1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes formulas 10. Holomorphic functions and...

In this paper, we consider an impulsive Sturm-Lioville problem on Sturmian time scales. We investigate the existence and uniqueness of solution problem. study some spectral properties self-adjointness boundary-value Later, construct Green function for Finally, eigenfunction expansion is obtained.

In this paper, we investigate a singular perturbation problem including a fourth order O.D.E. with general linear boundary conditions. Firstly, we obtain the necessary conditions of solution of O.D.E. by making use of fundamental solution, then by compatibility of these conditions with boundary conditions, we determine that, for given perturbation problem, whether boundary layer is formed or not.

The boundary knot method is a recent truly meshfree boundary-type radial basis function (RBF) collocation scheme, where the nonsingular general solution is used instead of the singular fundamental solution to evaluate the homogeneous solution, while the dual reciprocity method is employed to the approximation of particular solution. Despite the fact that there are not nonsingular RBF general so...

A fundamental classification problem of data mining and machine learning is that of minimizing a strongly convex, piecewise quadratic function on the n-dimensional real space Rn. We show finite termination of a Newton method to the unique global solution starting from any point in Rn. If the function is well conditioned, then no stepsize is required from the start, and if not, an Armijo stepsiz...

The boundary knot method (BKM) is a recent meshfree boundary-type radial basis function (RBF) collocation technique. Compared with the method of fundamental solution, the BKM uses the nonsingular general solution instead of the singular fundamental solution to evaluate the homogeneous solution, while as such the dual reciprocity method (DRM) is still employed to approximate the particular solut...