in this paper, we consider the second-kind chebyshev polynomials (skcps) for the numerical solution of the fractional optimal control problems (focps). firstly, an introduction of the fractional calculus and properties of the shifted skcps are given and then operational matrix of fractional integration is introduced. next, these properties are used together with the legendre-gauss quadrature fo...

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such are useful in various physical and geometric applications. Our formulas compute the difference indices two Hessians related to different graphs or sets boundary conditions. Several applications such as iteration lower bounds proved.