Computer simulations of decoherence in quantum spin systems require the solution of the time-dependent Schrödinger equation for interacting quantum spin systems over extended periods of time. We use exact diagonalization, the Chebyshev polynomial technique, four Suzuki-formula algorithms, and the short-iterative-Lanczos method to solve a simple model for decoherence of a quantum spin system by ...

In this paper, the Chebyshev spectral collocation method(CSCM) for one-dimensional linear hyperbolic telegraph equation is presented. Chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. A straightforward implementation of these methods involves the use of spectral differentiation matrices. Firstly, we transform ...

The IBM Power4 processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of non-stand...

In this paper, by the Chebyshev-type inequalities we define three mappings, investigate their main properties, give some refinements for Chebyshev-type inequalities, obtain some applications.

We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x = T3(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials and b + degC = 3N . If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a ≤ 3. Most results are derived from continued fractions and their matrix represe...

A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(logN)2/ log logN) operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the di...

Ahs~aet-A new technique is presented for the design of digital FIR filters, with a prescribed degree of flatness in the passband, and a prescribed (equiripple) attenuation in the stopband. The design is based entirely on an appropriate use of the well-known Rem&-exchange algorithm for the design of weighted Chebyshev FIR filters. The extreme versatility of this algorithm is combined with certai...

A compensation scheme is presented to compensate the effect of dead-zone nonlinearity in a class of uncertain discrete-time nonlinear systems. Chebyshev Neural Network (CNN) is utilized to compensate the dead-zone nonlinearity and the unknown nonlinear functions are also approximated. The control design is attained by introducing dead-zone nonlinearity and using it in the controller design with...

BACKGROUND Magnetic particle imaging (MPI) is a new tomographic imaging technique capable of imaging magnetic tracer material at high temporal and spatial resolution. Image reconstruction requires solving a system of linear equations, which is characterized by a "system function" that establishes the relation between spatial tracer position and frequency response. This paper for the first time ...

where x(t) ∈ D ⊂ Rp and f (x) ∈ R is an unknown function but assumed to be bounded function in x. When the structure of the uncertainty is unknown, function approximation is usually employed to estimate the unknown function. In recent years, neural networks have gained a lot of attention in function approximation theory in connection with adaptive control. Multi-layer neural networks have the c...