We have designed a cubic spline wavelet-like decomposition for the Sobolev space H (I) where I is a bounded interval. Based on a special point value vanishing property of the wavelet basis functions, a fast discrete wavelet transform (DWT) is constructed. This DWT will map discrete samples of a function to its wavelet expansion coefficients in at most 7N log N operations. Using this transform, we propose a collocation method for the initial boundary value problem of nonlinear partial differential equations (PDEs). Then, we test the efficiency of the DWT and apply the collocation method to solve linear and nonlinear PDEs.

Abstract. Let • • • C K_ ] c Vq c Vx c • • • be a multiresolution analysis ofL2 generated by the mth order 5-spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces ... , W_ \, Wo, Wx, ... . Consequently, the two finite sequences that describe the two-scale relations of Nm(x) and i//m(x) in terms of Nm(2x -j), ;6Z, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases {Nm(x -j)} and {y/m(x -j)} , relative to {Nm(x -j)} and {y/m(x -j)}, respectively.

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In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ andφ in L 2 (R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψ jk := 2 j/2 ψ(2 j • − k) (j, k ∈ Z) form a Riesz basis for L 2 (R). If, in addition, φ lies in the Sobolev space H m (R), then the derivatives 2 j/2 ψ (m) (2 j • − k) (j, k ∈ Z) also form a Riesz basis for L 2 (R). Consequently, {ψ jk : j, k ∈ Z} is a stable wavelet basis for the Sobolev space H m (R). The pair of φ andφ are not required to be biorthogonal or semi-orthogonal. In particular, φ andφ can be a pair of B-splines. The added flexibility on φ andφ allows us to construct wavelets with relatively small supports.

Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation satisfied by the basis function.

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Artificially inducing 2n gametes through chromosome doubling is an effective way to obtain polyploids. In this study, Eucalyptus urophylla microsporogenesis and flower development were investigated to guide 2n pollen induction. We also investigated suitable conditions for colchicine treatment. Our results showed that E. urophylla 2n pollen was spherical and had a large volume (mean diameter 28.57 ± 0.46 lm), while normal untreated pollen (mean diameter 19.68 ± 0.11 lm) was tetrahedron. The highest rate of 2n pollen production was 28.71 % when the flower buds, which ranged in size from 3.5 to 4.0 mm, underwent treatment with 0.5 % colchicine solution for 6 h. Further studies suggested that diplotene to diakinesis and metaphase I to telophase I were suitable meiotic stages for chromosomes doubling, due to asynchronous development of microsporogenesis between the anthers in a single flower bud. These data help to illuminate research in other areas, such as triploid eucalyptus production by chromosome doubling of female gametes.

Excavation equipment recognition attracts increasing attentions in recent years due to its significance in underground pipeline network protection and civil construction management. In this paper, a novel classification algorithm based on acoustics processing is proposed for four representative excavation equipments. New acoustic statistical features, namely, the short frame energy ratio, concentration of spectrum amplitude ratio, truncated energy range, and interval of pulse are first developed to characterize acoustic signals. Then, probability density distributions of these acoustic features are analyzed and a novel classifier is presented. Experiments on real recorded acoustics of the four excavation devices are conducted to demonstrate the effectiveness of the proposed algorithm. Comparisons with two popular machine learning methods, support vector machine and extreme learning machine, combined with the popular linear prediction cepstral coefficients are provided to show the generalization capability of our method. A real surveillance system using our algorithm is developed and installed in a metro construction site for real-time recognition performance validation.