Euler’s criterion

For an odd prime p, exactly half of the elements of Zp are quadratic residues, and each one has two square roots w. The quadratic residues are characterized by the Legendre symbol 0, which is defined as +1 if y is a quadratic residue and -1 otherwise (and 0 for y = 0 mod p). The Legendre symbol can be computed efficiently with Euler s criterion = y(P l) 2... 

Euler s criterion implies (y) = ( ) = -1, becausep = q = 3 mod 4. This yields two further important facts ... 

In text-books on differential equations this condition is shown to be necessary and sufficient in order that certain equations may be solved, or integrated as it is called. Equation (7) is called Euler s criterion of integrability. An equation that satisfies this condition is said to be a complete or an exact differential. 

The Glansdorff-Prigogine general evolution criterion involves the minimum of global entropy production in such a constitutive theory where the potentials are homogeneous Euler s functions. We show below the strictly convex property of dissipation potentials guarantee the minimum, and the function... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

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