The papers [MRT05], [CGS07], [XXG12], and [XXCB14] combine the techniques of the Fast Multipole Method of [GR87], [CGR98] with the transformations of matrix structures, traced back to [P90]. The resulting numerically stable algorithms approximate the solutions of Toeplitz, Hankel, Toeplitz-like, and Hankel-like linear systems of equations in nearly linear arithmetic time, versus the classical c...

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices can date back to the early twentieth century. It forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vander...

that is as easy as playing cards. Let Vn denote the Vandermonde matrix with (i, j)th entry vi j = x j i (0 ≤ i, j ≤ n). Since the determinant of Vn is a polynomial in x0, x1, . . . , xn , it suffices to prove the identity for positive integers x0, x1, . . . , xn with x0 ≤ x1 ≤ · · · ≤ xn . We define a Vandermonde card to possess a suit and a value, where a card of Suit i has a value from the se...

Variable-step variable-order 3-stage Hermite–Birkhoff–Obrechkoff methods of order 4 to 14, denoted by HBO(4-14)3, are constructed for solving non-stiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. These methods use y′ and y′′ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads...

We use the language of signal flow graph representation of digital filter structures to solve three purely mathematical problems, including fast inversion of certain polynomial–Vandermonde matrices, deriving an analogue of the Horner and Clenshaw rules for polynomial evaluation in a (H, m)–quasiseparable basis, and computation of eigenvectors of (H, m)– quasiseparable classes of matrices. While...

Background: Hepatitis activity index (HAI) and fibrosing stage are two important findings during the evaluation of liver samples in patients with chronic hepatitis B. The aim of this study was to determine the HAI and fibrosing stage in patients with anti-HBe positive chronic hepatitis B. Methods: Liver biopsy slides of 72 patients were evaluated at the Department of Pathology in two teaching ...

We study the computational potential of spiking neural (SN) P systems. Under various conditions, several intractable problems have been proven to be solvable by these systems in polynomial or even constant time. We study first their formal aspects such as the input encoding, halting versus spiking, and descriptional complexity. Then we establish a formal platform for complexity classes of unifo...