We treat finite oscillatory integrals of the form ∫ b a F (x) exp[ikG(x)]dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is ...

In this paper we study the geometric properties, existence, regularity and related issues for a family of surfaces which are boundaries of sets minimizing certain integral norms. These surfaces can be interpreted as a non-infinitesimal version of classical minimal surfaces. Our work is motivated by the structure of interphases that arise in classical phase field models when very long space corr...

Path integral formulation of Brownian trajectories is employed to re-derive the steepest descent path (SDP). An analogy between the mechanics of overdamped tra-jectories and a Hamiltonian system is found and exploited. The SDP is a special path selected from the curves that connect two predetermined xed conngurations. One xed connguration is the reactant and the second is the product. It is the...

The most familiar illustrations of downstream topographic waves are the streamline plots from the original linear wave studies of Queney (1947,1948). For steady flow past a twodimensional ridge, Queney’s downstream radiation patterns were obtained through approximations of the Fourier integral which describes the dispersion of linear gravity waves. In the case of constant stratification with ro...

This paper investigates the impact of fast parameter identification methods, which do not require any forward simulations, on model-based glucose control, using retrospective data in the Christchurch Hospital Intensive Care Unit. The integral-based identification method has been previously clinically validated and extensively applied in a number of biomedical applications; and is a crucial elem...

The widely used backpropagation algorithm based on stochastic gradient descent suffers from typically slow convergence to either local or global minimum error. This backpropagation algorithm bears great resemblance to a classic proportional integral derivative (PID) control system. Fractional calculus shows promise for improving stability and response in feedback control through the use of non-...

Hyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asympt...

Oscillatory integrals are present in many applications, and their numerical approximation is the subject of this paper. Contrary to popular belief, their computation can be achieved efficiently, and in fact, the more oscillatory the integral, the more accurate the approximation. We review several existing methods, including the asymptotic expansion, Filon method, Levin collocation method and nu...

Oscillatory integrals are present in many applications, and their numerical approximation is the subject of this paper. Contrary to popular belief, their computation can be achieved efficiently, and in fact, the more oscillatory the integral, the more accurate the approximation. We review several existing methods, including the asymptotic expansion, Filon method, Levin collocation method and nu...