Let p > 3 be a prime, u, v, d ∈ Z, gcd(u, v) = 1, p u2 − dv2 and (−3d p ) = 1, where ( p ) is the Legendre symbol. In the paper we mainly determine the value of u−v √ d u+v √ d (p−( p3 ))/3 (mod p) by expressing p in terms of appropriate binary quadratic forms. As applications, for p ≡ 1 (mod 3) we obtain a general criterion for m(p−1)/3 (mod p) and a criterion for εd to be a cubic residue of p...

Based on analysis of basic cubic spline interpolation, the clamped cubic spline interpolation is generalized in this paper. The methods are presented on the condition that the first derivative and second derivative of arbitrary node are given. The Clamped spline and Curvature-adjusted cubic spline are also generalized. The methods are presented on the condition that the first derivatives of arb...

Based on analysis of basic cubic spline interpolation, the clamped cubic spline interpolation is generalized in this paper. The methods are presented on the condition that the first derivative and second derivative of arbitrary node are given. The Clamped spline and Curvature-adjusted cubic spline are also generalized. The methods are presented on the condition that the first derivatives of arb...

. In this paper, we develop a quadratic spline collocation method for integrating the nonlinear partial differential equations (PDEs) of a plug flow reactor model. The method is proposed in order to be used for the operation of control design and/or numerical simulations. We first present the Crank-Nicolson method to temporally discretize the state variable. Then, we develop and analyze the pro...

By applying finite difference formula to time discretization and the cubic B-splines for spatial variable, a numerical method for solving the inverse system of Burgers equations is presented. Also, the convergence analysis and stability for this problem are investigated and the order of convergence is obtained. By using two test problems, the accuracy of presented method is verified. Additional...

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solutio...

Multiscale transforms for real-valued data, based on interpolatory subdivision operators, have been studied in recent years. They are easy to define and can be extended other types of example, manifold-valued data. In this chapter, we linear multiscale transforms, certain linear, non-interpolatory termed “even-reversible.” For such prove, using Wiener’s lemma, the existence an inverse operator ...

This paper discusses the application of orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence name collocation. The used solve various cases Burgers’ equations, including modified equation. KdV–Burgers’ equation considered as a tes...

the most commonly used utility functional forms are reviewed and their properties compared. then, using data collected from west azarbaijan province, utility functions are estimated and absolute risk aversion coefficients measured. exponential and expopower utility functions classified all farmers as risk averse, but quadratic and cubic utility functions classified 75% and 65% of farmers as...