A system of weakly magnetically coupled oscillating blades is studied experimentally, computationally, and theoretically. It is found that when the uncoupled natural frequencies of the blades are nearly equal, the normal modes produced by the coupling are almost impossible to find experimentally if the random variation level in the system parameters is on the order of ͑or larger than͒ the relativ...

Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is the main methodological contribution of the paper. The one of our integrals has a similar structur...

This evaluation proves a conjecture in [9, p. 370] and solves the problem of finding explicitly the number of rational points (mod/?) on the surface z = (x + l)(y + ΐ)(x + y), a problem some algebraic geometers had worked on without success. Character sum analogues of the important formulas for orthogonal polynomials are potentially as useful as those for hypergeometric series, so a systematic ...

Let μ be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, · · · , x, . . . } to get a sequence {Pn}∞n=0 of orthogonal polynomials with respect to μ. In this paper we explain a method of deriving a generating function ψ(t, x) for μ. The power series expansion of ψ(t, x) in t produces the explicit form of p...

Using the basis of Hermite–Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive constraints for a function and its Fourier transform to be both real and positive. We propose a constructive method based on the algebra of Hermite polynomials. Applications are extended to the 2-dimensional case (i.e. Fourier–Bessel transforms and the algebra of Laguerre poly...

Abstract Integrals involving products of Hermite polynomials with the weight factor exp (−x2) over the interval (−∞,∞) are considered. A result of Azor, Gillis and Victor (SIAM J. Math. Anal. 13 (1982) 879–890] is derived by analytic arguments and extended to higher order products. An asymptotic expansion in the case of a product of four Hermite polynomials Hn(x) as n → ∞ is obtained by a discr...

In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

Abstract. Through an algebraic method using the Dunkl–Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the AN−1and BN Calogero models with bosonic, fermionic and distinguishable particles are investigated. The Rodrigues formulas of column type that algebraically generate the monic nonsymmetric multivariable Hermite and Laguerre polynomials corresponding to...

We present a symbolic algorithm generating finite-element schemes with interpolating Hermite polynomials intended for solving the boundary-value problems with self-adjoint second-order differential equation and implemented in the Maple computer algebra system. Recurrence relations for the calculation in analytical form of the interpolating Hermite polynomials with nodes of arbitrary multiplicit...

In this paper Kharitonov's t h e o m for the robust stability of interval polynomials is proved using the second method of Lyapunov. The Hermite matrix is taken as the matrix of the quadratic form which is used as a Lyapunov function to prove Hurwitz stability. It is shown that if the four Hemite matrices correspondii to the four Kharitonov extreme polynomials are positive definite, the Hermite...