the principle of dimensionality reduction with pca is the representation of the dataset ‘x’in terms of eigenvectors ei ∈ rn  of its covariance matrix. the eigenvectors oriented in the direction with the maximum variance of x in rn carry the most      relevant information of x. these eigenvectors are called principal components [8]. assume that n images in a set are originally represented in mat...

The first 2 PARI-GP files below give the sets of vertices and edges of the Voronoi graph first in dimensions 2 to 6, then in dimension 7. The numerical data are extracted from Jaquet’s thesis [Ja]. The third file, based on Chapters 9 and 14 of [Mar], is devoted to minimal classes in dimensions 2 to 4. We present below a short account of Voronoi’s theory and minimal classes. 1. The perfection ra...

We present methods to compute verified square roots of a square matrix A. Given an approximation X to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of X. Our approach is based on the Krawczyk method which we modify in two different ways in such a manner that the computational comp...

On a closed Riemann surface Ra of genus g there exist g linearly independent differentials of the first kind, wu • • • , ws,.and their integrals around 2g canonical cycles or retrosections, Oi, • • • , a„, bi, ••• , ba, are usually put together to form a gX2g matrix (an; ßn) = (A; B), i, j =1, • • • , g, where a,-,is the integral of w{ over a,. Riemann showed that the entries in this matrix wer...

The efficiency of interior-point algorithms for linear programming is related to the effort required to factorize the matrix used to solve for the search direction at each iteration. When the linear program is in symmetric form (i.e., the constraints are Ax < b, x > 0 ), then there are two mathematically equivalent forms of the search direction, involving different matrices. One form necessitat...

The lower bound is the same as before, due to the fact that Tr(n) is Kr+1-free. Define the adjacency matrix A = A(G) = (aij) for a graph G of order n. Let V = {v1, · · · , vn}. Then A is a n-by-n 0 − 1 matrix such that aij = 1 if and only if vivj ∈ E. Thus A is symmetric. We will be interested in a quadratic form ⟨Ax,x⟩ where x denotes a vector of length n. This is often called the Lagrangian o...

efficient mode shape extraction of fluid-structure systems is of particular interest in engineering. an efficient modified version of unsymmetric lanczos method is proposed in this paper. the original unsymmetric lanczos method was applied to general form of unsymmetric matrices, while the proposed method is developed particularly for the fluid-structure matrices. the method provides us with si...

In this paper, we provide a proof for the Hanson-Wright inequalities for sparse quadratic forms in subgaussian random variables. This provides useful concentration inequalities for sparse subgaussian random vectors in two ways. Let X = (X1, . . . , Xm) ∈ R be a random vector with independent subgaussian components, and ξ = (ξ1, . . . , ξm) ∈ {0, 1} be independent Bernoulli random variables. We ...

Complex symmetric matrices arise from many applications, such as chemical exchange in nuclear magnetic resonance and power systems. Singular value decomposition (SVD) reveals a great deal of properties of a matrix. A complex symmetric matrix has a symmetric SVD (SSVD), also called Takagi Factorization, which exploits the symmetry [3]. Let A be a complex symmetric matrix, its Takagi factorizatio...