LINEAR INDEPENDENCE OF THE INTEGER TRANS-LATES OF COMPACTLY SUPPORTED DISTRIBU-TIONS AND REFINABLE VECTORS

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.

SOME RESULTS ON BUNDLE SYSTEMS FOR A COUNTABLE DISCRETE AMENABLE GROUP ACTION

We consider the style number, independence number and entropy for a frame bundle dynamical system. The base system of which is a countable discrete amenable group action on a compact metric space. We obtain the existence of cover measures, an ergodic theorem about mean linear independence and the style number, and a variational principle for style numbers and independence numbers. We also study the relationship between the entropy of base systems and that of their bundle systems.

Why Are There as Many Elements in the Cantor Set as There Are Real Numbers?

There are many important concepts in linear algebra, such as linear correlation and linear independence, eigenvalues and eigenvectors, and so on. The article provides a graphical explanation of how to distinguish between the concepts of linear correlation and linear independence. The conclusion points out that linear independence means that there are no two (base) vectors with the same direction in a vector graph;otherwise, it is a linear correlation.

NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.

The optimum linear data fusion of independent observation data

Data Fusion is one of the attractive topic in sonar signal processing. Decision level data fusion of multi-sensor (multi-array) system is described in this paper. The optimum linear data fusion algorithm for N independent observation data is derived. It is proved that the estimation error of optimum data fusion is not greater than that of individual components. The expression of estimation error and weight coefficients are presented. The results of numerical calculation and some examples are illustrated. The effect of input signal to noise ratio for the data fusion is described.

On the Linear Independence and Partition of Unity of Arbitrary Degree Analysis-Suitable T-splines

Analysis-suitable T-splines are a topological-restricted subset of T-splines,which are optimized to meet the needs both for design and analysis(Li and Scott ModelsMethods Appl Sci 24:1141-1164,2014;Li et al.Comput Aided Geom Design 29:63-76,2012;Scott et al.Comput Methods Appl Mech Eng 213-216,2012).The paper independently derives a class of bi-degree(d_(1),d_(2))T-splines for which no perpendicular T-junction extensions intersect,and provides a new proof for the linearly independence of the blending functions.We also prove that the sum of the basis functions is one for an analysis-suitable T-spline if the T-mesh is admissible based on a recursive relation.

Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L2 function.Firstly,background and motivation for the conjecture are provided.Secondly,the main progress of this linear independence in the past twenty five years is reviewed.Finally,the partial results of the high dimensional case and other cases for the conjecture are briefly presented.

LINEAR INDEPENDENCE OF REFINABLEDISTRIBUTIONS SUPPORTED ON[0, 2]×[0, 2]

In this paper a necessary and sufficient condition on symbol is given for linear independence of refinable distributions supported on [0,2]×[0,2].

REAL-VALUED PERIODIC WAVELETS: CONSTRUCTION ANDRELATION WITH FOURIER SERIES

In this paper, we construct the real-valued periodic orthogonal wavelets. The method presented here is new. The decomposition and reconstruction formulas involve only 4 terms respectively. It demonstrates that the formulas are simpler than that in other kinds of periodic wavelets. Our wavelets are useful in applications since it is real valued. The relation between the periodic wavelets and the Fourier series is also discussed.

Transcendence of Certain Trigonometric Series

In this paper the transcendence of values of certain trigonometric series with algebraic coefficients and its derivatives at algebraic points is proved under a general condition dependent only on the coefficients. The proof of the theorem is based on a criterion for linear independence over a number field.

SOME PROPERTIES FOR ANALYSIS-SUITABLE T-SPLINES

Analysis-suitable T-splines （AS T-splines） are a mildly topological restricted subset of T-splines which are linear independent regardless of knot values [1-3]. The present paper provides some more iso-geometric analysis （IGA） oriented properties for AS T- splines and generalizes them to arbitrary topology AS T-splines. First, we prove that the blending functions for analysis-suitable T-splines are locally linear independent, which is the key property for localized multi-resolution and linear independence for non-tensor- product domain. And then, we prove that the number of T-spline control points contribute each Bezier element is optimal, which is very important to obtain a bound for the number of non zero entries in the mass and stiffness matrices for IGA with T-splines. Moreover, it is found that the elegant labeling tool for B-splines, blossom, can also be applied for analysis-suitable T-splines.