For many statistical pattern recognition methods, distributions of sample vectors are assumed to be normal, and the quadratic discriminant function derived from the probability density function of multivariate normal distribution is used for classification. However, the computational cost is O(n) for n-dimensional vectors. Moreover, if there are not enough training sample patterns, covariance m...

Let F(z) be a vector-valued function F C CN, which is analytic at z 0 and meromorphic in a neighborhood of z 0, and let its Maclaurin series be given. In a recent work [J. Approx. Theory, 76 (1994), pp. 89-111] by the author, vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series, were developed, and some of their convergence properties were analyzed in ...

Finite-time Lyapunov exponents and vectors are used to define and diagnose boundarylayer type, two-timescale behavior and to construct a method for determining an associated slow manifold when one exists. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for the state-space. The slow-fast splitting defined by finite-time Lyapunov exponents and vectors is int...

The fuzzy structured element (FSE) theory is a very useful toolfor dealing with fuzzy multi-criteria decision making (MCDM)problems by transforming the criterion value vectors of eachalternative into the corresponding criterion function vectors. Inthis paper, some concepts related to function vectors are firstdefined, such as the inner product of two function vectors, thecosine of the included ...

We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted approximation process. An approximation to the subspace to be deflated is successively refined in the course ...

1. Matrices, Vectors and their Basic Operations 1.1. Matrices 1.2. Vectors 1.3. Addition and Scalar Multiplication of Matrices 1.4. Multiplication of Matrices 2. Determinants 2.1. Square Matrices 2.2. Determinants 2.3. Cofactors and the Inverse Matrix 3. Systems of Linear Equations 3.1. Linear Equations 3.2. Cramer’s Rule 3.3. Eigenvalues of a Complex Square Matrix 3.4. Jordan Canonical Form 4....

Local Fisher Discriminant Analysis (LFDA) is a feature extraction method which combines the ideas of Fisher discriminant analysis (FDA) and locality preserving projection (LPP). It works well for multimodal problems. But LFDA suffers from the under-sampled problem of the linear discriminant analysis (LDA). To deal with this problem, we propose a regularized orthogonal local Fisher discriminant ...

The autoencoder is an artificial neural network that learns hidden representations of unlabeled data. With a linear transfer function it is similar to the principal component analysis (PCA). While both methods use weight vectors for linear transformations, the autoencoder does not come with any indication similar to the eigenvalues in PCA that are paired with eigenvectors. We propose a novel su...

In this paper, we study a distance-regular graph Γ = (X,R) with an intersection number a2 6= 0 having a strongly closed subgraph Y of diameter 2. Let E0, E1, . . . , ED be the primitive idempotents corresponding to the eigenvalues θ0 > θ1 > · · · > θD of Γ. Let V = C be the vector space consisting of column vectors whose rows are labeled by the vertex set X. Let W be the subspace of V consistin...

the eigenvalues of a graph is the root of its characteristic polynomial. a fullerene f is a 3-connected graphs with entirely 12 pentagonal faces and n/2 -10 hexagonal faces, where n isthe number of vertices of f. in this paper we investigate the eigenvalues of a class of fullerenegraphs.