k Abstract. Let fip . . , Hk be odd positive integers and »i = Z¡=.x(p¡ + 1). Let {«(.}|=j be an extended Tchebycheff system on [a, b]. Let L be a positive linear functional on U = span( {u,}). We prove that L has a unique representation in the form k M,— 1 £<p) = E Z "i/PW(f,). « < h < ■ ■ • < tk < b, 1=1 ;=0 _ k for all p G U. The proof uses the topological degree of a mapping F: D C R k —► R...

This paper describes the development of an approximate method for propagating uncertainty through stochastic dynamical systems using a quadrature rule integration based method. The development of quadrature rules for Gaussian mixture distributions is discussed. A numerical solution to this problem is considered that uses a Gram-Schmidt process. Simulation results are presented where the quadrat...

Recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form J"£" x"e~xf(x) dx and J"o° x"Ep(x)f(x) dx, where Ef(x) is the exponential int...

We construct and analyze generalized Gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities. The rules have quadrature points inside the interval of integration and the weights are all strictly positive. Such rules date back to the study of Chebyshev sets, but their use in applications has only recently been appreciated. We provide error estimates an...

We prove the secret key rate formulas and derive security threshold parameters of multi-carrier continuous-variable quantum key distribution (CVQKD). In a multicarrierCVQKD scenario, the Gaussian input quantum states of the legal parties are granulatedinto Gaussian subcarrier CVs (continuous-variables). The multicarrier communicationformulates Gaussian sub-channels from the ...

Interval quadrature formulae of Gaussian type on R and R+ for exponential weight functions of the form w(x) = exp(−Q(x)), where Q is a continuous function on its domain and such that all algebraic polynomials are integrable with respect to w, are considered. For a given set of nonoverlapping intervals and an arbitrary n, the uniqueness of the n-point interval Gaussian rule is proved. The result...