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We present a new refinement strategy for locally refined B-splines which ensures the local linear independence of the basis functions. The strategy also guarantees the spanning of the full spline space on the underlying locally refined mesh. The resulting mesh has nice grading properties which grant the preservation of shape regularity and local quasi uniformity of the elements in the refining process. [ABSTRACT FROM AUTHOR]

We generate hierarchical T-meshes by repeatedly inserting new line segments, in order to adapt both the size and the shape of the cells to the specific requirements of the underlying application. The associated spaces of bilinear spline functions are spanned by the locally refined (LR) B-splines of Dokken et al. (2013) , which are products of univariate B-splines defined over local knot vectors. Our method guarantees that these functions are linearly independent for the obtained class of hierarchical T-meshes. Furthermore, they even possess the property of local linear independence, since exactly 4 functions take non-zero values on each cell of the mesh, and they form a partition of unity without the need to perform additional scaling or truncation. The correctness of our refinement algorithm is verified by enumerating the (newly introduced) local configurations, which represent the possible topologies of the mesh in the vicinity of a cell. It is then shown that the method works well in all the resulting situations. The fairly large number (385) of cases sheds some light on the challenges associated with linear independence in the context of LR B-spline refinement. Additionally we apply our method to meshes that possess the additional property local semi-regularity (first studied by Weller and Hagen, 1995), where only 49 topologically different situations may arise. However, we observe that the resulting class of hierarchical T-meshes is less flexible, which leads to a substantial amount of excess refinement in applications. Finally we note that the same construction can be applied to generate basis functions for C s –smooth spline spaces of degree p = 2 s + 1 , again with the property of local linear independence. [ABSTRACT FROM AUTHOR]

The main result of this short note is a generic version of Paz' conjecture on the length of generating sets in matrix algebras. Consider a generic g -tuple A _ = ( A 1 , … , A g ) of n × n matrices over an infinite field. We show that whenever g 2 d ≥ n 2 , the set of all words of degree 2 d in A _ spans the full n × n matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of special kinds of directed multigraphs with few edge-disjoint walks. [ABSTRACT FROM AUTHOR]

We describe a construction of LR-spaces whose bases are composed of locally linearly independent B-splines which also form a partition of unity. The construction conforms to given refinement requirements associated to subdomains. In contrast to the original LR-paper ( Dokken et al., 2013 ) and similarly to the hierarchical B-spline framework ( Forsey and Bartels, 1988 ) the construction of the mesh is based on a priori choice of a sequence of nested tensor B-spline spaces. [ABSTRACT FROM AUTHOR]

Characterizing the linear and local linear independence of the functions that span a linear space is a key task if the space is to he used computationally. Given a control net, the spanning functions of one spatial coordinate of a generalized subdivision surface are called nodal functions. They are the limit, under subdivision, of associating the value one with one control net node and zero with all others. No characterization of independence of nodal functions has been published to date, even for the two most popular generalized subdivision algorithms, Catmull–Clark subdivision and Loop's subdivision. This paper provides a road map for the verification of linear and local linear independence of generalized subdivision functions. It proves the conjectured global independence of the nodal functions of both algorithms, disproves local linear independence (for higher valences), and establishes linear independence on every surface region corresponding to a facet of the control net. Subtle exceptions, even to global independence, underscore the need for a detailed analysis to provide a sound basis for a number of recently developed computational approaches. [ABSTRACT FROM AUTHOR]

The paper considers properties of compactly supported, locally linearly independent refinable function vectors Φ=(φ1,…,φr)T, r∈N. In the first part of the paper, we show that the interval endpoints of the global support of φν, ν=1,…,r, are special rational numbers. Moreover, in contrast with the scalar case r=1, we show that components φν of a locally linearly independent refinable function vector Φ can have holes. In the second part of the paper we investigate the problem whether any shift-invariant space generated by a refinable function vector Φ possesses a basis which is linearly independent over (0,1). We show that this is not the case. Hence the result of Jia, that each finitely generated shift-invariant space possesses a globally linearly independent basis, is in a certain sense the strongest result which can be obtained. [Copyright &y& Elsevier]

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L2 solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory. [ABSTRACT FROM AUTHOR]

In this paper, the global and local linear independence of any compactly supported distributions by using time domain spaces, and of refinable vectors by invariant linear spaces are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we describe an adaptive refinement strategy for LR B-splines. The presented strategy ensures, at each iteration, local linear independence of the obtained set of LR B-splines. This property is then exploited in two applications: the construction of efficient quasi-interpolation schemes and the numerical solution of elliptic problems using the isogeometric Galerkin method. • A simple adaptive refinement for locally linearly independent LR B-splines is presented. • An efficient and effortless quasi-interpolation scheme with LR B-splines is described. • The new refinement is also adopted to perform isogeometric analysis with LR B-splines. [ABSTRACT FROM AUTHOR]