Legendre symbol

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... 

The Jacobi symbol is another generalization of the Legendre symbol its multiplicative extension to arbitrary odd numbers as the lower parameter. Thus the Jacobi symbol of y modulo n is denoted by ( ) it only depends on y mod n and takes the values 1 and -1 on and 0 outside and it is multiplicative in both its parameters, i.e., ( )( ) = ( ) and (p( ) = always hold. 

Secondly, the two roots Wp of a quadratic residue mod p have different Legendre symbols, and similarly for q. Hence the four roots of a quadratic residue mod n are mapped to all the four different values (1, 1), (1, -1), (-1, 1), and (-1, -1) by Xn-In particular, two of them have the Jacobi symbol +1 they form a pair w and exactly one of them is a quadratic residue. 

In view of the importance of Legendre transformations in equilibrium thermodynamics, we shall make, following Hermann (Hermann, 1984), an alternative formulation of the fundamental thermodynamic relation. We introduce a five-dimensional space (we shall use hereafter the symbol N to denote it) with coordinates (e,n,e ji, s) and present the fundamental thermodynamic relation as a two-dimensional manifold imbedded in the five-dimensional space N by the mapping... 

We shall call this manifold a Gibbs-Legendre manifold and denote it by the symbol N. The advantage of this formulation is that the space N is naturally... 

Let. .MA" V. o be the intersection of. t(A with the plane x 0. We note that its restriction to the plane x is a manifold of the states that we denote xeth(y ) and call equilibrium states. These are the states for which cp reaches its extremum if considered as a function of x. We shall denote the manifold of equilibrium states by the symbol Meth- Restriction of Mm x, 0 to the plane y is the manifold representing y expressed in terms of x (i.e. the function (4)), and its restriction to the plane (i/. yj is the Gibbs-Legendre manifold Af representing the dual form s s (y ) of the fundamental thermodynamic relation s = s(y) in N that is implied by the fundamental thermodynamic relation h = h(x) in M. This completes our presentation of the passage h(x) —> s(y). 

It is to be noted that the order m is restricted to positive values (and zero) we are using the rather clumsy symbol m to represent the order of the associated Legendre function so that we may later identify m with the magnetic quantum number previously... 

A few examples of the application of the 3-1 symbols to calculations with real spherical harmonics will be shown. Frequently surface harmonics occur that are normalized like Legendre polynomials such harmonics will be denoted by the letter ( ,... 

The symbol corresponds to the spherical Bessel function of order , Vm denotes the position of the atom m relative to the center of mass (there are n atoms within one molecule). Pi is a Legendre polynomial, mm is the angle between Vm and r, and S m the Kronecker delta. 

The physical meaning of the individual symbols is as follows k t) is the time-dependent rate constant of the radiative depletion of the excited state, p v,t) is the normalized time-dependent emission spectmm, i.e., the denominator k f)p v, t))) describes the total fluorescence, S t), P2 is the Legendre polynomial of the second order which correlates with the mutual angular orientations of /energy states. He also employed the master equation, which describes the time change of the conditional probabilityp(/, i2 tj(j, J2o t = 0) that the fluorophore is in energy state i and... 

The symbol is used to designate a normalized associated Legendre polynomial in 0 that is,... 

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