Dimension Of Such Spaces Research Articles

Dimension estimate of polynomial growth harmonic forms

Let Hpl(M) be the space of polynomial growth harmonic forms. We proved that the dimension of such spaces must be finite and can be estimated if the metric is uniformly equivalent to one with a nonnegative curvature operator. In particular, this implies that the space of harmonic forms of fixed growth order on the Euclidean space with any periodic metric must be finite dimensional.

The Alexandroff Dimension of Digital Quotients of Euclidean Spaces

Alexandroff T 0 -spaces have been studied as topological models of the supports of digital images and as discrete models of continuous spaces in theoretical physics. Recently, research has been focused on the dimension of such spaces. Here we study the small inductive dimension of the digital space X(W) constructed in [15] as a minimal open quotient of a fenestration W of R n . There are fenestrations of R n giving rise to digital spaces of Alexandroff dimension different from n , but we prove that if W is a fenestration, each of whose elements is a bounded convex subset of R n , then the Alexandroff dimension of the digital space X(W) is equal to n .

Matrix spaces with bounded number of eigenvalues

Let V be a linear space of n × n complex matrices with the property that the cardinality of the spectrum of every A ∈ V is at most k. We determine the maximum dimension of such spaces for k ∈ {1, 2, n − 1}. The structure of such spaces of the maximum dimension is also described.

On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.

Linear spaces of nilpotent matrices

We consider several questions on spaces of nilpotent matrices. We present sufficient conditions for triangularizability and give examples of irreducible spaces. We give a necessary and sufficient condition, in terms of the trace, for all linear combinations of a given set of operators to be nilpotent. We also consider the question of the dimension of a space L of nilpotents on F n . In particular, we give a simple new proof of a theorem due to M. Gerstenhaber concerning the maximal dimension of such spaces.

Spaces of bivariate spline functions over triangulation

We consider the spaces of bivariate Cμ-splines of degree k defined over arbitrary triangulations of a polygonal domain. We get an explicit formula for the dimension of such spaces when k≥3μ+2 and construct a local basis for them. The dimension formula is valid for any polygonal domain even it is complex connected, and the formula is sharp since it evaluates the lower-bound which was given by Schumaker in [11].

Homology of smooth splines: generic triangulations and a conjecture of Strang

For Δ \Delta a triangulated d d -dimensional region in R d {{\mathbf {R}}^d} , let S m r ( Δ ) S_m^r(\Delta ) denote the vector space of all C r {C^r} functions F F on Δ \Delta that, restricted to any simplex in Δ \Delta , are given by polynomials of degree at most m m . We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds Δ \Delta in the plane, getting lower bounds on the dimension of S m r ( Δ ) S{}_m^r(\Delta ) for all r r . For r = 1 r = 1 , we prove a conjecture of Strang concerning the generic dimension of the space of C 1 {C^1} splines over a triangulated manifold in R 2 {{\mathbf {R}}^2} . Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1

We consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessingr continuous derivatives globally. Morgan and Scott constructed a basis in the case wherer=1 andd≥5. The purpose of this paper is to extend the dimension part of their result tor≥0 andd≥4r+l. We use Bezier nets as a crucial tool in deriving the dimension of such spaces.

Boolean-rank-preserving operators and Boolean-rank-1 spaces

We study the extent to which certain theorems on linear operators on field-valued matrices carry over to linear operators on Boolean matrices. We obtain analogues and near analogues of several such theorems. One of these leads us to consider linear spaces of m × n Boolean matrices whose nonzero members all have Boolean rank 1. We obtain a structure theorem for such spaces that enables us to determine the maximum Boolean dimension of such spaces and their maximum cardinality.

Algebraic complexity of the computation of a pair of bilinear forms

The problem mentioned in the titled reduces to the estimation of the rank of a collection of matrices. The rank of a collection of matrices A1,...,Ae, denoted rk(A1,...,Ae), is the least number of such one-dimensional matrices that their linear combinations will represent each matrix of the given collection. For an operator A on ℂ n there exists a space V and a diagonal operator B such that ; we denote the minimal dimension of such spaces V by d(V)