The trigonometric network, introduced in this paper, is a multilayer feedforward neural network with sinusoidal activation functions. Unlike the N-dimensional Fourier series, the basis functions of the proposed trigonometric network have no strict harmonic relationship. An effective training algorithm for the network is developed. It is shown that the trigonometric network performs better than ...

This paper develops a method for constructing rational parameterizations of knots, based on a trigonometric parameterization. It also introduces the class of torus knots and describes a method for constructing trigonometric and rational parameterizations of these knots. This research was conducted at the Mt. Holyoke REU, and was funded by the NSF through grant number DMS-9732228.

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interest...

a relationship based on the modified couple stress theory is developed to investigate the flexural sensitivity of an atomic force microscope (afm) with assembled cantilever probe (acp). this acp comprises a horizontal cantilever, two vertical extensions and two tips located at the free ends of the extensions which form a caliper. an approximate solution to the flexural vibration problem is obta...

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...

In this paper, the Ritz-Galerkin method in Bernstein polynomial basis is applied for solving the nonlinear problem of the magnetohydrodynamic (MHD) flow of third grade fluid between the two plates. The properties of the Bernstein  polynomials together with the Ritz-Galerkin method are used to reduce the solution of the MHD Couette flow of non-Newtonian fluid in a porous medium to the solution o...

in this paper, the stability characteristics of single-walled carbon nanotubes (swcnts) under the action of axial load are investigated. to this end, a nonlocal flügge shell model is developed to accommodate the small length scale effects. the analytical rayleigh-ritz method with beam functions is applied to the variational statement derived from the flügge-type buckling equations. molecular dy...

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) q′′(t) + Mq(t) = f(q(t)) with a principal frequency matrix M ∈ Rd×d. If M is symmetric and positive semi-definite and f(q) = −∇U(q) for a smooth function U(q), then this is a multi-frequency oscillatory Hamiltonian system with ...

Pollution has become a very serious threat to our environment. Monitoring pollution is the rst step toward planning to save the environment. The use of dierential equations, monitoring pollution has become possible. In this paper, a Ritz-collocation method is introduced to solve non-linear oscillatory systems such as modelling the pollution of a system of lakes. The method is based upon Bernoul...