On k-circulant matrices with the generalized third-order Pell numbers

In this paper, we obtain explicit forms of the sum of entries, the maximum column sum matrix norm, the maximum row sum matrix norm, Euclidean norm, eigenvalues and determinant of k-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this k-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

A Non-Contact Fault Diagnosis Method for Bearings and Gears Based on Generalized Matrix Norm Sparse Filtering

Fault diagnosis of mechanical equipment is mainly based on the contact measurement and analysis of vibration signals. In some special working conditions, the non-contact fault diagnosis method represented by the measurement of acoustic signals can make up for the lack of contact testing. However, its engineering application value is greatly restricted due to the low signal-to-noise ratio (SNR) of the acoustic signal. To solve this deficiency, a novel fault diagnosis method based on the generalized matrix norm sparse filtering (GMNSF) is proposed in this paper. Specially, the generalized matrix norm is introduced into the sparse filtering to seek the optimal sparse feature distribution to overcome the defect of low SNR of acoustic signals. Firstly, the collected acoustic signals are randomly overlapped to form the sample fragment data set. Then, three constraints are imposed on the multi-period data set by the GMNSF model to extract the sparse features in the sample. Finally, softmax is used to as a classifier to categorize different fault types. The diagnostic performance of the proposed method is verified by the bearing and planetary gear datasets. Results show that the GMNSF model has good feature extraction ability performance and anti-noise ability than other traditional methods.

On the Convergence of Orbits in Sequence Space l^2

This paper discusses the convergence of orbits for diagonal operators defined on . In particular, the basis elements of are obtained using the linear combinations of the elements of the orbit. Furthermore, via the classical result of the determinant of the Vandermonde matrix, it is shown that, the more the elements of the orbit are used, the faster the convergence of the orbit to the basis elements of . Keywords: Diagonal operators; Convergence of Orbits of operators; Vandermonde matrix; Norm topology

The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers

In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.