in this paper, the rayleigh's quotient and the inverse vector iteration method are presented. the latter approach helps to obtain the natural frequencies and mode shapes of a structure. inverse vector iteration method with shifting enables to determine the higher modes. some basic theorems of linear algebra are presented and extended to study the free vibration of structures. the variation...

Frames have turned out to be an essential tool for many applications such as, for example, data transmission, due to their robustness not only against noise but also against losses and due to their freedom in design [4, 6]. Their main advantage lies in the fact that a frame can be designed to be redundant while still providing a reconstruction formula. Since the frame operator Sg = ∑n i=1 〈g, f...

The representations currently used by local search and some evolutionary algorithms have the disadvantage that these algorithms are partially blind to “ridges” in the search space. Both heuristics search and gradient search algorithms can exhibit extremely slow convergence on functions that display ridge structures. A class of rotated representations are proposed and explored; these rotated rep...

For g < n, let b1, . . . , bn−g be n− g independent vectors in R with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when n → +∞ concerning the property that such a random basis is reduced in the sense of Lenstra, Lenstra & Lovász. The proof passes by the study of t...

Mixed precision hardware has recently become commercially available, and more than 25% of the supercomputers in TOP500 list now have mixed capabilities. Using lower algorithms can be beneficial terms reducing both computation communication costs. There are many current efforts towards developing numerical linear algebra algorithms, which lead to speedups real applications. Motivated by this, we...

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

Abstract Multi-spectral CT (MSCT) is increasingly used in industrial non-destructive testing and medical diagnosis because of its outstanding performance like material distinguishability. The process obtaining MSCT data can be modeled as a nonlinear system the basis decomposition comes down to inverse problem system. For different spectra data, geometric inconsistent parameters cause geometrica...

The advancement of quantum computers undermines the security classical blockchain, necessitating either a post-quantum upgrade existing architecture or creation an inherently blockchain. Here we propose practically realizable model fully blockchain based on generalized Gram-Schmidt procedure utilizing dimensional lifting. In this model, information transactions stored in multi-qubit state are s...