Some lower bounds on Shelah rank in the free group

We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.     

Upper and lower bounds for the Perron root of a nonnegative matrix

We derive upper and lower bounds for the Perron root of a nonnegative matrix by using generalized Gershgorin inclusion regions. Our bounds seem particularly effective for certain sparse matrices.     

Equality conditions for lower bounds on the smallest singular value of a bidiagonal matrix

Several lower bounds have been proposed for the smallest singular value of a square matrix, such as Johnson’s bound, Brauer-type bound, Li’s bound and Ostrowski-type bound. In this paper, we focus on a bidiagonal matrix and investigate the equality conditions for these bounds. We show that the former three bounds give strict lower bounds if all the bidiagonal elements are non-zero. For the Ostrowski-type bound, we present an easily verifiable necessary and sufficient condition for the equality to hold.     

非负矩阵Perron根的下界序列

非负矩阵Perron根的估计是非负矩阵理论研究的重要课题之一.如果其上下界能够表示为非负矩阵元素的易于计算的函数,那么这种估计价值更高.本文结合非负矩阵的迹分两种情况给出Perron根的下界序列,并且给出数值例子加以说明.     

On lower bounds for singular values of matrix sums

It is pointed out by a counterexample that the result on lower bounds for the absolute singular values of sum of matrices given in [1] is not valid. Moreover, in this note a new result is established on such lower bounds.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

On lower bounds on the size of designs in compact symmetric spaces of rank 1

The concept of t-designs in compact symmetric spaces of rank 1 is a generalization of the theory of classical t-designs. In this paper we obtain new lower bounds on the cardinality of designs in projective compact symmetric spaces of rank 1. With one exception our bounds are the first improvements of the classical bounds by more than one. We use the linear programming technique and follow the approach we have proposed for spherical codes and designs. Some examples are shown and compared with the classical bounds.     

Lower bounds for the spread of a matrix

A characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread. Comparisons are then made with several known bounds.     

Upper bounds for the spread of a matrix

In this paper, we exhibit new and sharper upper bounds of the spread of a matrix.     

Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and its applications

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.     

Cover-based bounds on the numerical rank of Gaussian kernels

A popular approach for analyzing high-dimensional datasets is to perform dimensionality reduction by applying non-parametric affinity kernels. Usually, it is assumed that the represented affinities are related to an underlying low-dimensional manifold from which the data is sampled. This approach works under the assumption that, due to the low-dimensionality of the underlying manifold, the kernel has a low numerical rank. Essentially, this means that the kernel can be represented by a small set of numerically-significant eigenvalues and their corresponding eigenvectors.We present an upper bound for the numerical rank of Gaussian convolution operators, which are commonly used as kernels by spectral manifold-learning methods. The achieved bound is based on the underlying geometry that is provided by the manifold from which the dataset is assumed to be sampled. The bound can be used to determine the number of significant eigenvalues/eigenvectors that are needed for spectral analysis purposes. Furthermore, the results in this paper provide a relation between the underlying geometry of the manifold (or dataset) and the numerical rank of its Gaussian affinities.The term cover-based bound is used because the computations of this bound are done by using a finite set of small constant-volume boxes that cover the underlying manifold (or the dataset). We present bounds for finite Gaussian kernel matrices as well as for the continuous Gaussian convolution operator. We explore and demonstrate the relations between the bounds that are achieved for finite and continuous cases. The cover-oriented methodology is also used to provide a relation between the geodesic length of a curve and the numerical rank of Gaussian kernel of datasets that are sampled from it.     

Remarks on the lower bounds for the average genus

Let G be a graph of maximum degree at most four. By using the overlap matrix method which is introduced by B. Mohar, we show that the average genus of G is not less than 1/3 of its maximum genus, and the bound is best possible. Also, a new lower bound of average genus in terms of girth is derived.     

Estimates for the inverse of a matrix and bounds for eigenvalues

This paper gives new bounds for the relationship between the diagonal elements of a square matrix and the corresponding diagonal elements of the matrix inverse, as well as bounds for the eigenvalues of the matrix. The results given here generalize those of Ostrowski and Ky Fan, and have their origin in engineering application.     

Applications of matrix methods to the theory of lower bounds in computational complexity

We present some criteria for obtaining lower bounds for the formula size of Boolean functions. In the monotone case we get the boundn ^(logn) for the function MINIMUM COVER using methods considerably simpler than all previously known. In the general case we are only able to prove that the criteria yield an exponential lower bound when applied to almost all functions. Some connections with graph complexity and communication complexity are also given.     

Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.