First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of an n_n matrix by an n_n 2 matrix in arithmetic time O(n | ), |=3.333953..., which is less by 0.041 than the previous record 3.375477... . Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as a function of the dimensions, and optimize the exponents of the complexity estimates. For a large class of input matrix pairs, we improve the known exponents. Finally we show three applications of our results:(a) we decrease from 2.851 to 2.837 the known exponent of the work bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n_n matrix, as well as for the solution to a nonsingular linear system of n equations, (b) we asymptotically accelerate the known sequential algorithms for the univariate polynomial composition mod x n , yielding the complexity bound O(n 1.667 ) versus the old record of O(n 1.688 ), and for the univariate polynomial factorization over a finite field, and article no. (c) we improve slightly the known complexity estimates for computing basic solutions to the linear programming problem with m constraints and n variables. Academic Press

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Emotional decoding and automatic identification of major depressive disorder (MDD) are helpful for the timely diagnosis of the disease. Electroencephalography (EEG) is sensitive to changes in the functional state of the human brain, showing its potential to help doctors diagnose MDD. In this paper, an approach for identifying MDD by fusing interhemispheric asymmetry and cross-correlation with EEG signals is proposed and tested on 32 subjects [16 patients with MDD and 16 healthy controls (HCs)]. First, the structural features and connectivity features of the θ-, α-, and β-frequency bands are extracted on the preprocessed and segmented EEG signals. Second, the structural feature matrix of the θ-, α-, and β-frequency bands are added to and subtracted from the connectivity feature matrix to obtain mixed features. Finally, the structural features, connectivity features, and the mixed features are fed to three classifiers to select suitable features for the classification, and it is found that our mode achieves the best classification results using the mixed features. The results are also compared with those from some state-of-the-art methods, and we achieved an accuracy of 94.13%, a sensitivity of 95.74%, a specificity of 93.52%, and an F1-score (f1) of 95.62% on the data from Beijing Anding Hospital, Capital Medical University. The study could be generalized to develop a system that may be helpful in clinical purposes.

We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by means of our new modification of Newton's iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, ChebyshevVandermonde-like, and Toeplitz-like matrices, where again the computations are confined to operations with short generators of the involved matrices.