In this paper, we propose three new tensor decompositions for even-order tensors corresponding respectively to the rank-one decompositions of some unfolded matrices. Consequently such new decompositions lead to three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank in this paper. We discuss the bounds between these new te...

Abstract This paper exhibits two families of unravelled abstract regular polytopes in Coxeter groups type B n \mathrm{B}_{n} . For one family, they have rank 4, while the other family has arbitrarily large rank.

We introduce the rank-growth function rkH(i) of a subgroup H of a finitely generated free group F . rkH(i) is defined to be the rank of the subgroup of H generated by elements of length less than or equal to i (with respect to the generators of F ), and it equals the rank of the fundamental group of the subgraph of the cosets graph of H, which consists of the paths starting at 1 that are of len...

The nonparanormal model assumes that variables follow a multivariate normal distribution after a set of unknown monotone increasing transformations. It is a flexible generalization of the normal model but retains the nice interpretability of the latter. In this paper we propose a rank-based tapering estimator for estimating the correlation matrix in the nonparanormal model in which the variable...

The higher rank numerical range is useful for constructing quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalues a1, . . . , an, then its rank-k numerical range Λk(A) is the intersection of convex polygons with vertices aj1 , . . . , ajn−k+1 , where 1 ≤ j1 < · · · < jn−k+1 ≤ n. In this paper, it is shown that the higher rank numeri...

background: advancement in the treatment of various types of cancer has led to greater patient survival. these treatments essentially have toxic effects on different kinds of cells, such as germ cells. infertility as one of the side effects of cancer treatment has changed the quality of life of young cancer survivors dramatically. melatonin is an antioxidant with receptors in the reproductive s...

We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved1 invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in {2, 3, 4, 6, 8}. Moreover, we giv...

We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morley has a normal nilpotent subgroup U and an abelian subgroup T such that G = U o T , if and only if, for any...

The article is devoted to different aspects of the question: ”What can be done with a matrix by a low rank perturbation?” It is proved that one can change a geometrically simple spectrum drastically by a rank 1 perturbation, but the situation is quite different if one restricts oneself to normal matrices. Also the Jordan normal form of a perturbed matrix is discussed. It is proved that with res...

A sparse precision matrix can be directly translated into a sparse Gaussian graphical model under the assumption that the data follow a joint normal distribution. This neat property makes high-dimensional precision matrix estimation very appealing in many applications. However, in practice we often face nonnormal data, and variable transformation is often used to achieve normality. In this pape...