Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

In this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish the characterizations for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families in $$L^2({\mathbb {R}}^+)$$ L 2 ( R + ) . Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

Multiple Sclerosis Recognition by Biorthogonal Wavelet Features and Fitness-Scaled Adaptive Genetic Algorithm

Aim: Multiple sclerosis (MS) is a disease, which can affect the brain and/or spinal cord, leading to a wide range of potential symptoms. This method aims to propose a novel MS recognition method.Methods: First, the bior4.4 wavelet is used to extract multiscale coefficients. Second, three types of biorthogonal wavelet features are proposed and calculated. Third, fitness-scaled adaptive genetic algorithm (FAGA)—a combination of standard genetic algorithm, adaptive mechanism, and power-rank fitness scaling—is harnessed as the optimization algorithm. Fourth, multiple-way data augmentation is utilized on the training set under the setting of 10 runs of 10-fold cross-validation. Our method is abbreviated as BWF-FAGA.Results: Our method achieves a sensitivity of 98.00 ± 0.95%, a specificity of 97.78 ± 0.95%, and an accuracy of 97.89 ± 0.94%. The area under the curve of our method is 0.9876.Conclusion: The results show that the proposed BWF-FAGA method is better than 10 state-of-the-art MS recognition methods, including eight artificial intelligence-based methods, and two deep learning-based methods.

Biorthogonal Wavelet on a Logarithm Curve ℂ

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curve C is constructed in this paper, such as a logarithm curve. The properties of biorthogonal wavelet filters on a smooth curve C are discussed, such as induced refinable equation and symmetry. Moreover, an example is given for discussing the biorthogonal scaling function and its dual on a logarithm curve C . Finally, a numerical application is given for dealing with financial data.

PYRAMID SCHEME FOR CONSTRUCTING BIORTHOGONAL WAVELET CODES OVER FINITE FIELDS

The existence of a biorthogonal decomposition of the space V of dimension n over the field GF(q) is constructively proved, namely, two representations of it are obtained as direct sums of subspaces V = W0⊕W1⊕. . .⊕WJ⊕VJ and V = W˜0⊕W˜1⊕. . .⊕W˜J⊕V˜J ,such that at the j-th level of the decomposition, for 0 < j 6 J, Vj−1 = Vj⊕Wj , V˜j−1 == V˜j ⊕ W˜j , the subspace Vj is orthogonal to W˜j , and the subspace Wj is orthogonal to V˜j . The partition of the space at the j-th level is made with the help of pairs of level filters (hj, gj) and (h˜j, g˜j), for the construction of which the corresponding algorithms have been developed and theoretically proved. A new family of biorthogonal wavelet codes is built on the basis of the multilevel wavelet decomposition scheme with coding rate 2−L, where L is the number of used decomposition levels, and examples of such codes are given.

Application of wavelets and conformal reflections to finding optimal scheme of fiber placement at 3d printing constructions from composition materials

The article is devoted to finding the optimal schemes of fiber placement at the production of constructions, reinforced with continuous fibers by 3D printing method. As the optimization of the objective function one of the criteria for the destruction of the composite was chosen. For the process acceleration of multiple solution of the system of partial differential equations describing the stress-strain state of the structure, a computational algorithm based on wavelets built through subdivision schemes is proposed. To set the local coordinate system, it is proposed to use analytical functions, which will be constructed using the well-known Dini and Cisotti formulas, just by specifying the direction of laying the fiber at the product boundary. The article also presents a lifting scheme (lifting scheme) allowing to construct biorthogonal wavelet systems with specified properties using some initial biorthogonal wavelet systems with filters.