and maximum depth estimators. Continuity is studied both as the pointwise continuity of data-analytic functions, and the weak continuity of statistical functionals—the latter having relevance for qualitative robustness. After a real-data example, some general criteria and counterexamples are given, as well as positive results holding for “typical” data. Finally, some consequences for diagnostics and practical use of the depth-based techniques are drawn.     

Regression Depth and Center Points

We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least , as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems. Received October 6, 1998, and in revised form July 26, 1999.     

The Depth of Subgroups of Suzuki Groups

We determine the combinatorial depth of certain subgroups of simple Suzuki groups Sz(q), among others, the depth of their maximal subgroups. We apply these results to determine the ordinary depth of these subgroups.     

Depth for complexes, and intersection theorems

This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster's big Cohen-Macaulay modules), and also a generalization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules. Received May 6, 1997; in final form December 3, 1997     

Depth of recursion and the ackermann function

The maximum depth of recursion refers to the number of levels of activation of a procedure which exist during the deepest call of the procedure. A re-examination of the maximum depth of recursion of the Ackermann function results in a new formula which takes a full account of the dependence of this property on the parameters. It is shown that the recursive use parameter of the Ackermann function contributes to the depth of recursion, and that this contribution may be reduced by rearranging the order of the parameters. The stack required by the Ackermann function was investigated using ALGOL 68-R.     

Depth of symmetric algebras of certain ideals

We compute the depth of the symmetric algebra of certain ideals in terms of the depth of the ring modulo the ideal generated by the entries of a minimal presentation matrix.

     

On the Asymptotic Depth of Multigraded Modules

The aim of this article is to establish, among other results, the asymptotic stability of the depth of the graded pieces of a nonstandard multigraded module. As a corollary, we get the asymptotic stability of the depth of the graded pieces of the multigraded Rees algebra defined by a finite set of ideals and their associated multigraded rings.