Micro-Raman measurements were performed to study the influence of Co doping on the lattice dynamic properties of the host ZnO. The structural evidence of secondary phase segregation was detected from two distinct phonon vibrational modes at around 472 and 678cm−1 in Zn1−xCoxO (0<x<0.6) ternary alloys with x above 0.098. In addition, an intense, broad, and symmetric phonon vibration was apparent at about 530cm−1 for alloys with x not more than 0.155. The authors suggest the shallow donor defects bound on the Co sites rather than the local vibrational mode involving Co motion as the origin.

A generalization of Mallat's classic multiresolution analysis (MRA), based on the theory of spectral pairs, was considered in two papers by Gabardo and Nashed. In this nonstandard setting, the translation set is no longer a subgroup or a translate of a subgroup of R, but is a spectrum associated with a onedimensional spectral pair. In this paper, we continue the study based on this nonstandard setting and give the characterization for nonuniform wavelets associated with a nonuniform MRA. These characterizations are consistent with both the known necessary and sufficient conditions for the existence of nonuniform MRA wavelets and the known characterization for standard dyadic wavelets associated with an MRA.

In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice Γ ⊆ Z n as a translation tiling set of R n . In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile K and to determine all of the lattice tilings of R n by K. Moreover, we also propose and determine algorithmically another type of translation tiling of R n by K, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set K ∩ (K + j), j ∈ Z n , is strictly positive. Proposition 1.1 [10, 17, 19]. The Lebesgue measure of any integral self-affine tile K(B, D) is a positive integer.However, an efficient algorithm to evaluate this measure is not yet available, to our knowledge. We say that a measurable set E ⊂ R n tiles R n using the discrete translation tiling set T if t∈T χ E+t = 1 almost everywhere on R n , where χ A denotes the characteristic function of A. In this case, we also say that E + T is a tiling of R n . One also knows that certain sets of integral translates of K tile R n (see [8,10,16]). In particular, Lagarias and Wang [18] as well as Conze, Hervé and Raugi [4] independently proved that every self-affine tile admits a lattice tiling.Proposition 1.2 [4, 18]. Every integral self-affine tile K(B, D) admits some lattice Γ ⊆ Z n as a tiling set for R n . Natural tiling and Lebesgue measure of K(B, D)Our first goal is to give a characterization for the integral points in K, which will be needed later on. Before doing so, we need to introduce some terminology.Definition 2.1. Given an integral expansive matrix B and associated standard digit set D, with 0 ∈ D, we have that, for any k ∈ Z n , and every m 11) where d 0 , . . . , d m−1 ∈ D, x m ∈ Z n . So for each k ∈ Z n , we can associate a unique sequence (d 0 , d 1 , d 2 , . . . ), and formally write k ∼ (d 0 , d 1 , d 2 , . . . ).It is easy to prove that, for each given k ∈ Z n , the sequence {x m : m 1} ⊂ Z n in (2.1) is bounded and can, therefore, take only finite distinct values.

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A generalization of Mallat's classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. As a generalization of Dai, Larson, and Speegle's theory of wavelet sets, we prove in this paper the existence of nonuniform wavelet sets associated with the same translation and dilation parameters.