Multi-Stage Convolutional Broad Learning with Block Diagonal Constraint for Hyperspectral Image Classification

By combining the broad learning and a convolutional neural network (CNN), a block-diagonal constrained multi-stage convolutional broad learning (MSCBL-BD) method is proposed for hyperspectral image (HSI) classification. Firstly, as the linear sparse feature extracted by the conventional broad learning method cannot fully characterize the complex spatial-spectral features of HSIs, we replace the linear sparse features in the mapped feature (MF) with the features extracted by the CNN to achieve more complex nonlinear mapping. Then, in the multi-layer mapping process of the CNN, information loss occurs to a certain degree. To this end, the multi-stage convolutional features (MSCFs) extracted by the CNN are expanded to obtain the multi-stage broad features (MSBFs). MSCFs and MSBFs are further spliced to obtain multi-stage convolutional broad features (MSCBFs). Additionally, in order to enhance the mutual independence between MSCBFs, a block diagonal constraint is introduced, and MSCBFs are mapped by a block diagonal matrix, so that each feature is represented linearly only by features of the same stage. Finally, the output layer weights of MSCBL-BD and the desired block-diagonal matrix are solved by the alternating direction method of multipliers. Experimental results on three popular HSI datasets demonstrate the superiority of MSCBL-BD.

MATRICES : SOME NEW PROPERTIES./ SB’S THEOREMS (SPECTRUM)

In this paper I have discussed about some new properties of Hermitian , Skew Hermitian matrices , diagonalisation , eigen values ,eigen vectors , spectrum , which will open up a new horizon to the students of Mathematics . Also , in this paper I have written two totally new theorems for students and researchers . SB’s Theorem 1 is on Normality of a block diagonal matrix and SB’s Theorem 2 is on Spectrum of eigen values . These ideas came to me in course of teaching . Hope , these two theorems will be of great help for the students of Physics and Chemistry as well .

Community Detection of Dynamic Complex Networks in Stock Markets Using Hybrid Methods (RMT‐CN‐LPAm+ and RMT‐BDM‐SA)

A stock market represents a large number of interacting elements, leading to complex hidden interactions. It is very challenging to find a useful method to detect the detailed dynamical complex networks involved in the interactions. For this reason, we propose two hybrid methods called RMT-CN-LPAm+ and RMT-BDM-SA (RMT, random matrix theory; CN, complex network; LPAm+, advanced label propagation algorithm; BDM, block diagonal matrix; SA, simulated annealing). In this study, we investigated group mapping in the S&P 500 stock market using these two hybrid methods. Our results showed the good performance of the proposed methods, with both the methods demonstrating their own benefits and strong points. For example, RMT-CN-LPAm+ successfully identified six groups comprising 485 involved nodes and 17 isolated nodes, with a maximum modularity of 0.62 (identified more groups and displayed more maximum modularity). Meanwhile, RMT-BDM-SA provided useful detailed information through the decomposition of matrix C into Cm (market-wide), Cg (group), and Cr (noise). Both hybrid methods successfully performed very detailed community detection of dynamic complex networks in the stock market.

On similarity of an arbitrary matrix to a block diagonal matrix

Let an n x n -matrix A have m < n (m ? 2) different eigenvalues ?j of the algebraic multiplicity ?j (j = 1,..., m). It is proved that there are ?j x ?j-matrices Aj, each of which has a unique eigenvalue ?j, such that A is similar to the block-diagonal matrix ?D = diag (A1,A2,..., Am). I.e. there is an invertible matrix T, such that T-1AT = ?D. Besides, a sharp bound for the number kT := ||T||||T-1|| is derived. As applications of these results we obtain norm estimates for matrix functions non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition, a new bound for the spectral variation of matrices is derived. In the appropriate situations it refines the well known bounds.

Computation of the Cutoff Wavenumbers of Metallic Waveguides with Symmetries by Using a Nonlinear Eigenproblem Formulation: A Group Theoretical Approach

The evaluation of the cutoff wavenumbers of metallic waveguides can be related to the numerical resolution of a suitable nonlinear eigenproblem defined on the domain C described by the contour of its transverse cross section. In this work, we show that the symmetries of C can be exploited to obtain a block diagonal matrix representation of the nonlinear eigenproblem, which enables a remarkable reduction in the computational effort involved in its resolution.

Methodological Provisions of Modeling Spatial Development of Agriculture in Russia: Undeservedly Forgotten, but Still in Demand

One of the main drawbacks of the existing economic and mathematical models of the spatial organization of agriculture in the country is the relatively weak relationship between the development, placement and specialization of its individual sub-sectors. Therefore, the results of their application do not always adequately reflect the spatial development of agriculture in the regions of the country. A model with a block-diagonal matrix structure, developed and success-fully tested in the 80s of the last century, was largely devoid of these shortcomings. Despite the fact that she was intended for a planned economy, many of the proposed methodological provi-sions in solving the task of optimizing the development, rational placement and deepening of agricultural specialization in the country have not lost their scientific and practical significance. So, in the process of developing and solving the problem, methods of mathematical statistics, linear programming and traditional methods of economic analysis were organically combined, and when substantiating various options for the spatial development of agriculture, methods of multi-criteria optimization were used. Particular attention was paid to the preparation of baseline information, a business case, the use of advanced technologies, the rational use of production resources, which made it possible to identify the bottlenecks in the spatial organization of agri-culture and propose optimal solutions.

The Tropical Matrix Groups with Symmetric Idempotents

In this paper we study the semigroup Mn(T) of all n×n tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of Mn(T) which contain symmetric idempotents.